Chapter 7: Thermal Properties of Matter
On a winter evening in 1820, a Danish professor named Hans Christian Oersted was setting up a demonstration for his students in Copenhagen. He had a battery, a wire, and — lying nearby on the same table — a small magnetic compass. As he connected the wire to the battery so that a current flowed through it, he happened to notice something no one had expected: the compass needle twitched and swung away from north. When he broke the circuit, the needle returned. Current in the wire was somehow steering a magnet several centimetres away.
That accidental twitch of a needle is not, at first sight, about heat at all — but the story carries a lesson that runs through this entire chapter. The behaviour of matter is governed by quantities we cannot see directly, and we come to know them only through the way ordinary objects respond: a needle turning, a column of liquid rising, a metal lid loosening, a puddle drying up. In this chapter the hidden quantity is temperature, and the responding objects are all around you. A thermometer’s mercury climbs, a railway track leaves gaps between its sections, a glass of iced water sweats, and steam scalds far worse than boiling water at the very same temperature.
Just as Oersted learned to read an invisible influence from the swing of a needle, you will learn in this chapter to read the thermal behaviour of matter — how it warms, expands, melts, boils, and passes heat from one place to another.
Diagram
Fig. 7.1 – A montage opening illustration: a thermometer with rising mercury, a railway track with an expansion gap, a steaming kettle, and a glass of iced water beaded with condensation, arranged around the title of the chapter.
By the end of this chapter, you will be able to answer each of these questions for yourself.
7.1 Introduction
Why Heat Needs a Fourth Fundamental Quantity
In your earlier study of mechanics, whether you were describing the equilibrium of a system or analysing the motion of a body, you needed only three fundamental physical quantities: length, mass and time. Every other mechanical quantity was built out of these three.
Think of how this works. Speed is length divided by time. Acceleration is length divided by the square of time. Force is mass multiplied by acceleration, and work is force multiplied by length. Not one of these required anything new — the three base quantities were enough.
Heat is different. The moment we start describing hotness, melting, boiling or the flow of heat from one body to another, the three mechanical quantities are no longer sufficient. We need a fourth fundamental quantity: temperature, which was introduced to you in Chapter 1. Its SI unit is the kelvin (K).
In this chapter we study the properties of matter and the phenomena that arise because bodies can be hot or cold, and because heat can move between them.
Hot and Cold Are Only Relative
The sensation of hotness or coldness is part of everyday experience. A mother checks whether her child has a fever by placing her hand on the child’s forehead. A cook sprinkles a few drops of water on a frying pan to judge whether it is hot enough to spread the dosa batter. Before a bath, we dip a finger into the water to see whether it is warm enough — although touching a hot object to test it is never advisable.
What is our skin actually reporting in all these cases? When we say that a body or some water is “hot”, we really mean that its temperature is higher than that of our hand. Heat therefore flows from the body into our skin, and we register this as hotness. If the body is colder than our hand, heat flows the other way and we call it cold.
This means our sense of touch can only compare two objects qualitatively. Hot and cold are relative terms, not absolute ones. You may recall the example from your Standard VIII science textbook: lukewarm water feels colder than hot water, but the very same lukewarm water feels hotter than cold water. The water has not changed at all — only the comparison has.
So we ascribe to every object a property called temperature, which fixes its degree of hotness. The higher the temperature, the hotter the body is.
From Sensation to Measurement
The precise temperature of a body can be known only when we have an accurate and easily reproducible way of measuring it quantitatively. This is the basic demand of scientific precision: a physical quantity must be expressed in numerical terms, on an agreed scale, so that any two observers anywhere obtain the same value. The device that measures temperature is the thermometer, and Fig. 7.2 shows the familiar idea of replacing the judgement of the hand by the reading of an instrument.
Diagram
Fig. 7.2 – A hand touching a beaker of water alongside a thermometer dipped in the same beaker, contrasting qualitative sensation with a quantitative reading.
What This Chapter Will Explain
Phenomena and properties that have to do with temperature changes and with the exchange of heat are called thermal phenomena or thermal properties of matter.
By the end of this chapter you will be able to explain several everyday puzzles that arise from these properties. You will understand why the direction of the wind near a sea shore reverses between day and night, why the metal lid of a glass bottle comes off easily after it is heated, and why two metal vessels stuck one inside the other can be separated by supplying heat to the outer vessel.
Along the way you will meet the ideas that follow from these questions in a definite order: how heat and temperature differ, how a thermometer is calibrated, what the absolute (kelvin) scale means, how bodies expand on heating, how much heat a substance needs for a given rise in temperature, what happens during melting and boiling, and how heat travels by conduction, convection and radiation. Numerical questions in board and entrance examinations are drawn almost entirely from these later sections; this introduction supplies the vocabulary — heat, temperature, thermometer, thermal property — on which all of them rest.
7.2 Temperature and Heat
Heat Is Energy in Transit
Heat is energy in transit. That short phrase carries an important idea: heat is not something a body stores and keeps, like a fixed quantity of water in a bucket. Heat is energy that is on the move, and it moves only because two bodies are at different temperatures.
When two bodies at different temperatures are brought in contact, they exchange heat. Energy leaves the hotter body and enters the colder one. After some time this transfer stops. When it does, we say the two bodies are in thermal equilibrium.
What decides whether two bodies are in thermal equilibrium is their temperature — the property you met in Section 7.1. As long as the temperatures differ, heat keeps flowing. When the temperatures become equal, the net flow stops.
Temperature is thus the physical quantity that defines the thermodynamic state of a system — the state of the system as far as its hotness and its heat exchanges are concerned.
Everyday Evidence of Thermal Equilibrium
You have surely noticed that a glass of ice-cold water left on a table slowly warms up, while a cup of hot tea kept on the same table cools down. The table and the room have not changed; only the two glasses have.
The explanation is the same in both cases. When the temperature of a body — the ice-cold water, or the hot tea — differs from that of the surrounding medium, heat transfer takes place between the body and its surroundings. The transfer continues until the body and the surroundings are at the same temperature. At that stage we say that the body and its surroundings have reached a state of thermal equilibrium, and there is no net transfer of heat from one to the other.
In fact, whenever two bodies are in contact, there is a transfer of heat between them owing to their temperature difference. The cold water gains heat from the room; the hot tea gives heat to the room. The direction is always fixed by which body is hotter.
What Heating Does Inside Matter
To understand why heat raises temperature, we must look at what matter is made of. Matter in any state — solid, liquid or gas — consists of particles: ions, atoms or molecules.
In a solid, these particles are not stationary. They vibrate about their fixed equilibrium positions, and because they are in motion they possess kinetic energy at any given temperature. They also possess potential energy, because of the interatomic forces that hold them together at their mean fixed positions. It is these strong forces that give a solid both a definite volume and a definite shape.
Diagram
Fig. 7.3 – Particles of a solid vibrating about fixed mean positions, with springs representing the interatomic forces between them.
When we heat a solid, we supply energy to it. The particles then vibrate with higher energy, and we observe that the temperature of the solid increases (except near its melting point, for a reason we shall see in a moment). The energy supplied does not disappear. It becomes internal energy of the solid, in the form of increased kinetic energy of its atoms and molecules, and this raises the temperature of the solid, as suggested in Fig. 7.4.
Diagram
Fig. 7.4 – The same solid before and after heating, with larger vibration amplitudes shown after heating.
Temperature is therefore a measure of the average kinetic energy of the atoms or molecules of the body. The greater the kinetic energy, the faster the molecules move, and the higher is the temperature of the body.
Melting, Boiling and the Spacing of Particles
Suppose we keep heating the solid until it starts to melt. Now something different happens. The heat supplied is used to weaken the bonds between the constituent particles, and the average kinetic energy of those particles does not change any further. This is why the temperature stops rising near the melting point.
During melting, the order of magnitude of the average distance between the molecules of the melt remains almost the same as it was in the solid. Because the bonds are weakened but the particles are still close together, liquids do not possess a definite shape but do possess a definite volume. The mean distance between the particles, and hence the density of the liquid, is more or less the same as that of the solid.
On heating further, the atoms and molecules in the liquid gain kinetic energy and the temperature of the liquid increases. If we continue heating the liquid, then at the boiling point the constituents can move about freely, overcoming the interatomic and intermolecular forces. The mean distance between the constituents now increases, so that the particles are far apart.
As per the kinetic theory of gases, for an ideal gas there are no forces between the molecules of the gas. Hence gases have neither a definite volume nor a definite shape — they simply fill whatever container they are put into.
These changes of spacing can be stated in numbers. The interatomic spacing in solids is about \(10^{-10}\ \text{m}\). The average spacing in liquids is almost twice that in solids. The average intermolecular spacing in gases at normal temperature and pressure (NTP) is about \(10^{-9}\ \text{m}\) — roughly ten times the solid spacing, which is why a gas is so much less dense than the solid or liquid it came from.
Temperature as Average Kinetic Energy per Particle
From the above discussion we can now state the microscopic meaning of the two central quantities of this chapter. Heat supplied to a substance increases the kinetic energy of the molecules or atoms of that substance. The average kinetic energy per particle of a substance defines its temperature.
That last condition is exam-critical, and it is worth pausing on. Temperature is an average per particle; thermal energy is a total over all the particles. A cup of tea at 90 °C is hotter than a bucket of water at 40 °C, yet the bucket contains far more thermal energy, simply because it contains far more particles.
Systems, Units and Dimensions
A glass of water, a gas enclosed in a container, and a block of copper metal are all examples of a system. A system is simply the definite portion of matter we choose to study; everything outside it is its surroundings. We can now say that heat, in the form of energy, is transferred between two (or more) systems, or between a system and its surroundings, by virtue of their temperature difference.
Since heat is a form of energy, its SI unit is the same as that of energy: the joule (J). The SI unit of temperature is the kelvin (K), and temperature is also commonly expressed in degree celsius (°C).
The CGS unit of heat energy is the erg, and
\[1\ \text{J} = 10^{7}\ \text{erg}\]
Another unit of heat energy, which you learnt in Standard VIII, is the calorie (cal). Its relation with the joule is
\[1\ \text{cal} = 4.184\ \text{J}\]
Finally, since heat is energy, it has the dimensions of energy:
\[[\text{Heat}] = [\text{L}^{2}\text{M}^{1}\text{T}^{-2}\text{K}^{0}]\]
while temperature, being a fundamental quantity of its own, has the dimensions
\[[\text{Temperature}] = [\text{L}^{0}\text{M}^{0}\text{T}^{0}\text{K}^{1}]\]
Notice how the dimension of temperature contains K to the first power and nothing else. That is exactly the statement made in Section 7.1 — temperature is a fourth fundamental quantity, and cannot be built out of length, mass and time.
7.3 Measurement of Temperature
Walls That Block Heat and Walls That Allow It
To measure temperature, we must first be able to control whether or not heat is allowed to move. In order to isolate two liquids or gases from each other and from the surroundings, we use containers and partitions made of materials such as wood, plastic and glass wool.
An ideal wall or partition separating two systems is one that does not allow any flow or exchange of heat energy from one system to the other. Such a perfect thermal insulator is never available in practice, but it is a very useful idea. It is called an adiabatic wall, and in diagrams it is generally shown as a thick cross-shaded region with slanting lines, as in Fig. 7.5.
When, instead, we wish to allow the exchange of heat energy between two systems, we use a partition such as a thin sheet of copper. This is termed a diathermic wall, and it is represented in diagrams as a thin dark region.
Diagram
Fig. 7.5 – A container divided into two compartments, showing an adiabatic wall as a thick cross-shaded partition and a diathermic wall as a thin dark partition.
The Principle That Makes Thermometry Possible
Let us now consider two sections of a container separated by an adiabatic wall, each holding a different gas. Let us call them system A and system B. Because the wall between them is adiabatic, they cannot exchange any heat with each other.
We now independently bring system A into thermal equilibrium with a third system C, and separately bring system B into thermal equilibrium with the same system C. This is done by allowing each of them, in turn, to exchange heat with C through a diathermic wall until the exchange stops.
Now comes the crucial step, shown in Fig. 7.6. If we remove the adiabatic wall separating systems A and B, we find that there is no transfer of heat from system A to system B, or from B to A. This indicates that systems A and B are also in thermal equilibrium with each other, even though they were never in contact before.
Diagram
Fig. 7.6 – Systems A and B, each separately brought into thermal equilibrium with system C through a diathermic wall, then placed in contact with each other.
The overall conclusion of this activity can be summarised as follows: if systems A and B are separately in thermal equilibrium with a system C, then A and B are also mutually in thermal equilibrium. And when two or more systems or bodies are in thermal equilibrium, their temperatures are the same. This principle is exactly what allows us to measure the temperature of a system using a thermometer — the thermometer plays the role of system C.
Why the Doctor Asks You to Wait
You will recognise this principle in a familiar situation. Before taking your temperature, the doctor shakes the clinical thermometer so that the mercury comes down to indicate some low temperature. You are then asked to keep the thermometer in your mouth, and to wait for some time before it is taken out.
The waiting is not a formality. Initially your body is at a higher temperature than the thermometer, so there is a transfer of heat energy from your body to the thermometer, and the mercury column rises. When the temperature shown on the thermometer becomes the same as that of your body, thermal equilibrium is said to be attained and the heat transfer stops. Only then does the reading represent your body temperature. Taking the thermometer out too early would give a reading that is still on its way up.
Setting Up a Temperature Scale
As stated earlier, to know the thermodynamic state of any system precisely, we need to know its temperature. The device used to measure temperature is a thermometer. Thermometry is the science of temperature and its measurement.
For measurement of temperature, we need to establish a temperature scale and adopt a set of rules for assigning numbers, with the corresponding units, to different degrees of hotness.
For the calibration of a thermometer, a standard temperature interval is selected between two easily reproducible fixed temperatures. This is exactly how the standard of length was once fixed: the metre was taken to be the distance between two fixed marks on a standard bar. In the same way, temperature needs two dependable marks.
The fact that substances change state from solid to liquid, and from liquid to gas, at fixed temperatures is used to define a reference temperature called a fixed point. The two fixed temperatures selected for this purpose are the melting point of ice (which is the freezing point of water) and the boiling point of water.
The next step is to sub-divide this standard temperature interval into smaller sub-intervals, by utilising some physical property that changes with temperature, and to call each sub-interval a degree of temperature. This procedure sets up an empirical scale of temperature.
The two fixed points are defined as follows.
Notice the condition “at one standard atmospheric pressure” in both definitions. It is not decoration. Boiling and freezing temperatures depend on pressure, so without fixing the pressure the marks would not be reproducible — and a scale built on shifting marks would be useless.
From Fixed Points to Numbers
Having decided the fixed point phenomena, it remains to assign numerical values to these fixed points, and to decide the number of divisions between them.
