10.1 Introduction
Chapter 10 — Thermal Properties of Matter
In the 1760s, a Scottish scientist named Joseph Black noticed something that puzzled him. When he placed a block of ice near a fire, it did not jump straight to warm water. Instead, the ice sat at its melting point and slowly turned to liquid — and all the while, a thermometer stuck in the mush refused to climb. Heat was clearly flowing in from the fire, yet the temperature simply would not rise until the last piece of ice had melted.
Where was all that heat going if not into making things hotter?
Black realised that heat and temperature are not the same thing. Temperature tells you how hot something is; heat is the energy that flows because of a difference in temperature. He showed that melting ice quietly swallows a large amount of heat without any change in temperature — heat that is spent breaking the solid apart rather than warming it. This hidden, or “latent,” heat is why a glass of iced drink stays cold for so long, and why the same idea, running in reverse, makes steam so dangerous.
Black’s careful measurements gave us two of the most useful ideas in this chapter: that every substance stores heat differently, and that changing state costs energy even when the temperature stands still.
Figure to come
Fig. 10.0 – A block of ice melting beside a flame, with a thermometer reading a steady 0 °C while heat arrows flow into the ice.
By the end of this chapter, you will be able to answer each of these questions using the physics of heat, temperature, expansion, and the way heat moves from one place to another.
All of us grow up with a rough, everyday sense of heat and temperature. We say a kettle of boiling water is “hot” and a box of ice is “cold,” and we usually get it right. If two objects are placed side by side, most people can tell which one is hotter just by touching them.
Temperature, in this everyday sense, is simply a measure of how “hot” a body is. A kettle full of boiling water has a higher temperature than a box containing ice, and we describe it as hotter.
But everyday language is not precise enough for physics. Words like hot and cold are comparative and depend on the observer — a metal spoon and a wooden spoon lying in the same room feel differently cool to the touch, even though they are at the same temperature. To build reliable science, we need to define heat and temperature carefully, measure them with instruments, and express them in fixed units.
That is what this chapter sets out to do. You will learn what heat actually is, how temperature is measured, and how the two are related but not the same. You will then study the various processes by which heat flows from one body to another, and what heat does to matter once it arrives.
Along the way, several familiar puzzles will be explained by physics. You will find out why a blacksmith heats an iron ring before fitting it onto the wooden rim of a cart wheel, and why the breeze at a beach often reverses its direction after the sun goes down. You will also see what happens when water boils or freezes: its temperature stays fixed during the change even though a large amount of heat is flowing into or out of it.
Figure to come
Fig. 10.1 – A split illustration: on the left a kettle of boiling water with steam rising (hot), on the right a box of ice cubes (cold), with a thermometer between them showing the two very different temperature readings.
These are not just curiosities. Each one is a clue to a deeper idea about heat, temperature, expansion, change of state, or the movement of heat — ideas that the rest of this chapter will develop step by step.
10.2 Temperature and Heat
The study of the thermal properties of matter begins with two ideas that everyday language often mixes up: temperature and heat. They are closely related, but they are not the same thing, and keeping them separate is the key to understanding this whole chapter.
Temperature. Temperature is a relative measure, or indication, of how hot or cold a body is. A hot utensil is said to be at a high temperature, and an ice cube at a low temperature. When one object has a higher temperature than another, we say it is hotter.
Notice that “hot” and “cold” are relative terms, much like “tall” and “short.” A person is not tall on their own — only tall compared to someone else. In the same way, an object is hotter or colder only in comparison with something else.
We can sense temperature by touch, but this sense is unreliable and works over only a narrow range. Our skin cannot tell us that boiling oil is at 200 °C or that liquid nitrogen is at −196 °C — it simply warns us of “very hot” or “very cold.” For scientific work we therefore need proper instruments and fixed units, which the next section takes up.
Heat. Now think about a common experience. A glass of ice-cold water left on a table on a hot summer day slowly warms up, while a cup of hot tea on the same table slowly cools down. In both cases the object and its surroundings are at different temperatures to begin with.
Because of this temperature difference, energy moves between the object and its surroundings. This transfer continues until the object and its surroundings reach the same temperature — a state we call thermal equilibrium, meaning there is no longer any net flow of energy between them.
The direction of this flow is worth noticing. For the glass of ice-cold water, energy flows from the warmer surroundings into the water. For the cup of hot tea, energy flows from the tea out to the cooler surroundings.
In every case, energy flows from the region at higher temperature to the region at lower temperature. This flowing energy is what we call heat.
The two quantities have different SI units, which reinforces that they are different physical ideas. The SI unit of heat, being a form of energy, is the joule (J). The SI unit of temperature is the kelvin (K), while the degree Celsius (°C) is a commonly used unit for everyday temperature.
Finally, note that when an object is heated, several things may happen. Its temperature may rise, it may expand, or it may change its state — from solid to liquid, or liquid to gas. The rest of this chapter studies each of these effects of heat on matter in turn.
10.3 Measurement of Temperature
Since our sense of touch is unreliable, temperature must be measured with an instrument called a thermometer. The idea behind every thermometer is simple: find some physical property of a material that changes steadily and predictably when the temperature changes, and read the temperature off that change.
Many properties of matter behave this way. The length of a metal rod, the electrical resistance of a wire, the pressure of a gas, and the volume of a liquid all vary with temperature. Any of these can, in principle, be used to build a thermometer.
The most common choice is the change in the volume of a liquid. In an ordinary liquid-in-glass thermometer, a liquid such as mercury or alcohol is sealed in a thin glass tube. As the temperature rises, the liquid expands and its level in the tube climbs; as the temperature falls, it drops. Mercury and alcohol are used because their volume changes almost linearly — that is, in equal steps for equal changes of temperature — over a wide range.
Figure to come
Fig. 10.2 – A liquid-in-glass thermometer, showing the bulb of mercury at the bottom, the fine capillary tube, and the graduated scale marked in degrees.
Assigning numbers: fixed points and scales. A thermometer is only useful if we can attach numbers to it. This is done by calibrating it — marking the tube so that a definite number can be assigned to each temperature on a chosen scale.
To define any temperature scale, we need two fixed reference points: two temperatures that can be reproduced reliably anywhere in the world. This is not as easy as it sounds. Since every substance changes size with temperature, there is no fixed “absolute” length or volume to measure against. Instead, we tie the fixed points to physical events that always happen at the same temperature.
Two very convenient events are the freezing and boiling of pure water at standard atmospheric pressure. The temperature at which pure water freezes is called the ice point, and the temperature at which it boils is called the steam point.
Using these two points, two familiar scales are built: the Celsius scale and the Fahrenheit scale.
On the Celsius scale, the ice point is set at 0 °C and the steam point at 100 °C, with 100 equal divisions between them. On the Fahrenheit scale, the same two points are set at 32 °F and 212 °F, with 180 equal divisions between them.
Converting between the scales. Because both scales measure the same physical temperature but use different zero points and different-sized degrees, we can convert between them. If we plot Fahrenheit temperature \(t_F\) against Celsius temperature \(t_C\), the points fall on a straight line, as shown in Fig. 10.1.
Figure to come
Fig. 10.1 – A straight-line graph of Fahrenheit temperature \(t_F\) (y-axis) versus Celsius temperature \(t_C\) (x-axis), passing through (0 °C, 32 °F) and (100 °C, 212 °F), with the intervals \(\Delta t_C = 100\) and \(\Delta t_F = 180\) marked.
The equation of this straight line gives the conversion relation:
\[\frac{t_F - 32}{180} = \frac{t_C}{100}\]
Here \(t_F\) is the Fahrenheit temperature (in °F) and \(t_C\) is the Celsius temperature (in °C). The number 32 shifts for the different zero point (water freezes at 32 °F, not 0 °F), while the ratio 180/100 accounts for the different-sized degrees — a Fahrenheit degree is smaller than a Celsius degree.
10.4 Ideal-Gas Equation and Absolute Temperature
The liquid-in-glass thermometer has a hidden weakness. Away from the two fixed points, a mercury thermometer and an alcohol thermometer do not quite agree, because mercury and alcohol expand by slightly different amounts as the temperature changes. Each liquid follows its own expansion pattern, so their readings drift apart in between.
Gases behave far more cooperatively. A thermometer that uses a gas gives the same reading no matter which gas is inside — provided the gas is at low density. Experiments show that all gases at low densities expand in the same way, which makes a gas an excellent and reliable substance for defining temperature.
To use a gas as a thermometer, we first need the rules that connect its pressure, volume, and temperature. For a fixed quantity (mass) of gas, three quantities describe its state: its pressure \(P\), its volume \(V\), and its temperature \(T\). Here \(T\) is the absolute temperature, related to the Celsius temperature \(t\) by \(T = t + 273.15\), where \(t\) is measured in °C.
Boyle’s law. Keep the temperature of a fixed amount of gas fixed, and change its pressure. You find that the volume changes so that the product of pressure and volume stays the same:
\[PV = \text{constant}\]
Charles’ law. Now keep the pressure of the fixed amount of gas constant instead, and change its temperature. This time the volume divided by the absolute temperature stays constant:
\[\frac{V}{T} = \text{constant}\]
Combining the two: the ideal-gas equation. Low-density gases obey both laws, and the two can be combined into a single relationship. Since \(PV = \text{constant}\) (at fixed temperature) and \(V/T = \text{constant}\) (at fixed pressure), it follows that the combination \(PV/T\) must also be a constant for a given quantity of gas. This combined relation is called the ideal gas law.
Written in a general form that applies not just to one fixed sample but to any quantity of any low-density gas, it becomes the ideal-gas equation:
\[\frac{PV}{T} = \mu R\]
or, rearranged,
\[PV = \mu R T \tag{10.2}\]
Here \(\mu\) is the number of moles of gas in the sample. A mole is simply a fixed count of molecules (a standard “packet” of matter), so \(\mu\) tells us how much gas is present. The quantity \(R\) is the universal gas constant, the same for every gas:
\[R = 8.31 \ \text{J mol}^{-1}\,\text{K}^{-1}\]
In this equation \(P\) is the pressure (in pascal, Pa), \(V\) the volume (in m³), \(T\) the absolute temperature (in kelvin, K), and \(\mu\) the number of moles (in mol). The fact that a single constant \(R\) works for all gases is what makes the gas thermometer so trustworthy.