In 1750, conventions were adopted to assign (i) the temperature at which pure ice melts at one atmosphere pressure — the ice point — to be 0°, and (ii) the temperature at which pure water boils at one atmosphere — the steam point — to be 100°, so that there are 100 degrees between the fixed points. This was the centigrade scale, centi meaning hundred in Latin. It was later redefined as the celsius scale, after the Swedish scientist Anders Celsius (1701–1744). It is a convention to express temperature on this scale as degree celsius (°C).
To measure temperature quantitatively, two different scales are generally used. They are described below.
Celsius scale: On this scale, the ice point is marked as 0 and the steam point is marked as 100, both taken at normal atmospheric pressure (\(10^{5}\) Pa or N/m\(^2\)). The interval between these points is divided into 100 equal parts. Each of these is known as a degree Celsius and is written as °C.
Fahrenheit scale: On this scale, the ice point is marked as 32 and the steam point is marked as 212, both taken at normal atmospheric pressure. The interval between these points is divided into 180 equal parts. Each division is known as a degree Fahrenheit and is written as °F.
Converting Between the Two Scales
A relationship for conversion between the two scales may be obtained from a graph of fahrenheit temperature (\(T_F\)) versus celsius temperature (\(T_C\)). As seen in Fig. 7.7, the graph is a straight line, whose equation is
\[\frac{T_F - 32}{180} = \frac{T_C}{100} \qquad \text{--- (7.1)}\]
Diagram
Fig. 7.7 – A plot of fahrenheit temperature (\(T_F\)) versus celsius temperature (\(T_C\)), a straight line through (0, 32) and (100, 212), marking \(\Delta T_C = 100^\circ\) against \(\Delta T_F = 180^\circ\).
Here \(T_F\) is the temperature in degree fahrenheit (°F) and \(T_C\) is the same temperature in degree celsius (°C). The 32 appears because the two scales start counting from different places, and the ratio 180 : 100 appears because they chop the same interval into different numbers of parts.
Thermometric Property and Thermometric Substance
A device used to measure temperature is based on the principle of thermal equilibrium. To measure the temperature, we use different measurable properties of materials which change with temperature. Some of these are the length of a rod, the volume of a liquid, the electrical resistance of a metal wire, and the pressure of a gas at constant volume.
Such changes in physical properties with temperature are used to design a thermometer. The physical property that is used in the thermometer for measuring the temperature is called the thermometric property, and the material employed for the purpose is termed the thermometric substance.
Temperature is measured by exploiting the continuous, monotonic variation of the chosen property with temperature. “Monotonic” means the property must keep increasing (or keep decreasing) as the temperature rises — it must never turn back. A calibration, however, is still required in order to define the temperature scale.
What Makes a Good Thermometer
There are different kinds of thermometers, each type being more suitable than others for a certain job. In each type, the physical property used to measure the temperature must vary continuously over a wide range of temperature, and it must be accurately measurable with simple apparatus.
An important characteristic of a thermometer is its sensitivity — that is, a large change in the thermometric property for a very small change in temperature. Two other characteristics are accuracy and reproducibility. Also, it is important that the system attains thermal equilibrium with the thermometer quickly, otherwise the measurement takes far too long.
If the values of a thermometric property are \(P_1\) and \(P_2\) at the ice point (0 °C) and the steam point (100 °C) respectively, and the value of this property is \(P_T\) at an unknown temperature \(T\), then \(T\) is given by
\[T = \frac{100\,(P_T - P_1)}{(P_2 - P_1)} \qquad \text{--- (7.2)}\]
Read this equation physically. The denominator \((P_2 - P_1)\) is the total change of the property over the whole 100-degree interval, and the numerator \((P_T - P_1)\) is how much of that change has already occurred at the unknown temperature. Their ratio is the fraction of the interval covered, and multiplying by 100 turns that fraction into a celsius reading. Note the assumption hidden inside it: the property is taken to vary linearly with temperature between the fixed points.
Ideally, there should be no difference between the temperatures recorded on two different thermometers. This ideal is most nearly achieved for thermometers based on gases as thermometric substances.
Types of Thermometers
Constant volume gas thermometer. In a constant volume gas thermometer, the pressure of a fixed volume of gas — measured by the difference in height of a mercury column — is used as the thermometric property. It is an accurate but bulky instrument.
Liquid-in-glass thermometer. This thermometer depends on the change in volume of a liquid with temperature. The liquid in a glass bulb expands up a capillary tube when the bulb is heated, as shown in Fig. 7.8. The liquid must be easily seen, and it must expand (or contract) rapidly and by a large amount for a small change in temperature over a wide range of temperature.
Diagram
Fig. 7.8 – A liquid-in-glass thermometer, showing the bulb, the narrow capillary bore and the graduated scale beside it.
The most commonly used liquids are mercury and alcohol, as they remain in the liquid state over a wide range. Mercury freezes at −39 °C and boils at 357 °C; alcohol freezes at −115 °C and boils at 78 °C. Outside these limits the liquid is no longer a liquid, so the thermometer simply cannot be used there.
Thermochromic liquids are liquids which change colour with temperature, but they have a limited range around room temperatures. For example, titanium dioxide and zinc oxide are white at room temperature, but change to yellow when heated.
Thermocouple. Normally, in research laboratories, a thermocouple is used to measure temperature. A thermocouple is a junction of two different metals or alloys — for example, copper and iron — joined together. When two such junctions at the two ends of two dissimilar metal rods are kept at two different temperatures, an electromotive force is generated, and this emf can be calibrated to measure the temperature.
7.4 Absolute Temperature and Ideal Gas Equation
7.4.1 Absolute zero and absolute temperature
What Experiments with Gases Show
Experiments carried out with gases at low densities point to two simple results. First, if the pressure is held constant, the volume of a given quantity of gas is directly proportional to the temperature (measured in ºC). Second, if the volume of a given quantity of gas is held constant, the pressure of the gas is directly proportional to the temperature (measured in ºC).
These two relations are shown graphically in Fig. 7.9 (a) and Fig. 7.9 (b). Mathematically, this relationship can be written as
\[PV \propto T_C\]
where \(P\) is the pressure of the gas (in Pa), \(V\) is its volume (in m\(^3\)), and \(T_C\) is its temperature on the celsius scale (in °C).
Diagram
Fig. 7.9 (a) – Graph of volume versus temperature (in °C) at constant pressure, showing straight lines for three different gases extrapolated backwards to meet the temperature axis at −273.15 °C.
Diagram
Fig. 7.9 (b) – Graph of pressure versus temperature (in °C) at constant volume, showing straight lines for three different gases extrapolated backwards to meet the temperature axis at −273.15 °C.
Thus the volume–temperature and pressure–temperature graphs for a gas are straight lines. They show that gases expand linearly with temperature as measured on a mercury thermometer; that is, an equal increase in temperature causes an equal increase in volume, or an equal increase in pressure.
This is a valuable fact. The similar thermal behaviour of all gases suggests that this relationship can itself be used to measure temperature, in a constant-volume gas thermometer, in terms of the pressure of the gas.
Although actual experimental measurements might differ a little from the ideal linear relationship, the linear relationship holds over a wide temperature range.
Extending the Lines Backwards
Look closely at the graphs in Fig. 7.9. It may be noted that the lines do not pass through the origin — they have a non-zero intercept along the y-axis. Also, the straight lines have different slopes for different gases, since different gases expand by different amounts.
Now let us make an assumption. If we assume that the gases do not liquefy even if we lower the temperature, we can extend the straight lines backwards to low temperatures. This backward extension is called extrapolation, and it is drawn as a dotted line in Fig. 7.9 because it goes beyond where measurements can actually be made.
Is it possible to reach a temperature at which the gases stop exerting any pressure, that is, where the pressure is zero? Similarly, in a constant pressure thermometer, as the temperature is lowered, the volume decreases. Supposing the gas does not liquefy even at very low temperature, at what temperature will its volume become zero?
Practically, it is not possible to keep a material in the gaseous state at very low temperature, and without its exerting any pressure. Every real gas turns into a liquid, and then a solid, long before that. But the mathematics of the straight line can still be followed to its end.
If we extrapolate the graph of pressure \(P\) versus temperature \(T_C\) (in ºC), the temperature at which the pressure of a gas would be zero is −273.15 ºC. The striking result is that all the lines for different gases cut the temperature axis at the same point, that is, at −273.15 ºC — even though their slopes are all different. This point is termed the absolute zero of temperature.
It is not possible to attain any temperature lower than this value, and even achieving absolute zero itself is not possible in practice. It may be noted that the point of zero pressure, or of zero volume, does not depend on any specific gas — which is precisely what makes it fit to be the zero of a universal scale.
From Two Fixed Points to One
The two fixed point scale, described in Section 7.3, had a practical shortcoming when it came to calibrating the scale. It was difficult to precisely control the pressure and to identify the fixed points — especially the boiling point, because the boiling temperature is very sensitive to changes in pressure. A small change in atmospheric pressure shifts the steam point, and with it the whole scale.
Hence, a one fixed point scale was adopted in 1954 to define a temperature scale. This scale is called the absolute scale, or the thermodynamic scale. It is named the kelvin scale, after Lord Kelvin (1824–1907).
But if only one fixed point is used, that point had better be a very reliable one. Such a point exists, and it is called the triple point.
It is possible for all the three phases of a material — solid, liquid and gas or vapour — to coexist in equilibrium. This is known as the triple point. To know the triple point, one has to see that all three phases coexist in equilibrium with no one phase dominating. This occurs, for each substance, at a single unique combination of temperature and pressure.
Thus, if the three phases of water — solid ice, liquid water and water vapour — coexist, then the pressure and the temperature are automatically fixed. Nature itself holds them there. This is termed the triple point of water, and it serves as a single fixed point to define a temperature scale.
The Kelvin Scale
The absolute scale of temperature is so termed since it is based on the properties of an ideal gas, and does not depend on the property of any particular substance. The zero of this scale is ideally the lowest temperature possible, although it has not been achieved in practice.
It is termed the Kelvin scale, with its zero at −273.15 °C and with temperature intervals the same as those on the celsius scale. It is written as K, without the degree sign.
Internationally, the triple point of water has been assigned the value 273.16 K, at a pressure equal to \(6.11 \times 10^{2}\) Pa or \(6.11 \times 10^{-3}\) atmosphere, as the standard fixed point for the calibration of thermometers.
The size of one kelvin is thus \(1/273.16\) of the difference between the absolute zero and the triple point of water. It is the same in size as one celsius degree. On the celsius scale, the triple point of water is 0.01 ºC and not zero.
Take care not to confuse two numbers that look almost alike: 273.15 is the shift between the celsius and kelvin scales (the position of absolute zero), while 273.16 K is the assigned temperature of the triple point of water. They differ by exactly the 0.01 ºC mentioned above.
Three identical thermometers, marked in kelvin, celsius and fahrenheit, are placed in a fixed temperature bath. Each thermometer shows the same rise in the level of mercury for the human body temperature, but each labels that same level with a different number, as depicted in Fig. 7.10. The mercury does not know which scale is printed beside it — the scale is only our way of naming the level.
Diagram
Fig. 7.10 – Comparison of the kelvin, celsius and fahrenheit temperature scales side by side, marking the steam point, human body temperature, ice point and absolute zero on each (thermometer readings not to scale).
The relation between the three scales of temperature is given by Eq. (7.3):
\[\frac{T_C}{100} = \frac{T_F - 32}{180} = \frac{T_K - 273.15}{100} \qquad \text{--- (7.3)}\]
where \(T_C\) is the temperature in °C, \(T_F\) is the temperature in °F, and \(T_K\) is the temperature in K.
7.4.2 Ideal Gas Equation
The Gas Laws in Absolute Temperature
The relation between the three properties of a gas — its pressure, volume and temperature — is called the ideal gas equation. You will learn more about the properties of gases in chemistry.
Using absolute temperatures, rather than celsius temperatures, the gas laws can be stated in a much cleaner form, as given below. The reason is visible in the graphs: on the celsius scale the lines had awkward non-zero intercepts, but on the kelvin scale they start from the origin, and a straight line through the origin means simple direct proportionality.
1) Charles’ law. In Fig. 7.9 (b), the volume–temperature graph passes through the origin if temperatures are measured on the kelvin scale, that is, if we take 0 K as the origin. In that case the volume \(V\) is directly proportional to the absolute temperature \(T\).
Thus \(V \propto T\)
\[or,\ \frac{V}{T} = \text{constant} \qquad \text{--- (7.4)}\]
2) Pressure (Gay Lussac’s) law. From Fig. 7.9, it can be seen that the pressure–temperature graph is similar to the volume–temperature graph.
Thus \(P \propto T\)
\[or,\ \frac{P}{T} = \text{constant} \qquad \text{--- (7.5)}\]
3) Boyle’s law. For a fixed mass of gas at constant temperature, the pressure is inversely proportional to the volume.
Thus \(P \propto \dfrac{1}{V}\)
\[PV = \text{constant} \qquad \text{--- (7.6)}\]
Each of these three laws holds only under the condition stated with it — one quantity is held fixed while the other two are allowed to vary. Dropping that condition is the commonest mistake in numerical problems on this topic.
Combining the Laws
Combining the above three equations, we get
\[\frac{PV}{T} = \text{constant} \qquad \text{--- (7.7)}\]
For one mole of a gas, the constant of proportionality is written as \(R\).
\[\therefore\ \frac{PV}{T} = R \quad or \quad PV = RT \qquad \text{--- (7.8)}\]
If a given mass of a gas consists of \(n\) moles, then Eq. (7.8) can be written as
\[PV = nRT \qquad \text{--- (7.9)}\]
This relation is called the ideal gas equation. Here \(P\) is the pressure of the gas in pascal (Pa), \(V\) is its volume in cubic metre (m\(^3\)), \(n\) is the number of moles, \(T\) is the absolute temperature in kelvin (K), and \(R\) is the constant of proportionality.
The value of the constant \(R\) is the same for all gases. It is therefore known as the universal gas constant, and its numerical value is 8.31 J K\(^{-1}\) mol\(^{-1}\).
The unit of \(R\) is worth a moment’s thought. \(PV\) has the dimensions of pressure multiplied by volume, which is force per unit area multiplied by volume — that is, energy. So \(R\) is an energy per mole per kelvin, and this is why it is measured in J K\(^{-1}\) mol\(^{-1}\).
In examination problems on this sub-section, marks are most often lost not in the algebra but at the very first step — forgetting to convert a celsius temperature into kelvin before putting it into any of Eqs. (7.4), (7.5), (7.7), (7.8) or (7.9). Boyle’s law, Eq. (7.6), is the only one of the set that does not contain \(T\) at all, and so is the only one where this trap does not exist.
7.5 Thermal Expansion
Why Matter Expands on Heating
When matter is heated, it normally expands, and when it is cooled, it normally contracts. To see why, we return to the particle picture built up in Section 7.2.
The atoms in a solid vibrate about their mean positions. When the solid is heated, they vibrate faster, and in doing so they force each other to move a little farther apart. This results in expansion of the solid. The molecules in a liquid or a gas move about with a certain speed. When they are heated, they move faster and force each other to move a little farther apart, and this results in the expansion of liquids and gases on heating.
Because the particles of a liquid are held less tightly than those of a solid, the same rise in temperature pushes them farther apart. Hence the expansion is more in liquids than in solids, and gases — whose molecules are hardly held back at all — expand even more.
A change in the temperature of a body therefore causes a change in its dimensions. The increase in the dimensions of a body due to an increase in its temperature is called thermal expansion.
There are three types of thermal expansion, depending on which dimension we choose to follow: 1) Linear expansion, 2) Areal expansion, 3) Volume expansion.
7.5.1 Linear Expansion
The expansion in length due to thermal energy is called linear expansion.
If the substance is in the form of a long rod of length \(l\), then for a small change \(\Delta T\) in temperature, the fractional change \(\Delta l / l\) in length, shown (much exaggerated) in Fig. 7.11, is directly proportional to \(\Delta T\).
Diagram
Fig. 7.11 – A rod of original length \(l\) and, below it, the same rod after heating with length \(l + \Delta l\); the linear expansion \(\Delta l\) is exaggerated for explanation.
\[\frac{\Delta l}{l} \propto \Delta T\]
\[or \quad \frac{\Delta l}{l} = \alpha\, \Delta T \qquad \text{--- (7.10)}\]
where \(\alpha\) is called the coefficient of linear expansion of the solid. Its value depends upon the nature of the material.
Notice that the equation is written for the fractional change in length, \(\Delta l / l\), and not for \(\Delta l\) alone. This matters: a 10 m rail and a 1 m rod of the same steel, heated by the same amount, do not expand by the same number of millimetres — but they do expand by the same fraction of their own length.
Rearranging Eq. (7.10), we get
\[\alpha = \frac{\Delta l}{l\,\Delta T} = \frac{l_T - l_0}{l_0\,(T - T_0)} \qquad \text{--- (7.11)}\]
where \(l_0\) = length of the rod at 0 °C, \(l_T\) = length of the rod when heated to \(T\) °C, \(T_0 = 0\) °C is the initial temperature, \(T\) = final temperature, \(\Delta l = l_T - l_0\) = change in length, \(\Delta T = T - T_0\) = rise in temperature.
Referring to Eq. (7.11), if \(l_0 = 1\) and \(T - T_0 = 1\) °C, then
\[\alpha = l_T - l_0 \quad \text{(numerically)}\]
This special case gives us the meaning of \(\alpha\) in words.
The unit of the coefficient of linear expansion is per degree celsius or per kelvin. Since one celsius degree and one kelvin are equal in size (Section 7.4.1), the numerical value of \(\alpha\) is the same whichever of the two is used — a small mercy in numerical problems.
The magnitude of \(\alpha\) is very small, and it varies only a little with temperature. For most practical purposes, \(\alpha\) can be assumed to be constant for a particular material. Therefore it is not necessary that the initial temperature be taken as 0 °C, and Equation (7.11) can be rewritten as
\[\alpha = \frac{l_2 - l_1}{l_1\,(T_2 - T_1)} \qquad \text{--- (7.12)}\]
where \(l_1\) = initial length at temperature \(T_1\) °C and \(l_2\) = final length at temperature \(T_2\) °C.
Table 7.1 lists average values of the coefficient of linear expansion for some materials in the temperature range 0 °C to 100 °C.
Table 7.1: Values of coefficient of linear expansion for some common materials.
| Materials | \(\alpha\) (K\(^{-1}\)) |
|---|---|
| Carbon (diamond) | \(0.1 \times 10^{-5}\) |
| Glass | \(0.85 \times 10^{-5}\) |
| Iron | \(1.2 \times 10^{-5}\) |
| Steel | \(1.3 \times 10^{-5}\) |
| Gold | \(1.4 \times 10^{-5}\) |
| Copper | \(1.7 \times 10^{-5}\) |
| Silver | \(1.9 \times 10^{-5}\) |
| Aluminium | \(2.5 \times 10^{-5}\) |
| Sulphur | \(6.1 \times 10^{-5}\) |
| Mercury | \(6.1 \times 10^{-5}\) |
| Water | \(6.9 \times 10^{-5}\) |
| Carbon (graphite) | \(8.8 \times 10^{-5}\) |
7.5.2 Areal Expansion
The increase \(\Delta A\) in the surface area of a body on heating is called areal expansion, or superficial expansion.
If a substance is in the form of a plate of area \(A\), then for a small change \(\Delta T\) in temperature, the fractional change in area \(\Delta A / A\), as shown in Fig. 7.12, is directly proportional to \(\Delta T\).
Diagram
Fig. 7.12 – A square plate with a shaded border showing the increase in area \(\Delta A\) on heating; the areal expansion \(\Delta A\) is exaggerated for explanation.
\[\frac{\Delta A}{A} \propto \Delta T\]
\[or \quad \frac{\Delta A}{A} = \beta\, \Delta T \qquad \text{--- (7.13)}\]
where \(\beta\) is called the coefficient of areal expansion of the solid. It depends on the material of the solid.
Rearranging Eq. (7.13), we get
\[\beta = \frac{\Delta A}{A\,\Delta T} = \frac{A_T - A_0}{A_0\,(T - T_0)} \qquad \text{--- (7.14)}\]
where \(A_0\) = area of the plate at 0 °C, \(A_T\) = area of the plate when heated to \(T\) °C, \(T_0 = 0\) °C is the initial temperature, \(T\) = final temperature, \(\Delta A = A_T - A_0\) = change in area, \(\Delta T = T - T_0\) = rise in temperature.
If \(A_0 = 1\) m\(^2\) and \(T - T_0 = 1\) °C, then
\[\beta = A_T - A_0 \quad \text{(numerically)}\]
Therefore we can state the meaning of \(\beta\) in the same style as before.
The unit of \(\beta\) is per degree celsius or per kelvin.
As in the case of \(\alpha\), the value of \(\beta\) also does not vary much with temperature. Hence, if \(A_1\) is the area of a metal plate at \(T_1\) °C and \(A_2\) is the area at a higher temperature \(T_2\) °C, then
\[\beta = \frac{A_2 - A_1}{A_1\,(T_2 - T_1)} \qquad \text{--- (7.15)}\]
7.5.3 Volume expansion
The increase in volume due to heating is called volume expansion, or cubical expansion.
If the substance is in the form of a cube of volume \(V\), then for a small change \(\Delta T\) in temperature, the fractional change \(\Delta V / V\) in volume, as shown in Fig. 7.13, is directly proportional to \(\Delta T\).
Diagram
Fig. 7.13 – A cube with a dashed outline showing its enlarged volume after heating; the volume expansion \(\Delta V\) is exaggerated for explanation.
\[\frac{\Delta V}{V} \propto \Delta T\]
\[or \quad \frac{\Delta V}{V} = \gamma\, \Delta T \qquad \text{--- (7.16)}\]
where \(\gamma\) is called the coefficient of cubical or volume expansion. It depends upon the nature of the material. Its unit is per degree celsius or per kelvin.
From Eq. (7.16), we can write
\[\gamma = \frac{\Delta V}{V\,\Delta T} = \frac{V_T - V_0}{V_0\,(T - T_0)} \qquad \text{--- (7.17)}\]
where \(V_0\) = volume at 0 °C, \(V_T\) = volume when heated to \(T\) °C, \(T_0 = 0\) °C is the initial temperature, \(T\) = final temperature, \(\Delta V = V_T - V_0\) = change in volume, \(\Delta T = T - T_0\) = rise in temperature.
If \(V_0 = 1\) m\(^3\) and \(T - T_0 = 1\) °C, then
\[\gamma = V_T - V_0 \quad \text{(numerically)}\]
If \(V_1\) is the volume of a body at \(T_1\) °C and \(V_2\) is the volume at a higher temperature \(T_2\) °C, then
\[\gamma_1 = \frac{V_2 - V_1}{V_1\,(T_2 - T_1)} \qquad \text{--- (7.18)}\]
where \(\gamma_1\) is the coefficient of volume expansion at temperature \(T_1\) °C.
Volume Expansion of Fluids
Since fluids possess a definite volume but take the shape of the container, only the change in volume is significant for them — a liquid has no fixed length or fixed surface area of its own to speak of. Equations (7.17) and (7.18) are valid for the cubical or volume expansion of fluids as well.
It is to be noted that since fluids are kept in containers, when one deals with the volume expansion of a fluid, the expansion of the container must also be considered. The container is itself a solid, and it too grows on heating.
If the expansion of the fluid results in a volume greater than the volume of the container, the fluid will overflow, provided the container is open. If the container is closed, the volume expansion of the fluid will cause an additional pressure on the walls of the container.
Normally, solids and liquids expand on heating, so their volume increases on heating. Since the mass is constant, and density is mass divided by volume, this results in a decrease in the density on heating.
You have learnt about the anomalous behaviour of water. Water expands on cooling from 4 °C to 0 °C. Hence its density decreases on cooling in this temperature range — the opposite of the normal behaviour just described.
In Table 7.2 are given typical average values of the coefficient of volume expansion \(\gamma\) for some materials in the temperature range 0 °C to 100 °C.
Table 7.2: Values of coefficient of volume expansion for some common materials.
| Materials | \(\gamma\) (K\(^{-1}\)) |
|---|---|
| Invar | \(2 \times 10^{-6}\) |
| Glass (ordinary) | \(2.5 \times 10^{-5}\) |
| Steel | \((3.3\text{–}3.9) \times 10^{-5}\) |
| Iron | \(3.55 \times 10^{-5}\) |
| Gold | \(4.2 \times 10^{-5}\) |
| Brass | \(5.7 \times 10^{-5}\) |
| Aluminium | \(6.9 \times 10^{-5}\) |
| Mercury | \(18.2 \times 10^{-5}\) |
| Water | \(20.7 \times 10^{-5}\) |
| Paraffin | \(58.8 \times 10^{-5}\) |
| Gasoline | \(95.0 \times 10^{-5}\) |
| Alcohol (ethyl) | \(110 \times 10^{-5}\) |
Read the table from top to bottom and a pattern appears: the solids sit near the top with small values, and the liquids gather at the bottom with values ten to fifty times larger. This is the numerical statement of what we said at the start of Section 7.5 — liquids expand more than solids.
Diagram
Fig. 7.14 – Graph of the coefficient of volume expansion \(\gamma\) (in \(10^{-5}\) K\(^{-1}\)) of copper as a function of temperature \(T\) (in K), rising steeply at low temperature and flattening to a constant value at high temperature.
7.5.4 Relation between Coefficients of Expansion
The three coefficients \(\alpha\), \(\beta\) and \(\gamma\) describe the same physical effect measured along one, two and three dimensions. They cannot therefore be independent of one another, and we now derive the relations between them.
i) Relation between \(\beta\) and \(\alpha\)
Consider a square plate of side \(l_0\) at 0 °C, whose side becomes \(l_T\) at \(T\) °C.
\[\therefore\ l_T = l_0\,(1 + \alpha T) \quad \text{from Eq. (7.11).}\]
If the area of the plate at 0 °C is \(A_0\), then \(A_0 = l_0^{\,2}\).
If the area of the plate at \(T\) °C is \(A_T\), then
\[A_T = l_T^{\,2} = l_0^{\,2}\,(1 + \alpha T)^2\]
\[or \quad A_T = A_0\,(1 + \alpha T)^2 \qquad \text{--- (7.19)}\]
Also, from Eq. (7.14),
\[A_T = A_0\,(1 + \beta T) \qquad \text{--- (7.20)}\]
The same final area has now been written in two ways — once through the growth of each side, and once through the definition of \(\beta\). Using Eqs. (7.19) and (7.20), we get
\[A_0\,(1 + \alpha T)^2 = A_0\,(1 + \beta T)\]
\[or \quad 1 + 2\alpha T + \alpha^2 T^2 = 1 + \beta T\]
Since the values of \(\alpha\) are very small (of the order of \(10^{-5}\)), the term \(\alpha^2 T^2\) is very small and may be neglected.
\[\therefore\ \beta = 2\alpha \qquad \text{--- (7.21)}\]
The result is general, because any solid can be regarded as a collection of small squares.
ii) Relation between \(\gamma\) and \(\alpha\)
Consider a cube of side \(l_0\) at 0 °C, whose side becomes \(l_T\) at \(T\) °C.
\[\therefore\ l_T = l_0\,(1 + \alpha T) \quad \text{from Eq. (7.11).}\]
If the volume of the cube at 0 °C is \(V_0\), then \(V_0 = l_0^{\,3}\).
If the volume of the cube at \(T\) °C is \(V_T\), then
\[V_T = l_T^{\,3} = l_0^{\,3}\,(1 + \alpha T)^3\]
\[or \quad V_T = V_0\,(1 + \alpha T)^3 \qquad \text{--- (7.22)}\]
Also, from Eq. (7.17),
\[V_T = V_0\,(1 + \gamma T) \qquad \text{--- (7.23)}\]
Using Eqs. (7.22) and (7.23), we get
\[V_0\,(1 + \alpha T)^3 = V_0\,(1 + \gamma T)\]
\[or \quad 1 + 3\alpha T + 3\alpha^2 T^2 + \alpha^3 T^3 = 1 + \gamma T\]
Since the values of \(\alpha\) are very small, the terms with higher powers of \(\alpha\) may be neglected.
\[\therefore\ \gamma = 3\alpha \qquad \text{--- (7.24)}\]
Again the result is general, because any solid can be regarded as a collection of small cubes.
Finally, the relation between \(\alpha\), \(\beta\) and \(\gamma\) is
\[\alpha = \frac{\beta}{2} = \frac{\gamma}{3} \qquad \text{--- (7.25)}\]
7.6 Specific Heat Capacity
7.6.1 Specific Heat Capacity of Solids and Liquids
An Experiment That Sets Up the Idea
Take 1 kg of water and 1 kg of paraffin, and heat each in turn for the same length of time with the same heater. The result is not the same. The temperature rise of the paraffin is about twice that of the water.
Since the heater gives equal amounts of heat energy to each liquid, and the masses are equal, the difference cannot lie in the heat or in the mass. It must lie in the substance itself. It seems that different substances require different amounts of heat to cause the same temperature rise of 1 °C in the same mass of 1 kg.