Measuring temperature with pressure. Equation 10.2 tells us that, for a gas, pressure and volume together are directly proportional to temperature: \(PV \propto T\). This is the key idea behind the constant-volume gas thermometer.
If we hold the volume of the gas fixed, then \(P \propto T\) — the pressure alone becomes directly proportional to the absolute temperature. So by keeping the volume constant and measuring the pressure, we can read off the temperature. A plot of pressure against temperature is then a straight line, as shown in Fig. 10.2.
Figure to come
Fig. 10.2 – A straight-line graph of pressure (y-axis) versus temperature (x-axis) for a low-density gas kept at constant volume, the line extending down toward −273.15 °C.
Extrapolating to absolute zero. Real gases do not follow the ideal gas law perfectly — at very low temperatures their behaviour deviates from the prediction. But over a large range the pressure-temperature line stays straight, and it looks as though the pressure would fall all the way to zero if we could keep cooling the gas while it stayed a gas.
If we extend (extrapolate) that straight line down until the pressure reaches zero, all such lines — for different gases and different amounts of gas — meet the temperature axis at the same point, as shown in Fig. 10.3. That point is the lowest temperature an ideal gas could ever reach.
Figure to come
Fig. 10.3 – Pressure-versus-temperature lines for three low-density gases (Gas A, B, C), all extrapolated as dashed lines to meet the temperature axis at the single point −273.15 °C (0 K).
This absolute minimum temperature is found to be −273.15 °C, and it is called absolute zero.
The Kelvin scale. Absolute zero is the natural starting point for a temperature scale, and it forms the basis of the Kelvin scale (also called the absolute temperature scale), named after the British scientist Lord Kelvin. On this scale, absolute zero (−273.15 °C) is taken as the zero point, written 0 K, as shown in the comparison in Fig. 10.4.
Figure to come
Fig. 10.4 – A side-by-side comparison of the Kelvin, Celsius, and Fahrenheit scales, aligning the steam point (373.15 K / 100 °C / 212 °F), ice point (273.15 K / 0 °C / 32 °F), and absolute zero (0 K / −273.15 °C / −459.69 °F).
One degree on the Kelvin scale is exactly the same size as one degree on the Celsius scale — the two scales differ only in where their zero sits. Because of this equal step size, the two are related by a simple shift:
\[T = t_C + 273.15 \tag{10.3}\]
where \(T\) is the temperature in kelvin (K) and \(t_C\) is the temperature in degrees Celsius (°C).
10.5 Thermal Expansion
You have almost certainly met thermal expansion without naming it. A sealed bottle with a tight metal lid often refuses to open — until you run hot water over the lid. The heat makes the metal lid expand a little, loosening its grip so it unscrews easily. This is thermal expansion at work in a solid.
Liquids do the same. The mercury in a thermometer rises when the bulb is dipped in warm water, because the mercury expands and is pushed up the tube. Take the thermometer out into cooler air, and the mercury level falls again as the liquid contracts.
Gases expand too, and often the most. A balloon that is only partly inflated in a cool room can swell to full size when moved into warm water. The reverse also happens: a fully inflated balloon dipped in cold water begins to shrink as the air inside contracts.
The common thread is clear from experience: most substances expand on heating and contract on cooling. A change in temperature causes a change in the dimensions of a body.
Depending on which dimension we track, thermal expansion is described in three ways. Expansion in length is called linear expansion, expansion in area is called area expansion, and expansion in volume is called volume expansion, as illustrated in Fig. 10.5.
Figure to come
Fig. 10.5 – Thermal expansion shown in three panels: (a) a rod lengthening (linear), (b) a flat plate growing in both length and breadth (area), and (c) a cube growing in all three dimensions (volume), each labelled with its fractional-change relation.
For the three cases, the fractional changes are related to the temperature rise \(\Delta T\) by:
\[\frac{\Delta l}{l} = \alpha_l\,\Delta T, \qquad \frac{\Delta A}{A} = 2\alpha_l\,\Delta T, \qquad \frac{\Delta V}{V} = 3\alpha_l\,\Delta T\]
We will see shortly why the area case carries a factor of 2 and the volume case a factor of 3.
Linear expansion
Take a substance in the shape of a long rod. For a small temperature change \(\Delta T\), the fractional change in its length, \(\Delta l / l\), is found to be directly proportional to \(\Delta T\):
\[\frac{\Delta l}{l} = \alpha_l\,\Delta T \tag{10.4}\]
Here \(\Delta l\) is the change in length (m), \(l\) is the original length (m), \(\Delta T\) is the temperature change (K or °C, since a change is the same size on both scales), and \(\alpha_l\) is the coefficient of linear expansion, with SI unit K⁻¹.
The coefficient \(\alpha_l\) measures how strongly a material expands per degree of temperature rise. It is a property of the material itself — steel, glass, and copper each have their own value.