We give this “thermal appetite” of a substance a name and a symbol. If \(\Delta Q\) stands for the amount of heat absorbed or given out by a substance of mass \(m\) when it undergoes a temperature change \(\Delta T\), then the specific heat capacity of that substance is given by
\[s = \frac{\Delta Q}{m\,\Delta T} \qquad \text{--- (7.26)}\]
where \(\Delta Q\) is the heat in joule (J), \(m\) is the mass in kilogram (kg), \(\Delta T\) is the temperature change in °C or K, and \(s\) is the specific heat capacity.
If \(m = 1\) kg and \(\Delta T = 1\) °C, then \(s = \Delta Q\). In other words, the specific heat capacity is numerically equal to the heat needed by one kilogram of the substance for a one-degree change of temperature.
Units and Table of Values
The SI unit of specific heat capacity is J/kg °C or J/kg K, and the C.G.S. unit is erg/g °C or erg/g K. Since a change of 1 °C is the same as a change of 1 K, the numerical value of \(s\) is unaffected by which of the two is used.
The specific heat capacity is a property of the substance and depends only weakly on its temperature. Except at very low temperatures, the specific heat capacity is almost constant for all practical purposes.
Table 7.3: Specific heat capacity of some substances at room temperature and atmospheric pressure.
| Substance | Specific heat capacity (J kg\(^{-1}\) K\(^{-1}\)) |
|---|---|
| Steel | 120 |
| Lead | 128 |
| Gold | 129 |
| Tungsten | 134.4 |
| Silver | 234 |
| Copper | 387 |
| Iron | 448 |
| Carbon | 506.5 |
| Glass | 837 |
| Aluminium | 903.0 |
| Kerosene | 2118 |
| Paraffin oil | 2130 |
| Alcohol (ethyl) | 2400 |
| Ethanol | 2500 |
| Water | 4186.0 |
Molar Specific Heat
If the amount of a substance is specified in terms of moles \(\mu\), instead of the mass \(m\) in kg, then the specific heat is called the molar specific heat (\(C\)), and is given by
\[C = \frac{1}{\mu}\,\frac{\Delta Q}{\Delta T} \qquad \text{--- (7.27)}\]
where \(\mu\) is the number of moles, \(\Delta Q\) is the heat in joule (J) and \(\Delta T\) is the temperature change in °C or K.
The SI unit of molar specific heat capacity is J/mol °C or J/mol K. Like specific heat, molar specific heat also depends on the nature of the substance and on its temperature.
The difference between \(s\) and \(C\) is only in how we count the substance — by kilogram or by mole. Both answer the same question: how much heat does this much substance need for one degree?
Why Water Stands Apart
From Table 7.3, it can be seen that water has the highest specific heat capacity compared to the other substances listed. It needs more than ten times as much heat per kilogram, per degree, as an equal mass of copper.
For this reason, water is used as a coolant in automobile radiators, as well as for fomentation using hot water bags.
7.6.2 Specific Heat Capacity of Gas
Why a Gas Needs Two Specific Heats
In the case of a gas, a slight change in temperature is accompanied by considerable changes in both the volume and the pressure. A gas, unlike a solid or a liquid, cannot be heated without deciding first what we will allow it to do.
If a gas is heated at constant pressure, its volume changes, and therefore some work is done on the surroundings during the expansion. This work requires additional heat, over and above the heat needed to raise the temperature. As a result, the specific heat at constant pressure (\(S_p\)) is greater than the specific heat at constant volume (\(S_v\)).
It is thus necessary to define two principal specific heat capacities for a gas.
Principal specific heat capacities of gases
- The principal specific heat capacity of a gas at constant volume (\(S_v\)) is defined as the quantity of heat absorbed or released for the rise or fall of temperature of unit mass of the gas through 1 K (or 1 °C), when its volume is kept constant.
- The principal specific heat capacity of a gas at constant pressure (\(S_p\)) is defined as the quantity of heat absorbed or released for the rise or fall of temperature of unit mass of the gas through 1 K (or 1 °C), when its pressure is kept constant.
Molar specific heat capacities of gases
The molar specific heat capacity of a gas at constant volume (\(C_v\)) is defined as the quantity of heat absorbed or released for the rise or fall of temperature of one mole of the gas through 1 K (or 1 °C), when its volume is kept constant.
The molar specific heat capacity of a gas at constant pressure (\(C_p\)) is defined as the quantity of heat absorbed or released for the rise or fall of temperature of one mole of the gas through 1 K (or 1 °C), when its pressure is kept constant.
The condition attached to each of these four definitions — “volume kept constant” or “pressure kept constant” — is not a detail. Drop it, and the quantity is not defined at all.
Relation between Principal and Molar Specific Heat Capacities
A relation between the principal specific heat capacity and the molar specific heat capacity is given by the following expression:
Molar specific heat capacity = Molecular weight × principal specific heat capacity.
i.e. \(C_p = M \times S_p\) and \(C_v = M \times S_v\)
where \(M\) is the molecular weight of the gas. The reason is simply that one mole of the gas has a mass equal to its molecular weight, so heating one mole means heating that many mass units.
Table 7.4 lists values of molar specific heat capacity for some commonly known gases.
Table 7.4: Molar specific heat capacity of some gases.
| Gas | \(C_P\) (J mol\(^{-1}\) K\(^{-1}\)) | \(C_V\) (J mol\(^{-1}\) K\(^{-1}\)) |
|---|---|---|
| He | 20.8 | 12.5 |
| H\(_2\) | 28.8 | 20.4 |
| N\(_2\) | 29.1 | 20.8 |
| O\(_2\) | 29.4 | 21.1 |
| CO\(_2\) | 37.0 | 28.5 |
Look down the two columns and you will see the prediction of the previous page confirmed for every gas listed: \(C_P\) is larger than \(C_V\), without exception.
7.6.3 Heat Equation
Let us read the meaning of a specific heat capacity out loud. If a substance has a specific heat capacity of 1000 J/kg °C, it means that a heat energy of 1000 J raises the temperature of 1 kg of that substance by 1 °C.
It follows that 6000 J will raise the temperature of 2 kg of the substance by 3 °C, since we need twice the heat for twice the mass and three times the heat for three times the temperature rise. And if the temperature of that 2 kg mass of the substance instead falls by 3 °C, the heat given out would also be 6000 J. The equation runs both ways: heat gained and heat lost obey the same rule.
In general, we can write the heat equation as
Heat received or given out (\(Q\)) = mass (\(m\)) × temperature change (\(\Delta t\)) × specific heat capacity (\(s\)).
\[or \quad Q = m \times \Delta T \times s \qquad \text{--- (7.28)}\]
where \(Q\) is in joule (J), \(m\) in kilogram (kg), \(\Delta T\) in °C or K, and \(s\) in J kg\(^{-1}\) K\(^{-1}\).
7.6.4 Heat Capacity (Thermal Capacity)
Specific heat capacity refers to unit mass. Often, however, we are dealing with one particular body — this copper block, this calorimeter — and we want to know what that whole body requires. That quantity is called the heat capacity.
The heat capacity, or thermal capacity, of a body is the quantity of heat needed to raise or lower the temperature of the whole body by 1 °C (or 1 K).
\(\therefore\) Thermal heat capacity can be written as
Heat received or given out = mass × 1 × specific heat capacity
\[\text{Heat capacity} = Q = m \times s \qquad \text{--- (7.29)}\]
Heat capacity (thermal capacity) is measured in J/°C.
Note the difference in the units carefully, because it is a favourite examination trap: specific heat capacity is measured in J kg\(^{-1}\) °C\(^{-1}\) and belongs to the material, while heat capacity is measured in J °C\(^{-1}\) and belongs to the object. Two copper blocks of different masses have the same specific heat capacity but different heat capacities.
7.7 Calorimetry
The Instrument
Calorimetry is an experimental technique for the quantitative measurement of heat exchange. In Section 7.6 we learnt how much heat a substance needs per degree; calorimetry is how that quantity is actually measured in the laboratory.
To make such a measurement, a calorimeter is used. Figure 7.15 shows a simple water calorimeter.
Diagram
Fig. 7.15 – A water calorimeter: a cylindrical copper vessel containing water, with a thermometer and a stirrer passing through an insulating lid, the vessel resting on an insulating support inside an outer jacket with an air gap.
It consists of a cylindrical vessel made of copper or aluminium, provided with a stirrer and a lid. The calorimeter is well-insulated, in order to prohibit any transfer of heat into or out of the calorimeter.
Every part of this arrangement has a reason behind it. The vessel is made of metal, a good conductor, so that the contents quickly reach a common temperature. The stirrer keeps the mixture uniformly mixed, so that the thermometer reads the temperature of the whole mixture and not just of the layer around its bulb. The lid, the insulating jacket and the air gap all serve one purpose: to make the calorimeter and its contents an isolated system, exchanging heat with each other and with nothing else.
The Principle Behind the Measurement
One important use of a calorimeter is to determine the specific heat of a substance, using the principle of conservation of energy. Here we are dealing with heat energy, and the system is isolated from its surroundings. Therefore, the heat gained is equal to the heat lost.
The Method of Mixtures
In the technique known as the “method of mixtures”, a sample ‘A’ of the substance is heated to a high temperature, which is accurately measured. The sample ‘A’ is then placed quickly in the calorimeter containing water. The contents are stirred constantly until the mixture attains a final common temperature.
The word “quickly” is doing real work here. Every second that the hot sample spends in the open air, it loses heat to the room — and that heat is then missing from the calorimeter, so the measurement is spoiled.
The heat lost by the sample ‘A’ will be gained by the water and by the calorimeter. The specific heat of the sample ‘A’ of the substance can be calculated as under.
Let \(m_1\) = mass of the sample ‘A’ \(m_2\) = mass of the calorimeter and the stirrer \(m_3\) = mass of the water in the calorimeter \(s_1\) = specific heat capacity of the substance of sample ‘A’ \(s_2\) = specific heat capacity of the material of the calorimeter (and stirrer) \(s_3\) = specific heat capacity of water \(T_1\) = initial temperature of the sample ‘A’ \(T_2\) = initial temperature of the calorimeter, stirrer and water \(T\) = final temperature of the combined system
All masses are in kg, all specific heat capacities in J kg\(^{-1}\) K\(^{-1}\), and all temperatures in °C (or K).
We have the data as follows. Each of the three expressions below is just the heat equation, Eq. (7.28), applied to one body, with \(\Delta T\) written so that it comes out positive.
Heat lost by the sample ‘A’ \(= m_1 s_1\,(T_1 - T)\)
Heat gained by the calorimeter and the stirrer \(= m_2 s_2\,(T - T_2)\)
Heat gained by the water \(= m_3 s_3\,(T - T_2)\)
Assuming no loss of heat to the surroundings, the heat lost by the sample goes into the calorimeter, the stirrer and the water. Thus, writing the heat equation,
\[m_1 s_1 (T_1 - T) = m_2 s_2 (T - T_2) + m_3 s_3 (T - T_2) \qquad \text{--- (7.30)}\]
Knowing the specific heat capacity of water (\(s_3 = 4186\) J kg\(^{-1}\) K\(^{-1}\)) and of copper (\(s_2 = 387\) J kg\(^{-1}\) K\(^{-1}\)), copper being the material of the calorimeter and the stirrer, one can calculate the specific heat capacity (\(s_1\)) of the material of sample ‘A’ from Eq. (7.30) as
\[s_1 = \frac{(m_2 s_2 + m_3 s_3)(T - T_2)}{m_1 (T_1 - T)} \qquad \text{--- (7.31)}\]
Also, one can find the specific heat capacity of water, or of any liquid, using the following expression, if the specific heat capacity of the material of the calorimeter and of the sample is known:
\[s_3 = \frac{m_1 s_1 (T_1 - T)}{m_3 (T - T_2)} - \frac{m_2 s_2}{m_3} \qquad \text{--- (7.32)}\]
Both Eq. (7.31) and Eq. (7.32) come from the single equation (7.30); it is only the unknown that changes. This is worth noticing, because in a numerical problem you should not try to memorise the two results separately — write the heat balance first, then solve for whichever quantity is missing.
Note: In the experiment, the heat from the solid sample ‘A’ is given to the liquid, and therefore the sample should be denser than the liquid, so that the sample does not float on the liquid. A floating sample would touch the air above the liquid and lose heat to it, and it would not be fully surrounded by the liquid it is supposed to be heating.
7.8 Change of State
Matter normally exists in three states: solid, liquid and gas. A transition from one of these states to another is called a change of state. The two common changes of state are solid to liquid, and liquid to gas — and, in the reverse direction, gas to liquid and liquid to solid.
These changes can occur when an exchange of heat takes place between the substance and its surroundings. Nothing else is needed: no chemical reaction, no addition of any new material. Heat alone does it.
Diagram
Fig. 7.16 – Variation of temperature with time for ice being heated steadily: a flat portion AB at 0 °C, a rising portion BC, and a second flat portion CD near 100 °C, with the curve rising again beyond D.
Analysis of Observations
1) From point A to point B:
There is no change in temperature from point A to point B. This means that the temperature of the ice bath does not change, even though heat is being continuously supplied. The temperature remains constant until the entire amount of the ice melts. The heat supplied is being utilised in changing the state from solid (ice) to liquid (water).
This is the flat stretch that surprises most students, and it is worth stating plainly: heat is going in, and yet the thermometer refuses to move. The heat is not being wasted — it is going into breaking the bonds of the solid, exactly as described in Section 7.2.
The change of state from solid to liquid is called melting, and that from liquid to solid is called solidification.
Both the solid and the liquid states of the substance co-exist in thermal equilibrium during the change of state from solid to liquid, or vice versa. This is why the beaker holds a mixture of ice and water all through the stretch AB.
The temperature at which the solid and the liquid states of the substance are in thermal equilibrium with each other is called the melting point of the solid (here ice), or the freezing point of the liquid (here water). It is characteristic of the substance, and it also depends on pressure.
The melting point of a substance at one standard atmospheric pressure is called its normal melting point.
At one standard atmospheric pressure, the freezing point of water and the melting point of ice is 0 °C, or 32 °F. The freezing point describes the liquid-to-solid transition, while the melting point describes the solid-to-liquid transition. The two words describe the same temperature approached from opposite directions.
2) From point B to point D:
The temperature begins to rise from point B to point C — that is, after the whole of the ice has been converted into water and we continue further heating. We see that the temperature begins to rise. The temperature keeps on rising till it reaches point C, that is, nearly 100 °C.
Then it again becomes steady. It is observed that the temperature remains constant until the entire amount of the liquid is converted into vapour. The heat supplied is now being utilised to change water from the liquid state to the vapour or gaseous state.
The change of state from liquid to vapour is called vapourisation, while that from vapour to liquid is called condensation.
Both the liquid and vapour states of the substance coexist in thermal equilibrium during the change of state from liquid to vapour.
The temperature at which the liquid and the vapour states of the substance coexist is called the boiling point of the liquid — here, of water, and hence the steam point. This is also the temperature at which water vapour condenses to form water.
- The boiling point of a substance at one standard atmospheric pressure is called its normal boiling point.
Boiling and Evaporation Are Not the Same
You must have seen that water spilled on the floor dries up after some time. Where does the water disappear? It is converted into water vapour and mixes with the air. We say that the water has evaporated. You also know that water can be converted into water vapour if you heat the water till its boiling point. What, then, is the difference between boiling and evaporation?
Both evaporation and boiling involve a change of state. But evaporation can occur at any temperature, whereas boiling takes place at a fixed temperature for a given pressure, unique for each liquid. Further, evaporation takes place only from the surface of the liquid, while boiling occurs throughout the whole liquid.
As you know, molecules in a liquid are moving about randomly. The average kinetic energy of the molecules decides the temperature of the liquid. However, all molecules do not move with the same speed. One with a higher kinetic energy may escape from the surface region by overcoming the interatomic forces. This process can take place at any temperature. This is evaporation.
If the temperature of the liquid is higher, the average kinetic energy is greater. Since the number of molecules is fixed, this implies that the number of fast-moving molecules is larger. Hence the rate of losing such molecules to the atmosphere will be larger. Thus, the higher the temperature of the liquid, the greater is the rate of evaporation.
Now notice a consequence. Since the faster molecules are the ones that are lost, the average kinetic energy of the liquid left behind is reduced, and hence the temperature of that liquid is lowered. Hence the phenomenon of evaporation gives a cooling effect to the remaining liquid.
Since evaporation takes place from the surface of a liquid, the rate of evaporation is greater if the area exposed is greater, and if the temperature of the liquid is higher.
You might have seen that if your mother wants her sari or her clothes to dry faster, she does not fold them. The more the area exposed, the faster is the drying, because the water evaporates faster. The presence of wind or a strong breeze, and the content of water vapour already in the atmosphere, are two other important factors determining the drying of clothes, but we do not refer to them here.
Before giving an injection to a patient, a spirit swab is normally used to disinfect the region. We feel a cooling effect on our skin, due to the evaporation of the spirit, as explained above.
Boiling Point Depends on Pressure
Diagram
Fig. 7.17 – A round bottom flask of water on a burner, with a thermometer and a steam outlet fitted through the cork, showing steam bubbles rising from the bottom.
7.8.1 Sublimation
Have you seen what happens when camphor is burnt? All substances do not pass through the three states solid–liquid–gas in turn. There are certain substances which normally pass from the solid to the vapour state directly, and vice versa.
The change from the solid state to the vapour state, without passing through the liquid state, is called sublimation, and the substance is said to sublime.
Dry ice (solid CO\(_2\)) and iodine sublime. During the sublimation process, both the solid and the vapour states of a substance coexist in thermal equilibrium. Most substances sublime at very low pressures.
7.8.2 Phase Diagram
A pressure–temperature (PT) diagram, often called a phase diagram, is particularly convenient for comparing the different phases of a substance. It answers, at a glance, the question: at this pressure and this temperature, which phase is the substance in?
A phase is a homogeneous composition of a material. A substance can exist in different phases even in the solid state. For example, you are familiar with two phases of carbon — graphite and diamond. Both are solids, but the regular geometric arrangement of the carbon atoms is different in the two cases.
Figure 7.18 shows the phase diagram of water and of CO\(_2\). Let us try to understand the diagram.
Diagram
Fig. 7.18 (a) – Phase diagram of water (not to scale): pressure P (atm) versus temperature T (°C), showing the solid, liquid and vapour regions separated by the fusion curve l–s, the vapourisation curve l–v and the sublimation curve s–v, meeting at the triple point, with the critical point marked at 218.0 atm and 374 °C.
Diagram
Fig. 7.18 (b) – Phase diagram of CO\(_2\) (not to scale): pressure P (atm) versus temperature T (°C), showing the same three curves and the triple point at 5.11 atm, with the critical point marked at 73.0 atm and 31.1 °C.
i) Vapourisation curve l – v: The curve labelled l – v represents those points where the liquid and the vapour phases are in equilibrium. Thus it is a graph of boiling point versus pressure. Note that the curves correctly show that at a pressure of 1 atmosphere the boiling point of water is 100 °C, and that the boiling point is lowered for a decreased pressure — which is the graphical form of the activity you performed above.
ii) Fusion curve l – s: The curve l – s represents the points where the solid and the liquid phases coexist in equilibrium. Thus it is a graph of the freezing point versus pressure. At one standard atmosphere pressure, the freezing point of water is 0 °C, as shown in Fig. 7.18 (a).
Also notice that at a pressure of one standard atmosphere, water is in the liquid phase if the temperature is between 0 °C and 100 °C, but is in the solid or the vapour phase if the temperature is below 0 °C or above 100 °C. Reading the diagram along a horizontal line at 1 atm therefore reproduces the whole of the graph in Fig. 7.16.
Note that the l – s curve for water slopes upward to the left; that is, the fusion curve of water has a slightly negative slope. This is true only of substances that expand upon freezing — and water, as you learnt in Section 7.5.3, is exactly such a substance. However, for most materials, like CO\(_2\), the l – s curve slopes upwards to the right, that is, the fusion curve has a positive slope. The melting point of CO\(_2\) is −56 °C at the higher pressure of 5.11 atm.
iii) Sublimation curve s – v: The curve labelled s – v is the sublimation point versus pressure curve. Water sublimates at a pressure less than 0.0060 atmosphere, while carbon dioxide, which in the solid state is called dry ice, sublimates even at atmospheric pressure, at a temperature as low as −78 °C.
iv) Triple point: The temperature and pressure at which the fusion curve, the vapourisation curve and the sublimation curve meet, and all the three phases of a substance coexist, is called the triple point of the substance. That is, the triple point of water is that point where water in the solid, liquid and gaseous states coexist in equilibrium, and this occurs only at a unique temperature and pressure — the property that made it so valuable as the single fixed point of the kelvin scale in Section 7.4.1.
The triple point of water is 273.16 K and \(6.11 \times 10^{-3}\) Pa, and that of CO\(_2\) is −56.6 °C and \(5.1 \times 10^{-5}\) Pa.
7.8.3 Gas and Vapour
The terms gas and vapour are sometimes used quite randomly, as though they were interchangeable. Therefore it is important to understand the difference between the two.
A gas cannot be liquefied by pressure alone, no matter how high the pressure is. In order to liquefy a gas, it must first be cooled to a certain temperature. This temperature is called the critical temperature.
The critical temperatures for some common gases and for water vapour are given in Table 7.5. Thus, nitrogen must be cooled below −147 °C in order to liquefy it by pressure.
Table 7.5: Critical Temperatures of some common gases and water vapour.
| Gas | Critical Temperature (°C) | Critical Temperature (K) |
|---|---|---|
| Air | −190 | 83 |
| N\(_2\) | −147 | 126 |
| O\(_2\) | −118 | 155 |
| CO\(_2\) | 31.1 | 241.9 |
| Water vapour | 374 | 647 |
Gas and vapour can thus be defined as follows.
Vapour can be liquefied simply by increasing the pressure, while a gas cannot. Vapour also exerts pressure, like a gas.
So the same substance may be called a gas or a vapour depending on its temperature. Water at 25 °C is far below its critical temperature of 374 °C, and what leaves a boiling kettle is therefore properly called water vapour, not water gas. Oxygen in the room, sitting far above its critical temperature of −118 °C, is a gas — and no amount of squeezing will liquefy it until it is cooled.
7.8.4 Latent Heat
Whenever there is a change in the state of a substance, heat is either absorbed or given out, but there is no change in the temperature of the substance. We have already seen this in the flat stretches AB and CD of Fig. 7.16. The heat that goes in without showing up on the thermometer is called latent heat — latent meaning hidden.
Thus, if a mass \(m\) of a substance undergoes a change from one state to the other, then the quantity of heat absorbed or released is given by
\[Q = mL \qquad \text{--- (7.33)}\]
where \(Q\) is the heat in joule (J), \(m\) is the mass in kilogram (kg), and \(L\) is known as the latent heat and is characteristic of the substance. Its SI unit is J kg\(^{-1}\). The value of \(L\) depends on the pressure, and is usually quoted at one standard atmospheric pressure.
Compare Eq. (7.33) with the heat equation, Eq. (7.28), \(Q = m\,\Delta T\,s\). The difference is instructive: the heat equation contains \(\Delta T\), and the latent heat equation does not. That single absence is the whole point — during a change of state, \(\Delta T\) is zero, so Eq. (7.28) would give zero heat, which is plainly wrong. Equation (7.33) is the equation that takes over there.
Depending on which change of state is taking place, latent heat carries two different names.
A plot of temperature versus heat energy, for a given quantity of water, is shown in Fig. 7.19. It carries the same information as Fig. 7.16, but with heat — rather than time — along the horizontal axis, which is what allows us to read energies directly off it.
Diagram
Fig. 7.19 – Temperature versus heat for water at one standard atmospheric pressure (not to scale): sloping line in the solid (ice) phase, a horizontal phase-change step of \(3.33 \times 10^{5}\) J/kg at the melting point, a sloping liquid (water) portion, a horizontal phase-change step of \(22.6 \times 10^{5}\) J/kg at the boiling point, and a sloping gas (steam) portion.
From Fig. 7.19 we see that when heat is added (or removed) during a change of state, the temperature remains constant. Also, the slopes of the phase lines are not all the same, which indicates that the specific heats of the various states are not equal — ice, water and steam each have their own value of \(s\).
For water, the latent heats of fusion and of vaporisation are
\[L_f = 3.33 \times 10^{5}\ \text{J kg}^{-1} \quad \text{and} \quad L_v = 22.6 \times 10^{5}\ \text{J kg}^{-1}\]
respectively. That is, \(3.33 \times 10^{5}\) J of heat is needed to melt 1 kg of ice at 0 °C, and \(22.6 \times 10^{5}\) J of heat is needed to convert 1 kg of water into steam at 100 °C.
Hence, steam at 100 °C carries \(22.6 \times 10^{5}\) J kg\(^{-1}\) more heat than water at 100 °C. This is why burns from steam are usually more serious than those from boiling water: the two are at the same temperature, but the steam has an enormous store of hidden energy that it dumps into the skin the moment it condenses there.
Melting points, boiling points and latent heats for various substances are given in Table 7.6.
Table 7.6: Temperature of change of state and latent heats for various substances at one standard atmosphere pressure.
| Substance | Melting point (°C) | \(L_f\) (\(\times 10^{5}\) J kg\(^{-1}\)) | Boiling point (°C) | \(L_v\) (\(\times 10^{5}\) J kg\(^{-1}\)) |
|---|---|---|---|---|
| Gold | 1063 | 0.645 | 2660 | 15.8 |
| Lead | 328 | 0.25 | 1744 | 8.67 |
| Water | 0 | 3.33 | 100 | 22.6 |
| Ethyl alcohol | −114 | 1.0 | 78 | 8.5 |
| Mercury | −39 | 0.12 | 357 | 2.7 |
| Nitrogen | −210 | 0.26 | −196 | 2.0 |
| Oxygen | −219 | 0.14 | −183 | 2.1 |
Read across any row of Table 7.6 and you will find \(L_v\) larger than \(L_f\) for every single substance listed.
7.9 Heat Transfer
Heat may be transferred from one point of a body to another in three different ways — by conduction, convection and radiation.
Heat transfers through solids by conduction. In this process, heat is passed on from one molecule to another molecule, but the molecules do not leave their mean positions. Liquids and gases are heated by convection. In this process, there is a bodily movement of the heated molecules themselves.
In order to transfer heat by conduction and by convection, a material medium is required. However, the transfer of heat by radiation does not need any medium. Radiation of heat energy takes place by electromagnetic (EM) waves, that travel with a speed of \(3 \times 10^{8}\) m s\(^{-1}\) in space or vacuum. The energy from the Sun comes to us by radiation.
It may be noted that conduction is a slow process of heat transfer, while convection is a rapid process. However, radiation is the fastest process, because in radiation the transfer of heat takes place at the speed of light.
7.9.1 Conduction
Conduction is the process by which heat flows from the hot end to the cold end of a solid body, without any net bodily movement of the particles of the body.
Heat passes through solids by conduction only. When one end of a metal rod is placed in a flame while the other end is held in the hand, the end held in the hand slowly gets hotter, although it itself is not in direct contact with the flame. We say that heat has been conducted from the hot end to the cold end.
What is happening at the molecular level? When one end of the rod is heated, the molecules there vibrate faster. As they collide with their slow-moving neighbours, they transfer some of their energy by collision to these molecules, which in turn transfer energy to their neighbouring molecules still farther down the length of the rod. Thus the energy of thermal motion is transferred by molecular collisions down the rod — the energy travels, but the molecules stay where they are.
The transfer of heat continues till the two ends of the rod are at the same temperature, in principle, but this will take an infinite time. Normally the various sections of the rod will attain a temperature which remains constant, but which is not the same throughout the length of the rod. This method of heat transfer is called conduction.
Those solid substances which conduct heat easily are called good conductors of heat — for example, silver, copper, aluminium and brass. All metals are good conductors of heat. Those substances which do not conduct heat easily are called bad conductors of heat — for example, wood, cloth, air and paper.
In general, good conductors of heat are also good conductors of electricity, and similarly bad conductors of heat are bad conductors of electricity also. In metals both heat and electric current are carried largely by the same free electrons, which is why the two lists agree so well.
7.9.1.1 Thermal Conductivity
The thermal conductivity of a solid is a measure of the ability of the solid to conduct heat through it. Thus good conductors of heat have a higher thermal conductivity than bad conductors.
Suppose that one end of a metal rod is heated, as in Fig. 7.20 (a). The heat flows by conduction from the hot end to the cold end. As a result, the temperature of every section of the rod starts increasing. Under this condition, the rod is said to be in a variable temperature state.
After some time, the temperature at each section of the rod becomes steady, that is, it does not change any further. Note carefully what this does and does not mean: the temperature of each cross-section of the rod is now constant in time, but it is not the same from one cross-section to the next. This is called the steady state condition.
Under steady state conditions, the temperature at points within the rod decreases uniformly with distance from the hot end to the cold end. The fall of temperature with distance between the ends of the rod, in the direction of flow of heat, is called the temperature gradient.
\[\therefore\ \text{Temperature gradient} = \frac{T_1 - T_2}{x}\]
where \(T_1\) = temperature of the hot end (°C or K), \(T_2\) = temperature of the cold end (°C or K), \(x\) = length of the rod (m).