Table 10.1 lists average values of \(\alpha_l\) for some materials over the range 0 °C to 100 °C. Comparing glass and copper is instructive: copper expands about five times more than pyrex glass for the same temperature rise. In general, metals expand more than most other solids and so have relatively high values of \(\alpha_l\).
Table 10.1 — Values of coefficient of linear expansion for some materials
| Material | \(\alpha_l\) (\(10^{-5}\) K⁻¹) |
|---|---|
| Aluminium | 2.5 |
| Brass | 1.8 |
| Iron | 1.2 |
| Copper | 1.7 |
| Silver | 1.9 |
| Gold | 1.4 |
| Glass (pyrex) | 0.32 |
| Lead | 0.29 |
Volume expansion
In the same way that we defined a coefficient for length, we define one for volume. For a temperature change \(\Delta T\), the fractional change in volume \(\Delta V / V\) defines the coefficient of volume expansion (or volume expansivity), \(\alpha_V\):
\[\alpha_V = \left(\frac{\Delta V}{V}\right)\frac{1}{\Delta T} \tag{10.5}\]
Here \(\Delta V\) is the change in volume (m³), \(V\) the original volume (m³), \(\Delta T\) the temperature change (K), and \(\alpha_V\) has SI unit K⁻¹.
Unlike \(\alpha_l\), the coefficient \(\alpha_V\) is not strictly constant — it generally depends on the temperature, as shown for copper in Fig. 10.6. Only at high temperatures does \(\alpha_V\) settle to a nearly constant value.
Figure to come
Fig. 10.6 – A graph of the coefficient of volume expansion \(\alpha_V\) of copper (y-axis) versus absolute temperature \(T\) (x-axis), rising from low values and levelling off to a constant at high temperature.
Table 10.2 gives \(\alpha_V\) for some common substances over 0–100 °C. Notice that thermal expansion of these solids and liquids is quite small, with special materials such as pyrex glass and invar (an iron-nickel alloy) having particularly low values. Among the liquids, alcohol (ethanol) has a larger \(\alpha_V\) than mercury, so it expands more than mercury for the same temperature rise.
Table 10.2 — Values of coefficient of volume expansion for some substances
| Material | \(\alpha_V\) (K⁻¹) |
|---|---|
| Aluminium | \(7 \times 10^{-5}\) |
| Brass | \(6 \times 10^{-5}\) |
| Iron | \(3.55 \times 10^{-5}\) |
| Paraffin | \(58.8 \times 10^{-5}\) |
| Glass (ordinary) | \(2.5 \times 10^{-5}\) |
| Glass (pyrex) | \(1 \times 10^{-5}\) |
| Hard rubber | \(2.4 \times 10^{-4}\) |
| Invar | \(2 \times 10^{-6}\) |
| Mercury | \(18.2 \times 10^{-5}\) |
| Water | \(20.7 \times 10^{-5}\) |
| Alcohol (ethanol) | \(110 \times 10^{-5}\) |
The strange case of water
Most substances expand steadily as they warm. Water breaks this rule over a small range. Between 0 °C and 4 °C, water contracts as it is heated instead of expanding — this is called its anomalous behaviour.
Cool water down from room temperature, and its volume shrinks as expected until it reaches 4 °C, as in Fig. 10.7(a). Cool it further, below 4 °C, and now the volume increases again, so the density decreases, as in Fig. 10.7(b).
Figure to come
Fig. 10.7 – Thermal expansion of water in two panels: (a) volume of 1 kg of water versus temperature, dipping to a minimum near 4 °C; (b) density versus temperature, peaking at 4 °C.
The consequence is striking: water has its maximum density at 4 °C. This single fact has a large effect on nature.
Consider a pond in winter. As the surface water cools toward 4 °C, it becomes denser and sinks, while warmer, lighter water from below rises to take its place. But once the surface water cools below 4 °C, it becomes less dense and stays on top, where it eventually freezes. Ice therefore forms at the surface first, and the denser 4 °C water below stays liquid — allowing fish and plants to survive the winter underneath.
If water behaved normally, the coldest water would sink and lakes would freeze from the bottom up, destroying much of their animal and plant life.
Expansion of gases
At ordinary temperatures, gases expand much more than solids and liquids. For liquids, \(\alpha_V\) hardly depends on temperature; for gases, it depends on temperature strongly. For an ideal gas we can find \(\alpha_V\) directly from the ideal-gas equation met in the previous section, \(PV = \mu R T\).
Hold the pressure \(P\) constant and let the temperature change by \(\Delta T\), producing a volume change \(\Delta V\). Then:
\[P\,\Delta V = \mu R\,\Delta T\]
Dividing this by \(PV = \mu R T\) cancels \(P\) on the left and \(\mu R\) on the right:
\[\frac{\Delta V}{V} = \frac{\Delta T}{T}\]
So the coefficient of volume expansion of an ideal gas at constant pressure is simply:
\[\alpha_v = \frac{1}{T} \quad \text{for an ideal gas} \tag{10.6}\]
At 0 °C (273 K), this gives \(\alpha_v = 3.7 \times 10^{-3}\) K⁻¹, far larger than the values for solids and liquids in Table 10.2. Equation (10.6) also shows that \(\alpha_v\) decreases as temperature rises. For a gas at room temperature and constant pressure, \(\alpha_v\) is about \(3300 \times 10^{-6}\) K⁻¹ — orders of magnitude larger than for a typical liquid.