7.9.1.2 Coefficient of Thermal Conductivity
Consider a cube of side \(x\), each face of cross-sectional area \(A\). Suppose its opposite faces are maintained at temperatures \(T_1\) and \(T_2\) (with \(T_1 > T_2\)), as shown in Fig. 7.20 (b). Experiments show that, under steady state conditions, the quantity of heat ‘\(Q\)’ that flows from the hot face to the cold face is
directly proportional to the cross-sectional area \(A\) of the face, i.e., \(Q \propto A\) — a wider path lets more heat through;
directly proportional to the temperature difference between the two faces, i.e., \(Q \propto (T_1 - T_2)\) — a bigger difference drives the flow harder;
directly proportional to the time \(t\) (in seconds) for which heat flows, i.e., \(Q \propto t\);
inversely proportional to the perpendicular distance \(x\) between the hot and the cold faces, i.e., \(Q \propto 1/x\) — a longer path opposes the flow.
Diagram
Fig. 7.20 (a) – Section of a metal bar in the steady state, of length \(x\), with the hot end at \(T_1\) and the cold end at \(T_2\), and arrows showing the direction of heat flow.
Diagram
Fig. 7.20 (b) – Section of a cube of side \(x\) in the steady state, with opposite faces of area \(A\) maintained at \(T_1\) and \(T_2\), and heat \(Q\) flowing across it.
Combining the above four factors, we have the quantity of heat
\[Q \propto \frac{A\,(T_1 - T_2)\,t}{x}\]
\[\therefore\ Q = \frac{k\,A\,(T_1 - T_2)\,t}{x} \qquad \text{--- (7.34)}\]
where \(k\) is a constant of proportionality and is called the coefficient of thermal conductivity. Its value depends upon the nature of the material. Here \(Q\) is in joule (J), \(A\) in m\(^2\), \(t\) in second (s) and \(x\) in metre (m).
If \(A = 1\) m\(^2\), \(T_1 - T_2 = 1\) °C (or 1 K), \(t = 1\) s and \(x = 1\) m, then from Eq. (7.34), \(Q = k\). This special case gives us the meaning of \(k\) in words.
From Eq. (7.34), we have
\[k = \frac{Q\,x}{A\,(T_1 - T_2)\,t} \qquad \text{--- (7.35)}\]
The SI unit of the coefficient of thermal conductivity \(k\) is J s\(^{-1}\) m\(^{-1}\) °C\(^{-1}\), or J s\(^{-1}\) m\(^{-1}\) K\(^{-1}\), and its dimensions are [L\(^{1}\) M\(^{1}\) T\(^{-3}\) K\(^{-1}\)].
From Eq. (7.34), we also have
\[\frac{Q}{t} = \frac{k\,A\,(T_1 - T_2)}{x} \qquad \text{--- (7.36)}\]
The quantity \(Q/t\), denoted by \(P_{\text{cond}}\), is the time rate of heat flow — that is, the heat flow per second — from the hotter face to the colder face, at right angles to the faces. Its SI unit is the watt (W). The SI unit of \(k\) can therefore also be written as W m\(^{-1}\) °C\(^{-1}\) or W m\(^{-1}\) K\(^{-1}\).
Using calculus, Eq. (7.36) may be written as
\[\frac{dQ}{dt} = -\,k\,A\,\frac{dT}{dx}\]
where \(\dfrac{dT}{dx}\) is the temperature gradient. The negative sign indicates that the heat flow is in the direction of decreasing temperature — heat runs downhill in temperature, never uphill on its own.
If \(A = 1\) m\(^2\) and \(\dfrac{dT}{dx} = 1\), then \(\dfrac{dQ}{dt} = k\) (numerically).
Hence the coefficient of thermal conductivity of a material may also be defined in a second, equivalent way.
The coefficients of thermal conductivity of some materials are given in Table 7.7.
7.9.1.3 Thermal Resistance
The conduction rate \(P_{\text{cond}}\) is the amount of energy transferred per unit time through a slab of area \(A\) and thickness \(x\), the two sides of the slab being at temperatures \(T_1\) and \(T_2\) (with \(T_1 > T_2\)), and is given by Eq. (7.36):
\[P_{\text{cond}} = \frac{Q}{t} = k\,A\,\frac{T_1 - T_2}{x} \qquad \text{--- (7.37)}\]
As discussed earlier, \(k\) depends on the material of the slab. A material that readily transfers heat energy by conduction is a good thermal conductor, and has a high value of \(k\).
Table 7.7: Coefficient of thermal conductivity (\(k\)).
| Substance | Coefficient of thermal conductivity (J s\(^{-1}\) m\(^{-1}\) K\(^{-1}\)) |
|---|---|
| Silver | 406 |
| Copper | 385 |
| Aluminium | 205 |
| Steel | 50.2 |
| Insulating brick | 0.15 |
| Glass | 0.8 |
| Brick and concrete | 0.8 |
| Water | 0.8 |
| Wood | 0.04–0.12 |
| Air at 0 °C | 0.024 |
The spread in this table is enormous — silver conducts heat nearly twenty thousand times better than still air. It is exactly this spread that makes both cooking pans and insulating jackets possible.
In western countries, where the temperature falls below 0 ºC in the winter season, insulating a house from its surroundings is very important. In our country, if we wish to carry cold drinks with us for a picnic, or to bring ice-cream from the shop to our house, we need to keep them in containers — made, say, of thermocol — that are poor thermal conductors.
Hence the concept of thermal resistance \(R_T\), similar to electrical resistance, is introduced. The opposition of a body to the flow of heat through it is called thermal resistance.
The greater the thermal conductivity of a material, the smaller is its thermal resistance, and vice versa. Thus bad thermal conductors are those which have a high thermal resistance.
From Eq. (7.37),
\[\frac{(T_1 - T_2)}{P_{\text{cond}}} = \frac{x}{k\,A} \qquad \text{--- (7.38)}\]
We know that when a current flows through a conductor, the ratio \(V/I\) is called the electrical resistance of the conductor, where \(V\) is the electrical potential difference between the ends of the conductor and \(I\) is the current, that is, the rate of flow of charge.
Look at Eq. (7.38) with that comparison in mind. In it, \((T_1 - T_2)\) is the temperature difference between the ends of the conductor — the quantity that drives the flow, just as \(V\) does — and \(P_{\text{cond}}\) is the rate of flow of heat, which plays the role of the current \(I\). Therefore, in analogy with electrical resistance, \((T_1 - T_2)/P_{\text{cond}}\) is called the thermal resistance \(R_T\) of the material, i.e.,
\[\text{Thermal resistance}\ R_T = \frac{x}{k\,A}\]
The SI unit of thermal resistance is °C s/kcal, or °C s/J, and its dimensional formula is [M\(^{-1}\) L\(^{-2}\) T\(^{3}\) K\(^{1}\)].
The lower the thermal conductivity \(k\), the higher is the thermal resistance \(R_T\). A material with a high \(R_T\) value is a poor thermal conductor and is a good thermal insulator.
Two related terms must be kept apart. Thermal resistivity \(\rho_T\) is the reciprocal of the thermal conductivity \(k\), and is a characteristic of the material alone. Thermal resistance \(R_T\) is that of the slab (or of the rod), and it depends both on the material and on the thickness of the slab (or the length of the rod). Doubling the thickness of a wall doubles its thermal resistance, but does not change its thermal resistivity at all.
7.9.1.4 Applications of Thermal Conductivity
- Cooking utensils are made of metals, but are provided with handles of bad conductors.
Since metals are good conductors of heat, heat can be easily conducted through the base of the utensil into the food. The handles of the utensils are made of bad conductors of heat — for example, wood or ebonite — so that they cannot conduct heat from the utensil to our hands.
- Thick walls are used in the construction of cold storage rooms.
Brick is a bad conductor of heat, so that it reduces the flow of heat from the surroundings into the rooms. Still better heat insulation is obtained by using hollow bricks. Air, being a poorer conductor than brick — as Table 7.7 confirms — further avoids the conduction of heat from outside.
- To prevent ice from melting, it is wrapped in a gunny bag.
A gunny bag is a poor conductor of heat and reduces the heat flow from outside to the ice. Moreover, the air filled in the interspaces of a gunny bag, being a very bad conductor of heat, further avoids the conduction of heat from outside.
When Low Conductivity Becomes a Problem
Low thermal conductivity can also be a disadvantage. When hot water is poured into a glass beaker, the inner surface of the glass expands on heating. Since glass is a bad conductor of heat, the heat from inside does not reach the outside surface so quickly. Hence the outer surface does not expand, and the resulting stress between the expanded inner surface and the unexpanded outer surface causes a crack in the glass.
7.9.2 Convection
We have seen that heat is transmitted through solids by conduction, wherein energy is transferred from one molecule to another, but the molecules themselves — though vibrating with a larger amplitude — do not leave their mean positions. But in convection, heat is transmitted from one point to another by the actual bodily movement of the heated (energised) molecules within the fluid.
In liquids and gases, heat is transmitted by convection because their molecules are quite free to move about. The mechanism of heat transfer by convection in liquids and gases is described below.
Consider water being heated in a vessel from below. The water at the bottom of the vessel is heated first, and consequently its density decreases — that is, the water molecules at the bottom are separated farther apart. This is thermal expansion, from Section 7.5, doing its work: hotter water is less dense than colder water.
These hot molecules have high kinetic energy and rise upward to the cold region, while the molecules from the cold region come down to take their place. Thus each molecule at the bottom gets heated and rises, then cools and descends. This action sets up a flow of water molecules called convection currents. The convection currents transfer heat to the entire mass of the water. Note that here the transfer of heat is by the bodily, physical movement of the water molecules themselves.
7.9.2.1 Applications of Convection
- Heating and cooling of rooms
The mechanism of heating a room by a heat convector or heater is entirely based on convection. The air molecules in immediate contact with the heater are heated up. These air molecules acquire sufficient energy and rise upward. The cool air at the top, being denser, moves down to take their place. This cool air in turn gets heated and moves upward.
In this way, convection currents are set up in the room, which transfer heat to different parts of the room. The same principle, but working in the opposite direction, is used to cool a room by an air-conditioner — which is why an air-conditioner is mounted high on the wall, so that the cool, dense air it produces can sink and set up the circulation.
- Cooling of transformers
Due to the current flowing in the windings of a transformer, an enormous amount of heat is produced. Therefore, a transformer is always kept in a tank containing oil.
The oil in contact with the transformer body heats up, creating convection currents. The warm oil comes in contact with the cooler tank, gives heat to it, and descends to the bottom. It again warms up and rises upward. This process is repeated again and again. The heat of the transformer body is thus carried away by convection to the cooler tank. The cooler tank, in turn, loses its heat by convection to the surrounding air.
7.9.2.2 Free and Forced Convection
When a hot body is in contact with air under ordinary conditions — like the air around a firewood fire — the air removes heat from the body by a process called free or natural convection. Here nothing drives the circulation except the difference in density created by the heating itself. Land and sea breezes are also formed as a result of free convection currents in air.
The convection process can be accelerated by employing a fan to create a rapid circulation of fresh air. This is called forced convection. The examples in Section 7.9.2.1 are of forced convection — namely, the heat convector, the air conditioner, and heat radiators in an IC engine.
7.9.3 Radiation
For the transfer of heat by radiation, molecules are not needed; that is, a medium is not required. The fact that the Earth receives large quantities of heat from the Sun shows that heat can pass through empty space — that is, through the vacuum between the Sun and the atmosphere that surrounds the Earth. In fact, the transfer of heat by radiation has the same properties as light (or an EM wave): it travels in straight lines, it can be reflected and absorbed, and it moves at \(3 \times 10^{8}\) m s\(^{-1}\).
A natural question arises: how does heat transfer occur in the absence of a medium, that is, in the absence of molecules to pass the energy along?
All objects possess thermal energy due to their temperature \(T\) (with \(T > 0\) K). The rapidly moving molecules of a hot body emit EM waves travelling with the velocity of light. These are called thermal radiations.
These waves carry energy with them, and transfer it to the low-speed molecules of a cold body on which they fall. This results in an increase in the molecular motion of the cold body, and its temperature rises.
Thus the transfer of heat by radiation is a two-fold process: the conversion of thermal energy into waves at the hot body, and the reconversion of those waves into thermal energy at the body on which they fall. Notice that nothing needs to happen in the space in between — which is exactly why the medium is not heated. We will learn about EM waves in Chapter 13.
7.10 Newton’s Laws of Cooling
If hot water in a vessel is kept on a table, it begins to cool gradually. From Section 7.9 we know why: the hot water loses heat to the cooler surroundings, chiefly in the form of heat radiation. But how fast does it cool, and on what does that speed depend? To study how a given body cools while exchanging heat with its surroundings, the following experiment is performed.
The Cooling Experiment
A calorimeter is filled up to two-thirds of its capacity with boiling water and is covered. A thermometer is fixed through a hole in the lid, and its position is adjusted so that the bulb of the thermometer is fully immersed in the water.
The calorimeter vessel is kept in a constant temperature enclosure, or simply in open air, since the room temperature will not change much during the experiment. This matters, because the whole point of the experiment is to compare the temperature of the water with a fixed surrounding temperature.
The temperature on the thermometer is noted at one minute intervals, until the temperature of the water decreases by about 25 °C. A graph of temperature \(T\) (along the y-axis) is then plotted against time \(t\) (along the x-axis). This graph is called the cooling curve, shown in Fig. 7.21 (a).
Diagram
Fig. 7.21 (a) – Temperature versus time graph (the cooling curve): a falling curve that is steep at first and flattens later, with a tangent drawn at a point A and the small triangle \(\Delta T\), \(\Delta t\) marked on it.
From this graph you can infer how the cooling of hot water depends on the difference of its temperature from that of its surroundings. You will notice that initially the rate of cooling is higher, and that it decreases as the temperature of the water falls. This is exactly why a very hot cup of tea loses its first ten degrees far more quickly than its last ten.
A tangent is drawn to the curve at suitable points on the curve. The slope of each tangent,
\[\lim_{\Delta t \to 0} \frac{\Delta T}{\Delta t} = \frac{dT}{dt}\]
gives the rate of fall of temperature at that temperature. In Fig. 7.21 (a), the tangent drawn at the point A indicates the rate of fall of temperature at A.
Taking (0, 0) as the origin, if a graph of \(dT/dt\) is plotted against the corresponding temperature difference \((T - T_0)\), the curve is a straight line, as shown in Fig. 7.21 (b). A straight line through the origin means direct proportionality — and that proportionality is the whole content of the law we are about to state.
Diagram
Fig. 7.21 (b) – Rate of change of temperature \(dT/dt\) (along the y-axis) plotted against the temperature difference \((T - T_0)\), giving a straight line through the origin.