Relation between \(\alpha_v\) and \(\alpha_l\)
There is a simple and important link between the volume and linear coefficients. Imagine a cube of side \(l\) that expands equally in all directions when its temperature rises by \(\Delta T\). Each side grows by:
\[\Delta l = \alpha_l\, l\, \Delta T\]
The new volume is \((l + \Delta l)^3\), so the change in volume is:
\[\Delta V = (l+\Delta l)^3 - l^3 \simeq 3l^2\,\Delta l \tag{10.7}\]
Here the tiny terms in \((\Delta l)^2\) and \((\Delta l)^3\) have been dropped, because \(\Delta l\) is very small compared with \(l\). Substituting \(\Delta l = \alpha_l\, l\, \Delta T\) and using \(l^3 = V\):
\[\Delta V = \frac{3V\,\Delta l}{l} = 3V\alpha_l\,\Delta T \tag{10.8}\]
Comparing this with the definition \(\Delta V / V = \alpha_v \Delta T\) gives the key result:
\[\alpha_v = 3\alpha_l \tag{10.9}\]
This is why the volume-expansion factor in Fig. 10.5 is 3, while (as Example 10.1 will show) the area-expansion factor is 2. Physically, a solid expands in length, area, and volume all at once, and these three coefficients are simply the same expansion counted along one, two, or three directions.
Thermal stress
What happens if a rod is heated but its ends are clamped so it cannot expand? The rod “wants” to lengthen but is held back, so the rigid supports push inward on it. This sets up a compressive strain, and the internal stress produced is called thermal stress.
Recall from the study of elasticity that stress and strain are linked by Young’s modulus \(Y\), where \(Y = \dfrac{\text{stress}}{\text{strain}}\). If the rod were free, it would strain by \(\Delta l / l = \alpha_l \Delta T\); preventing that strain forces an equal elastic strain, and hence a stress, into the material.
As an example, consider a steel rail of length 5 m and cross-sectional area 40 cm², prevented from expanding while its temperature rises by 10 °C. With \(\alpha_{l(\text{steel})} = 1.2 \times 10^{-5}\) K⁻¹, the compressive strain is:
\[\frac{\Delta l}{l} = \alpha_{l(\text{steel})}\,\Delta T = 1.2 \times 10^{-5} \times 10 = 1.2 \times 10^{-4}\]
Taking Young’s modulus of steel as \(Y_{\text{steel}} = 2 \times 10^{11}\) N m⁻², the thermal stress is:
\[\frac{\Delta F}{A} = Y_{\text{steel}}\left(\frac{\Delta l}{l}\right) = 2.4 \times 10^{7} \ \text{N m}^{-2}\]
This corresponds to an external force of:
\[\Delta F = A\,Y_{\text{steel}}\left(\frac{\Delta l}{l}\right) = 2.4 \times 10^{7} \times 40 \times 10^{-4} \simeq 10^{5}\ \text{N}\]
A force of about \(10^5\) N is enormous — if two such steel rails, fixed at their outer ends, meet at their inner ends, this force can easily bend the rails. This is the practical reason the expansion gaps mentioned earlier are so important.
10.6 Specific Heat Capacity
Put some water in a vessel and heat it on a burner. Before long, bubbles start rising, the water particles move faster and faster, and eventually the water boils. Clearly, supplying heat raises the temperature — but on what, exactly, does the amount of heat needed depend? We can answer this with a simple three-step experiment.
Throughout the experiment we use the same burner, which supplies heat at a steady rate. That lets us use a stopwatch cleverly: since heat = (rate of heating) × (time), a longer time on the same burner simply means more heat was supplied. So comparing times is the same as comparing amounts of heat.
Step 1 — change in temperature. Heat a fixed amount of water and raise its temperature by 20 °C; note the time. Now take the same amount of water and raise it by 40 °C. You will find it takes about twice as long — so raising the temperature by twice as much needs twice the heat. Heat needed grows with the temperature change \(\Delta T\).
Step 2 — mass. Now take double the amount of water and, with the same burner, raise its temperature by 20 °C. The time taken is again twice that of the first step. So doubling the mass doubles the heat needed. Heat needed grows with the mass \(m\).
Step 3 — nature of the substance. Replace the water with the same amount of some oil, say mustard oil, and raise its temperature by 20 °C. This time it takes less time, so less heat is needed than for the same mass of water through the same rise. The heat needed also depends on what the substance is.
Figure to come
Fig. 10.6a – Three side-by-side heating set-ups on identical burners with stopwatches: (1) water raised by 20 °C, (2) same water raised by 40 °C taking double time, (3) double water raised by 20 °C taking double time, and oil raised by 20 °C taking less time.