Newton’s Law of Cooling
The above activity shows that a hot body loses heat to its surroundings in the form of heat radiation, and that the rate of loss of heat depends on the difference in the temperature of the body and its surroundings. Newton was the first to study, in a systematic manner, the relation between the heat lost by a body in a given enclosure and its temperature.
According to Newton’s law of cooling, the rate of loss of heat \(dT/dt\) of the body is directly proportional to the difference of temperature \((T - T_0)\) of the body and the surroundings, provided the difference in temperatures is small. Mathematically, this may be expressed as
\[\frac{dT}{dt} \propto (T - T_0)\]
\[\therefore\ \frac{dT}{dt} = \text{C}\,(T - T_0) \qquad \text{--- (7.39)}\]
where C is the constant of proportionality, \(T\) is the temperature of the body and \(T_0\) is the temperature of the surroundings.
The condition “provided the difference in temperatures is small” is part of the law, and it must never be dropped. If the body is very much hotter than its surroundings, the loss of heat by radiation no longer varies simply as the first power of the temperature difference, and Eq. (7.39) ceases to hold.
7.11 Summary
Temperature and Heat
- Heat is energy in transit. It flows between two systems, or between a system and its surroundings, only because of a difference in their temperatures.
- Two bodies are in thermal equilibrium when there is no net transfer of heat between them; this happens only when their temperatures are equal.
- Temperature measures the degree of hotness of a body. Microscopically, the average kinetic energy per particle of a substance defines its temperature. Temperature is not the amount of thermal energy in a body.
- Temperature is the fourth fundamental quantity, needed in addition to length, mass and time.
- SI unit of heat: joule (J). SI unit of temperature: kelvin (K); also used, degree celsius (°C).
- \(1\ \text{J} = 10^{7}\ \text{erg}\) and \(1\ \text{cal} = 4.184\ \text{J}\).
- Dimensions: heat \([\text{L}^{2}\text{M}^{1}\text{T}^{-2}\text{K}^{0}]\); temperature \([\text{L}^{0}\text{M}^{0}\text{T}^{0}\text{K}^{1}]\).
- Interatomic spacing: solids \(\sim 10^{-10}\) m; liquids about twice that of solids; gases at NTP \(\sim 10^{-9}\) m.
Measurement of Temperature
- An adiabatic wall allows no exchange of heat between the systems it separates; a diathermic wall freely allows it.
- Zeroth law of thermodynamics: if systems A and B are each separately in thermal equilibrium with a system C, then A and B are in thermal equilibrium with each other. This law makes thermometry possible.
- A thermometric property is a physical property that varies continuously and monotonically with temperature (length of a rod, volume of a liquid, resistance of a wire, pressure of a gas at constant volume).
- Fixed points: the ice point (pure water freezes, at one standard atmospheric pressure) and the steam point (pure water boils, at one standard atmospheric pressure).
- Celsius scale: ice point 0, steam point 100, interval divided into 100 parts. Fahrenheit scale: ice point 32, steam point 212, interval divided into 180 parts.
- Conversion between the scales: \[\frac{T_F - 32}{180} = \frac{T_C}{100} \qquad (7.1)\]
- If a thermometric property has values \(P_1\) at the ice point and \(P_2\) at the steam point, and \(P_T\) at an unknown temperature \(T\), then \[T = \frac{100\,(P_T - P_1)}{(P_2 - P_1)} \qquad (7.2)\]
- A good thermometer has high sensitivity, accuracy and reproducibility, and reaches thermal equilibrium quickly.
- Types: constant volume gas thermometer, liquid-in-glass thermometer (mercury: freezes at −39 °C, boils at 357 °C; alcohol: freezes at −115 °C, boils at 78 °C), resistance thermometer, thermocouple, thermistor.
Absolute Temperature and the Ideal Gas Equation
- Extrapolating the pressure–temperature (or volume–temperature) graphs of all gases gives the same intercept, −273.15 ºC, called the absolute zero of temperature. No lower temperature is attainable.
- The triple point of a substance is the unique combination of temperature and pressure at which its solid, liquid and vapour phases coexist in equilibrium.
- The kelvin (absolute) scale has its zero at −273.15 °C, with intervals equal to those of the celsius scale. The triple point of water is assigned 273.16 K, at a pressure of \(6.11 \times 10^{2}\) Pa or \(6.11 \times 10^{-3}\) atmosphere. One kelvin is \(1/273.16\) of the difference between absolute zero and the triple point of water. On the celsius scale, the triple point of water is 0.01 ºC.
- Relation between the three scales: \[\frac{T_C}{100} = \frac{T_F - 32}{180} = \frac{T_K - 273.15}{100} \qquad (7.3)\]
- Charles’ law: \(\dfrac{V}{T} = \text{constant}\), for a fixed mass of gas at constant pressure. — (7.4)
- Pressure (Gay Lussac’s) law: \(\dfrac{P}{T} = \text{constant}\), for a fixed mass of gas at constant volume. — (7.5)
- Boyle’s law: \(PV = \text{constant}\), for a fixed mass of gas at constant temperature. — (7.6)
- Combining these: \(\dfrac{PV}{T} = \text{constant}\) — (7.7); for one mole, \(PV = RT\) — (7.8); for \(n\) moles, \[PV = nRT \qquad (7.9)\]
- \(R\) is the universal gas constant, the same for all gases, with the value 8.31 J K\(^{-1}\) mol\(^{-1}\). Temperature in all gas-law equations must be in kelvin.
Thermal Expansion
- Thermal expansion is the increase in the dimensions of a body due to a rise in its temperature. It is of three types: linear, areal and volume expansion.
- Linear expansion: \(\dfrac{\Delta l}{l} = \alpha\,\Delta T\) — (7.10), with \[\alpha = \frac{l_2 - l_1}{l_1\,(T_2 - T_1)} \qquad (7.12)\]
- Areal expansion: \(\dfrac{\Delta A}{A} = \beta\,\Delta T\) — (7.13), with \[\beta = \frac{A_2 - A_1}{A_1\,(T_2 - T_1)} \qquad (7.15)\]
- Volume expansion: \(\dfrac{\Delta V}{V} = \gamma\,\Delta T\) — (7.16), with \[\gamma_1 = \frac{V_2 - V_1}{V_1\,(T_2 - T_1)} \qquad (7.18)\]
- \(\alpha\) is the increase in length per unit original length at 0 °C per degree rise; \(\beta\) and \(\gamma\) are defined similarly for area and volume. All are measured per °C or per K.
- Neglecting higher powers of \(\alpha\): \(\beta = 2\alpha\) — (7.21) and \(\gamma = 3\alpha\) — (7.24), so \[\alpha = \frac{\beta}{2} = \frac{\gamma}{3} \qquad (7.25)\] that is, \(\alpha : \beta : \gamma = 1 : 2 : 3\).
- Liquids expand more than solids, and gases more than liquids. For fluids, the expansion of the container must also be taken into account; the overflow is the difference of the two expansions.
- Since mass is constant, expansion on heating lowers the density. Water is anomalous: it expands on cooling from 4 °C to 0 °C, so its density decreases in this range.
Specific Heat Capacity and Calorimetry
- Specific heat capacity is the heat per unit mass required to change the temperature of a substance by one degree: \[s = \frac{\Delta Q}{m\,\Delta T} \qquad (7.26)\] Its SI unit is J kg\(^{-1}\) K\(^{-1}\). Water has the highest specific heat capacity, 4186.0 J kg\(^{-1}\) K\(^{-1}\), and is therefore used as a coolant.
- Molar specific heat: \(C = \dfrac{1}{\mu}\dfrac{\Delta Q}{\Delta T}\) — (7.27), in J mol\(^{-1}\) K\(^{-1}\).
- For a gas, two specific heats are defined, since heating at constant pressure requires additional heat for the work done during expansion. Hence \(S_p > S_v\), and correspondingly \(C_p > C_v\). Also \(C_p = M \times S_p\) and \(C_v = M \times S_v\), where \(M\) is the molecular weight.
- Heat equation: \[Q = m \times \Delta T \times s \qquad (7.28)\]
- Heat capacity (thermal capacity) of a body is the heat needed to change the temperature of the whole body by one degree: \[\text{Heat capacity} = Q = m \times s \qquad (7.29)\] measured in J/°C.
- Principle of calorimetry: in an isolated system, heat lost = heat gained. For the method of mixtures, \[m_1 s_1 (T_1 - T) = m_2 s_2 (T - T_2) + m_3 s_3 (T - T_2) \qquad (7.30)\] from which \[s_1 = \frac{(m_2 s_2 + m_3 s_3)(T - T_2)}{m_1 (T_1 - T)} \qquad (7.31)\]
- The solid sample must be denser than the liquid, so that it does not float.
Change of State
- Melting is solid \(\to\) liquid; solidification is liquid \(\to\) solid; vapourisation is liquid \(\to\) vapour; condensation is vapour \(\to\) liquid. During each change of state, the two states coexist in thermal equilibrium and the temperature stays constant.
- The melting point and the boiling point are characteristic of the substance and depend on pressure; measured at one standard atmospheric pressure they are called the normal melting point and the normal boiling point. For water these are 0 °C and 100 °C.
- Boiling point increases with an increase in pressure and decreases with a decrease in pressure.
- Evaporation occurs from the surface, at any temperature; boiling occurs throughout the liquid, at a fixed temperature for a given pressure. Evaporation cools the remaining liquid, and its rate increases with temperature and with the exposed surface area.
- Sublimation is the direct change from solid to vapour, without passing through the liquid state (e.g., dry ice, iodine).
- A phase diagram (P–T diagram) shows the vapourisation curve l–v, the fusion curve l–s, and the sublimation curve s–v, meeting at the triple point. The fusion curve of water has a slightly negative slope, which is true only of substances that expand on freezing.
- Critical temperature is the temperature above which a substance cannot be liquefied by pressure alone. A substance in the gaseous phase above its critical temperature is a gas; below its critical temperature it is a vapour. A vapour can be liquefied by pressure alone; a gas cannot.
- Latent heat is the heat required to change the state of unit mass of a substance without changing its temperature: \[Q = mL \qquad (7.33)\] with \(L\) in J kg\(^{-1}\). Latent heat of fusion \(L_f\) applies at the melting point; latent heat of vapourization \(L_v\) applies at the boiling point.
- For water, \(L_f = 3.33 \times 10^{5}\) J kg\(^{-1}\) and \(L_v = 22.6 \times 10^{5}\) J kg\(^{-1}\). Steam at 100 °C therefore carries \(22.6 \times 10^{5}\) J kg\(^{-1}\) more heat than water at 100 °C, which is why steam burns are more serious.
- \(L_v\) is always larger than \(L_f\), because separating the molecules completely takes more energy than merely loosening them, and because work must be done against the atmosphere during expansion.
Heat Transfer
- Heat is transferred by conduction (through solids, without bodily movement of particles), by convection (in fluids, by bodily movement of heated particles) and by radiation (by EM waves, needing no medium, travelling at \(3 \times 10^{8}\) m s\(^{-1}\)). Conduction is slow, convection rapid, radiation the fastest.
- In the steady state, the temperature of each cross-section of a rod is constant in time but different from section to section. Temperature gradient \(= \dfrac{T_1 - T_2}{x}\).
- Conduction: \[Q = \frac{k\,A\,(T_1 - T_2)\,t}{x} \qquad (7.34)\] \[k = \frac{Q\,x}{A\,(T_1 - T_2)\,t} \qquad (7.35)\] \[P_{\text{cond}} = \frac{Q}{t} = \frac{k\,A\,(T_1 - T_2)}{x} \qquad (7.36),\ (7.37)\] In calculus form, \(\dfrac{dQ}{dt} = -\,k\,A\,\dfrac{dT}{dx}\), the negative sign showing that heat flows towards decreasing temperature.
- \(k\) is the coefficient of thermal conductivity: SI unit J s\(^{-1}\) m\(^{-1}\) K\(^{-1}\) (or W m\(^{-1}\) K\(^{-1}\)); dimensions [L\(^{1}\) M\(^{1}\) T\(^{-3}\) K\(^{-1}\)].
- Thermal resistance: \[R_T = \frac{(T_1 - T_2)}{P_{\text{cond}}} = \frac{x}{k\,A} \qquad (7.38)\] SI unit °C s/J (or °C s/kcal); dimensions [M\(^{-1}\) L\(^{-2}\) T\(^{3}\) K\(^{1}\)]. Low \(k\) means high \(R_T\), i.e., a good thermal insulator. Thermal resistivity \(\rho_T = 1/k\) belongs to the material; \(R_T\) belongs to the particular slab or rod.
- Convection may be free (natural), as in land and sea breezes, or forced, as with a fan, an air conditioner or a heat convector.
- Radiation is a two-fold process: thermal energy is converted into EM waves at the hot body, and reconverted into thermal energy at the body on which they fall. The intervening medium is not heated.
Newton’s Law of Cooling
- Newton’s law of cooling: the rate of loss of heat of a body is directly proportional to the difference in temperature between the body and its surroundings, provided this difference is small. \[\frac{dT}{dt} \propto (T - T_0) \qquad \therefore\ \frac{dT}{dt} = \text{C}\,(T - T_0) \qquad (7.39)\] where C is the constant of proportionality and \(T_0\) is the temperature of the surroundings.
- The cooling curve of temperature versus time is steep at first and flattens as the body approaches the temperature of its surroundings; a graph of \(dT/dt\) against \((T - T_0)\) is a straight line through the origin.
7.12 Check Your Concepts
A cup of tea at 90 °C and a bucket of water at 40 °C stand in the same room. Which is at the higher temperature, and which holds more thermal energy? Explain why the two answers are not the same.
Why can our sense of touch compare hotness only qualitatively? Use the idea of heat flow between the skin and the object to explain how lukewarm water can feel both hot and cold.
Why does the temperature of a solid stop rising while it is melting, even though heat is being supplied continuously? Where is the supplied energy going?
Why is the zeroth law of thermodynamics essential for a thermometer to work at all? What would a thermometer reading mean if the law were not true?
A doctor asks the patient to keep the clinical thermometer in the mouth for some time before it is removed. Justify this instruction using the idea of thermal equilibrium.
Why was the two fixed point scale replaced by a one fixed point scale in 1954? What makes the triple point of water a more dependable fixed point than the steam point?
Different gases give straight lines of different slopes on the pressure–temperature graph, yet all the lines meet the temperature axis at the same point. What conclusion does this force us to draw about that point?
Why must the temperature be expressed in kelvin, and not in degree celsius, when Charles’ law or the pressure law is applied? Which one of the three gas laws is free of this restriction, and why?
A metal plate has a circular hole cut in it. State, with reasoning, whether the diameter of the hole increases or decreases when the plate is heated.