Putting the three observations together: the heat required to warm a substance depends on its mass \(m\), its temperature change \(\Delta T\), and the nature of the substance.
Heat capacity
We first capture how a particular body responds to heat. When a body absorbs (or gives off) heat \(\Delta Q\) and its temperature changes by \(\Delta T\), its heat capacity \(S\) is defined as:
\[S = \frac{\Delta Q}{\Delta T} \tag{10.10}\]
where \(\Delta Q\) is the heat supplied to change the temperature from \(T\) to \(T + \Delta T\). The SI unit of heat capacity is J K⁻¹.
Specific heat capacity
Add equal amounts of heat to equal masses of different substances, and their temperatures rise by different amounts. This tells us that every substance has its own value for the heat needed to change the temperature of one unit of mass by one unit of temperature. That value is the specific heat capacity.
Dividing the heat capacity by the mass gives:
\[s = \frac{S}{m} = \frac{1}{m}\frac{\Delta Q}{\Delta T} \tag{10.11}\]
Here \(s\) is the specific heat capacity, \(\Delta Q\) the heat absorbed or given off (J), \(m\) the mass (kg), and \(\Delta T\) the temperature change (K). The SI unit of specific heat capacity is J kg⁻¹ K⁻¹.
Note the condition in the box — “no change of state.” Specific heat applies while the substance stays in one phase (all solid, all liquid, or all gas). What happens during melting or boiling is a separate story, taken up in the next sections. Specific heat depends both on the nature of the substance and on its temperature.
Rearranging Eq. (10.11) gives the working formula used in most numerical problems:
\[\Delta Q = m\, s\, \Delta T\]
Molar specific heat capacity
Sometimes it is more natural to measure the amount of a substance in moles (\(\mu\)) rather than in kilograms — especially for gases, where we care about the number of molecules. Dividing the heat capacity by the number of moles instead of the mass gives the molar specific heat capacity:
\[C = \frac{S}{\mu} = \frac{1}{\mu}\frac{\Delta Q}{\Delta T} \tag{10.12}\]
Here \(C\) is the molar specific heat capacity, \(\mu\) the number of moles (mol), and \(\Delta Q\), \(\Delta T\) as before. Like \(s\), the value of \(C\) depends on the nature of the substance and its temperature. The SI unit of molar specific heat capacity is J mol⁻¹ K⁻¹.
Two molar specific heats for a gas
For solids and liquids, one value of specific heat is usually enough. For gases, there is a complication: a gas can be heated while keeping its pressure constant, or while keeping its volume constant, and the two give different results. So a gas needs two molar specific heats.
If the gas is heated at constant pressure, the value is the molar specific heat capacity at constant pressure, denoted \(C_p\). If it is heated at constant volume, it is the molar specific heat capacity at constant volume, denoted \(C_v\).
Tables 10.3 and 10.4 collect measured values. Table 10.3 lists specific heat capacities of common substances (at atmospheric pressure and ordinary temperature), and Table 10.4 lists molar specific heats of some gases.
Table 10.3 — Specific heat capacity of some substances at room temperature and atmospheric pressure
| Substance | \(s\) (J kg⁻¹ K⁻¹) | Substance | \(s\) (J kg⁻¹ K⁻¹) |
|---|---|---|---|
| Aluminium | 900.0 | Ice | 2060 |
| Carbon | 506.5 | Glass | 840 |
| Copper | 386.4 | Iron | 450 |
| Lead | 127.7 | Kerosene | 2118 |
| Silver | 236.1 | Edible oil | 1965 |
| Tungsten | 134.4 | Mercury | 140 |
| Water | 4186.0 |
Table 10.4 — Molar specific heat capacities of some gases
| Gas | \(C_p\) (J mol⁻¹ K⁻¹) | \(C_v\) (J mol⁻¹ K⁻¹) |
|---|---|---|
| He | 20.8 | 12.5 |
| H₂ | 28.8 | 20.4 |
| N₂ | 29.1 | 20.8 |
| O₂ | 29.4 | 21.1 |
| CO₂ | 37.0 | 28.5 |
Look carefully at Table 10.3: water has the highest specific heat capacity of all the substances listed — 4186 J kg⁻¹ K⁻¹, much larger than metals. This single fact explains a surprising number of everyday and natural phenomena.
Water’s high specific heat also shapes climate. Land heats up and cools down quickly, while a large body of water warms and cools slowly. So during summer, coastal water stays cooler than the land, and the breeze coming off the sea feels cooling. In desert regions, where there is little water, the ground heats rapidly by day and loses that heat rapidly at night — which is why deserts are scorching in the daytime and cold after dark. (The way this uneven heating actually drives the wind is explained later, under convection.)
10.7 Calorimetry
We have seen that heat flows from a hotter body to a colder one. If we can keep track of that flow carefully — making sure none of it leaks away — we can actually measure heat. This measurement is the business of calorimetry.
To do this cleanly, we need a system that does not trade heat with the outside world. Such a system is called an isolated system.
Now imagine an isolated system whose parts start at different temperatures. Heat will flow from the hotter parts to the colder parts until everything settles at one common temperature. Because the system is isolated, no heat escapes — so all the heat lost by the hot parts must reappear as heat gained by the cold parts.
This is simply the conservation of energy applied to heat: energy is not created or destroyed, only transferred. It gives us the central rule of calorimetry.
The word itself is straightforward: calorimetry means the measurement of heat. When a hot body is placed in contact with a colder body and no heat is allowed to escape, the heat lost by the hot body equals the heat gained by the cold body. By measuring temperatures and masses, we can turn this balance into a number.
The calorimeter
The device used to make these measurements is called a calorimeter. Its design is aimed entirely at one goal: stop heat from leaking to the surroundings, so that “heat lost = heat gained” holds as exactly as possible.
A calorimeter consists of a metallic vessel with a stirrer, both made of the same material — usually copper or aluminium. The vessel sits inside a wooden jacket packed with an insulating material such as glass wool. This outer jacket acts as a heat shield, greatly reducing heat loss from the inner vessel. A small opening in the jacket lets a mercury thermometer reach into the vessel to read the temperature, as shown in Fig. 10.7a.
Figure to come
Fig. 10.7a – Cross-section of a calorimeter: inner metallic vessel with a stirrer, surrounded by an insulating wooden jacket filled with glass wool, and a thermometer inserted through an opening in the lid.
Finding specific heat by the method of mixtures
A common use of the calorimeter is to find the specific heat capacity of a solid. A hot solid of known mass is dropped into water of known mass and temperature inside the calorimeter, and the final common temperature is measured. Applying “heat lost by the solid = heat gained by the water and calorimeter” then gives the unknown specific heat. The next example shows the method in full.
10.8 Change of State
Matter normally exists in three states: solid, liquid, and gas. A shift from one of these states to another is called a change of state. The two most common changes are solid-to-liquid and liquid-to-gas (and their reverses).
These changes happen when heat is exchanged between a substance and its surroundings — heat coming in usually drives a substance toward the gas state, while heat going out drives it toward the solid state. To see exactly how, we track temperature while steadily heating a substance.
Watching ice turn to water
Take some ice cubes in a beaker and note their temperature. Heat them slowly on a constant heat source, stirring continuously, and record the temperature every minute. Plotting temperature against time gives the graph in Fig. 10.9.
Figure to come
Fig. 10.9 – A temperature-versus-time graph for ice being heated: a rising line, then a flat plateau at the melting point while ice melts, then a rising line, then a flat plateau at the boiling point while water boils (not to scale).
The surprising result is the flat stretch: as long as any ice remains, the temperature does not change, even though heat is being supplied without pause. Where is that heat going? It is being used to convert solid ice into liquid water, not to raise the temperature.
The change of state from solid to liquid is called melting (or fusion), and the reverse, from liquid to solid, is called freezing.
Throughout melting, the temperature stays constant until the last bit of solid has melted. During this time the solid and liquid exist together, in the thermal equilibrium introduced earlier — there is no net heat flow between them because they are at the same temperature. The fixed temperature at which this happens is the melting point.
The melting point measured at standard atmospheric pressure has a special name.
Regelation: melting under pressure
Melting point depends on pressure, and a simple demonstration shows it. Take a slab of ice and hang a metal wire over it, loading each end of the wire with a heavy block of about 5 kg, as in Fig. 10.10. Over time, the wire slowly cuts through the slab — yet the slab does not split in two.
Figure to come
Fig. 10.10 – A block of ice with a thin metal wire draped over it, a heavy weight hanging from each end, the wire slowly passing through the ice while the ice refreezes above it.
Here is what happens. Directly under the wire, the pressure is very high, and high pressure lowers the melting point of ice — so the ice there melts at a temperature below 0 °C. The wire sinks into the meltwater. Once the wire has passed, the pressure above it drops back to normal, the melting point rises again, and the water refreezes into solid ice.
This refreezing after pressure-melting is called regelation.
From water to steam
Return to the heating experiment. Once all the ice has melted, continued heating makes the temperature climb again (Fig. 10.9), until it reaches nearly 100 °C — where it becomes steady once more. Now the supplied heat is being used to change liquid water into vapour (gas).
The change of state from liquid to vapour is called vaporisation. Just as in melting, the temperature stays constant until the entire liquid has turned to vapour, with the liquid and vapour coexisting in thermal equilibrium. The fixed temperature at which this occurs is the boiling point.
This constant-temperature behaviour finally answers a question from the start of the chapter. Boiling water sits at about 100 °C and refuses to get hotter, no matter how long you keep the flame on, because the incoming heat is spent turning liquid into vapour — changing the state of the water rather than raising its temperature.
A closer look at boiling
Watch water heating in a round-bottom flask fitted with a thermometer and a steam outlet, as in Fig. 10.11. First, the air dissolved in the water escapes as tiny bubbles. Next, bubbles of steam form at the hot bottom, but as they rise into cooler water near the top they condense and vanish. Finally, when the whole body of water reaches 100 °C, steam bubbles survive all the way to the surface — this is boiling. The steam is invisible inside the flask, but as it leaves and meets cooler air it condenses into tiny droplets, giving a foggy look.