A bimetallic strip made of brass and iron bends when it is heated. Towards which metal does it curve, and why? Name one appliance in which this behaviour is put to use.
Explain what physical fact the numbers 2 and 3 stand for in the relation \(\alpha : \beta : \gamma = 1 : 2 : 3\). Why is it legitimate to neglect \(\alpha^2 T^2\) while deriving \(\beta = 2\alpha\)?
When a liquid in an open glass vessel is heated, the volume that overflows is less than the calculated expansion of the liquid alone. Explain why.
Water is used as the coolant in an automobile radiator, and also as the filling of a hot water bag. Explain how a single property of water accounts for both of these very different uses.
Distinguish between specific heat capacity and heat capacity. State the SI unit of each, and say which of the two depends on the size of the body.
Why is \(S_p\) greater than \(S_v\) for a gas? Where does the additional heat supplied at constant pressure go, if not into raising the temperature?
In the method of mixtures, explain the reason for each of the following: (i) the calorimeter is well insulated, (ii) the contents are stirred constantly, (iii) the solid sample must be denser than the liquid.
Distinguish between evaporation and boiling with respect to the temperature at which each occurs, and the region of the liquid from which each occurs.
Explain why evaporation cools the liquid that is left behind, even when the surroundings supply no heat.
Why does water boil below 100 °C on a high mountain? Answer using the vapourisation curve of the phase diagram of water.
The fusion curve of water slopes upward to the left, while that of CO\(_2\) slopes upward to the right. Which property of water is responsible for this difference?
Using the phase diagram of CO\(_2\), explain why dry ice sublimes at atmospheric pressure instead of melting into a liquid.
A substance is in the gaseous phase. What further information do you need before deciding whether it should be called a gas or a vapour? Explain using the idea of critical temperature.
Steam at 100 °C and boiling water at 100 °C are at the same temperature, yet a steam burn is far more serious. Explain, with the relevant quantity named.
Why is the latent heat of vapourization of a substance always greater than its latent heat of fusion? Give both reasons.
Distinguish clearly between the variable temperature state and the steady state of a conducting rod, and explain why the conduction formulae apply only to the steady state.
Explain the analogy between thermal resistance and electrical resistance. Which thermal quantity plays the role of the potential difference, and which plays the role of the current?
Distinguish between thermal resistivity and thermal resistance. If a wall is made twice as thick, which of the two changes, and how?
Why is a vessel of water always heated from below and not from above? What would happen to the convection currents if it were heated from the top?
Explain why the direction of the wind near a sea shore reverses between day and night. Name the property of water and the mode of heat transfer involved.
Distinguish between free convection and forced convection, giving one example of each.
Heat reaches the Earth from the Sun without heating the space in between. Explain how this is possible, describing the two-fold nature of the process of radiation.
A vacuum flask has double walls with the air pumped out, and the facing surfaces silvered. Explain which mode of heat transfer each of these two features is designed to block.
Why does the cooling curve of hot water become flatter as time passes? What does Newton’s law of cooling predict about the rate of cooling as the body approaches room temperature?
State the condition under which Newton’s law of cooling is valid. What happens to the law if the body is very much hotter than its surroundings?
Clearly state the difference between heat and temperature.
How is a thermometer calibrated?
What are the different scales of temperature? What is the relation between them?
What is absolute zero?
Derive the relation between the three coefficients of thermal expansion.
State applications of thermal expansion.
Why do we generally consider two specific heats for a gas?
Are the freezing point and the melting point the same with respect to change of state? Comment.
Define (i) Sublimation (ii) Triple point.
Define coefficient of thermal conductivity. Derive its expression.
Give any four applications of thermal conductivity in everyday life.
Explain the term thermal resistance. State its SI unit and dimensions.
State Newton’s law of cooling and explain how it can be experimentally verified.
What is thermal stress? Give an example of the disadvantages of thermal stress in practical use.
Which materials can be used as thermal insulators, and why?
7.13 Practice with Numericals
The temperature of a liquid is 45 °C. Express this temperature in degree fahrenheit and in kelvin.
The temperature of a body rises from 30 °C to 78 °C. Express the initial temperature, the final temperature, and the rise in temperature, all in degree fahrenheit.
At what temperature is the reading on the celsius scale numerically equal to the reading on the fahrenheit scale?
In a mercury-in-glass thermometer, the length of the mercury column is 30 mm at the ice point and 210 mm at the steam point. Find the temperature when the length of the column is 111 mm.
A resistance thermometer has a resistance of 80.0 \(\Omega\) at the ice point and 120.0 \(\Omega\) at the steam point. Find the temperature at which its resistance is 98.0 \(\Omega\).
A constant volume gas thermometer registers a pressure of \(1.20 \times 10^{4}\) Pa at the ice point and \(1.64 \times 10^{4}\) Pa at the steam point. What pressure will it register at 60 °C?
0.40 mol of an ideal gas is held at a temperature of 27 °C and a pressure of \(1.0 \times 10^{5}\) Pa. Calculate the volume occupied by the gas. (Take \(R = 8.31\) J K\(^{-1}\) mol\(^{-1}\).)
A fixed mass of gas at 27 °C and \(1.0 \times 10^{5}\) Pa is heated at constant volume until its pressure becomes \(1.5 \times 10^{5}\) Pa. Find its final temperature in °C.
A steel bridge girder is 60 m long at 10 °C. Find the increase in its length on a summer day when the temperature is 45 °C. (\(\alpha_{steel} = 1.3 \times 10^{-5}\) K\(^{-1}\))
A copper rod is 1.50 m long at 20 °C. It is heated until its length becomes 1.5015 m. Find the final temperature of the rod. (\(\alpha_{copper} = 1.7 \times 10^{-5}\) K\(^{-1}\))
An aluminium sheet has an area of 0.50 m\(^2\) at 25 °C. Find the increase in its area when it is heated to 125 °C. (\(\alpha_{aluminium} = 2.5 \times 10^{-5}\) K\(^{-1}\))
The coefficient of volume expansion of a metal is \(6.9 \times 10^{-5}\) K\(^{-1}\). Find its coefficients of linear and areal expansion.
A glass beaker of volume 500 cm\(^3\) is filled to the brim with a liquid at 15 °C. The whole arrangement is heated to 65 °C. Given \(\gamma_{liquid} = 49 \times 10^{-5}\) /°C and \(\gamma_{glass} = 2.5 \times 10^{-5}\) /°C, calculate the volume of liquid that spills out.
How much heat is required to raise the temperature of 2.5 kg of iron from 30 °C to 130 °C? (\(s_{iron} = 448\) J kg\(^{-1}\) K\(^{-1}\))
An electric heater supplies heat at a steady rate of 500 W to 1.5 kg of water. Neglecting all losses, find the time taken to raise the temperature of the water from 25 °C to 65 °C. (\(s_{water} = 4186\) J kg\(^{-1}\) K\(^{-1}\))
Find the thermal capacity of an aluminium block of mass 0.35 kg. (\(s_{aluminium} = 903.0\) J kg\(^{-1}\) K\(^{-1}\))
A 0.20 kg block of metal at 95 °C is dropped into 0.40 kg of water at 22 °C, contained in a copper calorimeter of mass 0.10 kg. The mixture settles at a steady temperature of 27 °C. Calculate the specific heat capacity of the metal. (\(s_{water} = 4186\) J kg\(^{-1}\) K\(^{-1}\), \(s_{copper} = 387\) J kg\(^{-1}\) K\(^{-1}\))
Calculate the heat required to convert 0.050 kg of ice at 0 °C completely into water at 40 °C. (\(L_f = 3.33 \times 10^{5}\) J kg\(^{-1}\), \(s_{water} = 4186\) J kg\(^{-1}\) K\(^{-1}\))
Find the total heat that must be removed from 0.30 kg of steam at 100 °C in order to condense it entirely into water at 100 °C, and then to cool that water to 20 °C. (\(L_v = 22.6 \times 10^{5}\) J kg\(^{-1}\), \(s_{water} = 4186\) J kg\(^{-1}\) K\(^{-1}\))
0.040 kg of ice at 0 °C is dropped into 0.20 kg of water at 40 °C. Assuming no heat is lost to the surroundings, find the final temperature of the mixture. (\(L_f = 3.33 \times 10^{5}\) J kg\(^{-1}\), \(s_{water} = 4186\) J kg\(^{-1}\) K\(^{-1}\))
An aluminium rod of length 0.80 m and cross-sectional area \(4.0 \times 10^{-4}\) m\(^2\) has its ends maintained at 100 °C and 20 °C. Calculate the rate of flow of heat through the rod in the steady state. (\(k_{aluminium} = 205\) J s\(^{-1}\) m\(^{-1}\) K\(^{-1}\))
A glass window of area 1.5 m\(^2\) and thickness 6.0 mm separates a room at 24 °C from the outside air at −6 °C. Calculate (i) the rate of heat loss through the window, and (ii) the thermal resistance of the window. (\(k_{glass} = 0.8\) J s\(^{-1}\) m\(^{-1}\) K\(^{-1}\))
A wooden panel of area 2.0 m\(^2\) and thickness 2.5 cm has its two faces at 30 °C and 10 °C. Taking \(k_{wood} = 0.12\) J s\(^{-1}\) m\(^{-1}\) K\(^{-1}\), find the heat conducted through the panel in 10 minutes.
A copper rod and a steel rod of equal length and equal cross-section conduct heat between the same two temperatures. Using Table 7.7, find the ratio of the rates of heat flow through them, and the ratio of their thermal resistances.
A body cools at the rate of 1.2 °C/min when its temperature is 65 °C. The temperature of the surroundings is 25 °C. At what rate will the body cool when its temperature is 45 °C?
A liquid cools at the rate of 0.8 °C/s when its temperature is 75 °C, and at the rate of 0.4 °C/s when its temperature is 55 °C. Determine the temperature of the surroundings, and hence the rate of cooling when the temperature of the liquid is 45 °C.
A glass flask has volume \(1 \times 10^{-4}\) m\(^3\). It is filled with a liquid at 30 ºC. If the temperature of the system is raised to 100 ºC, how much of the liquid will overflow? (Coefficient of volume expansion of glass is \(1.2 \times 10^{-5}\) (ºC)\(^{-1}\), while that of the liquid is \(75 \times 10^{-5}\) (ºC)\(^{-1}\).)
Which will require more energy — heating a 2.0 kg block of lead by 30 K, or heating a 4.0 kg block of copper by 5 K? (\(s_{lead} = 128\) J kg\(^{-1}\) K\(^{-1}\), \(s_{copper} = 387\) J kg\(^{-1}\) K\(^{-1}\))
The specific latent heat of vaporization of water is \(2.26 \times 10^{6}\) J/kg. Calculate the energy needed to change 5.0 g of water into steam at 100 ºC.
A metal sphere cools at the rate of 0.05 ºC/s when its temperature is 70 ºC, and at the rate of 0.025 ºC/s when its temperature is 50 ºC. Determine the temperature of the surroundings, and find the rate of cooling when the temperature of the metal sphere is 40 ºC.
The volume of a gas varied linearly with absolute temperature if its pressure is held constant. Suppose the gas does not liquefy even at very low temperatures; at what temperature will the volume of the gas be ideally zero?
In olden days, while laying the rails for trains, small gaps used to be left between the rail sections to allow for thermal expansion. Suppose the rails are laid at room temperature 27 ºC. If the maximum temperature in the region is 45 ºC and the length of each rail section is 10 m, what should be the gap left, given that \(\alpha = 1.2 \times 10^{-5}\) K\(^{-1}\) for the material of the rail section?
A blacksmith fixes an iron ring on the rim of the wooden wheel of a bullock cart. The diameter of the wooden rim and of the iron ring are 1.5 m and 1.47 m respectively, at a room temperature of 27 ºC. To what temperature should the iron ring be heated so that it can fit the rim of the wheel? (\(\alpha_{iron} = 1.2 \times 10^{-5}\) K\(^{-1}\))
In a random temperature scale X, water boils at 200 °X and freezes at 20 °X. Find the boiling point of a liquid on this scale if it boils at 62 °C.
A gas at 900 °C is cooled until both its pressure and volume are halved. Calculate its final temperature.
An aluminium rod and an iron rod show a 1.5 m difference in their lengths when heated at all temperatures. What are their lengths at 0 °C, if the coefficient of linear expansion for aluminium is \(24.5 \times 10^{-6}\) /°C and for iron is \(11.9 \times 10^{-6}\) /°C?
What is the specific heat of a metal if 50 cal of heat is needed to raise 6 kg of the metal from 20 °C to 62 °C?
The rate of flow of heat through a copper rod with a temperature difference of 30 °C is 1500 cal/s. Find the thermal resistance of the copper rod.
An electric kettle takes 20 minutes to heat a certain quantity of water from 0 °C to its boiling point. It requires 90 minutes to turn all the water at 100 °C into steam. Find the latent heat of vaporisation. (Specific heat of water = 1 cal/g °C)
Find the temperature difference between the two sides of a steel plate 4 cm thick, when heat is transmitted through the plate at the rate of 400 kcal per minute per square metre at steady state. The thermal conductivity of steel is 0.026 kcal/m s K.
A metal sphere cools from 80 °C to 60 °C in 6 min. How much time will it take to cool from 60 °C to 40 °C if the room temperature is 30 °C?
7.14 Objective Questions
- The range of temperature in a clinical thermometer, which measures the temperature of the human body, is
- 70 ºC to 100 ºC
- 34 ºC to 42 ºC
- 0 ºF to 100 ºF
- 34 ºF to 80 ºF
- A glass bottle completely filled with water is kept in the freezer. Why does it crack?
- Bottle gets contracted
- Bottle is expanded
- Water expands on freezing
- Water contracts on freezing
- If two temperatures differ by 25 °C on the Celsius scale, the difference in temperature on the Fahrenheit scale is
- 65°
- 45°
- 38°
- 25°
- If \(\alpha\), \(\beta\) and \(\gamma\) are coefficients of linear, area and volume expansion of a solid, then
- \(\alpha:\beta:\gamma\) 1:3:2
- \(\alpha:\beta:\gamma\) 1:2:3
- \(\alpha:\beta:\gamma\) 2:3:1
- \(\alpha:\beta:\gamma\) 3:1:2
- Consider the following statements —
- The coefficient of linear expansion has dimension K\(^{-1}\)
- The coefficient of volume expansion has dimension K\(^{-1}\) a. I and II are both correct b. I is correct but II is wrong c. II is correct but I is wrong d. I and II are both wrong
- Water falls from a height of 200 m. What is the difference in temperature between the water at the top and the bottom of a water fall, given that the specific heat of water is 4200 J kg\(^{-1}\) °C\(^{-1}\)?
- 0.96 °C
- 1.02 °C
- 0.46 °C
- 1.16 °C