Figure to come
Fig. 10.11 – A round-bottom flask of boiling water on a burner, with a thermometer and a steam outlet fixed through the cork, and steam condensing into a foggy cloud at the outlet.
Boiling point depends on pressure
Boiling point is not fixed — it changes with pressure. In the flask, if the steam outlet is closed for a few seconds, the pressure inside rises and boiling stops. More heat (and a higher temperature) is then needed before boiling restarts. So boiling point increases when pressure increases.
The reverse can also be shown. Let the water cool to about 80 °C, seal the flask, invert it, and pour ice-cold water over it. The vapour inside condenses, lowering the pressure above the water, and the water starts boiling again — now at a temperature below 100 °C. So boiling point decreases when pressure decreases.
As with melting, the boiling point measured at standard atmospheric pressure has its own name.
Sublimation: skipping the liquid
Not every substance passes through all three states in turn. Some go straight from solid to vapour, and back, without ever becoming liquid. This direct solid-to-vapour change is called sublimation, and the substance is said to sublime.
The triple point
Figure to come
Fig. 10.8a – Two pressure-versus-temperature phase diagrams (not to scale): (a) for water and (b) for CO₂, each showing solid, liquid, and vapour regions bounded by the sublimation curve (BO), fusion curve (AO), and vaporisation curve (CO), meeting at the triple point O.
10.8.1 Latent Heat
In the previous section we saw something puzzling: during a change of state, heat keeps flowing into a substance, yet its temperature does not rise. That heat is not lost — it is stored in the change of state itself. This “hidden” heat, which goes into changing state rather than changing temperature, is called latent heat (from a word meaning “hidden”).
The amount of heat per unit mass transferred while a substance changes state is its latent heat for that process. Let us trace it through a familiar example.
Suppose we add heat to a block of ice starting at −10 °C. At first, the temperature of the ice climbs until it reaches the melting point, 0 °C. At 0 °C the temperature stalls: adding more heat no longer warms the ice but instead melts it, changing solid to liquid. Only after all the ice has melted does further heat begin to raise the temperature of the water. The same pattern repeats at the boiling point — heat added to boiling water produces vapour without any rise in temperature.
The latent heat formula
The heat needed for a change of state depends on two things: the nature of the substance (its “heat of transformation,” meaning how much heat each kilogram needs to change state) and the mass changing state. If a mass \(m\) of a substance changes completely from one state to another, the heat required is:
\[Q = m L \qquad \text{or} \qquad L = \frac{Q}{m} \tag{10.13}\]
Here \(Q\) is the heat absorbed or released (J), \(m\) is the mass changing state (kg), and \(L\) is the latent heat of the substance for that process (SI unit J kg⁻¹). Notice that \(L\) does not involve a temperature change — this is exactly the point, since the temperature stays constant throughout.
The value of \(L\) is a characteristic of the substance and also depends on pressure, so it is usually quoted at standard atmospheric pressure. Because a substance can change state in two common ways, there are two latent heats.
For a solid-liquid change, it is the latent heat of fusion, \(L_f\). For a liquid-gas change, it is the latent heat of vaporisation, \(L_v\). These are also called the heat of fusion and the heat of vaporisation.
Reading the temperature-heat graph
If we plot temperature against heat supplied for water, we get the graph in Fig. 10.12. The two flat portions are the change-of-state stages, where heat is being added but temperature holds still.
Figure to come
Fig. 10.12 – A temperature-versus-heat graph for water at 1 atm (not to scale): a sloped solid (ice) portion, a flat melting plateau at 0 °C labelled \(3.33 \times 10^{5}\) J/kg, a sloped liquid (water) portion, a flat boiling plateau at 100 °C labelled \(22.6 \times 10^{5}\) J/kg, and a sloped gas (steam) portion.
Two features are worth noting. First, along each flat portion the temperature stays constant as heat is added or removed — the signature of a change of state. Second, the sloped portions (solid, liquid, gas) do not all have the same steepness, which tells us the specific heats of ice, water, and steam are different from one another.
Table 10.5 lists the latent heats of some substances, together with their freezing and boiling points, at 1 atm pressure.
Table 10.5 — Temperatures of the change of state and latent heats for various substances at 1 atm pressure
| Substance | Melting Point (°C) | \(L_f\) (\(10^{5}\) J kg⁻¹) | Boiling Point (°C) | \(L_v\) (\(10^{5}\) J kg⁻¹) |
|---|---|---|---|---|
| Ethanol | −114 | 1.0 | 78 | 8.5 |
| Gold | 1063 | 0.645 | 2660 | 15.8 |
| Lead | 328 | 0.25 | 1744 | 8.67 |
| Mercury | −39 | 0.12 | 357 | 2.7 |
| Nitrogen | −210 | 0.26 | −196 | 2.0 |
| Oxygen | −219 | 0.14 | −183 | 2.1 |
| Water | 0 | 3.33 | 100 | 22.6 |
Why steam burns are worse
For water, the two latent heats are \(L_f = 3.33 \times 10^{5}\) J kg⁻¹ and \(L_v = 22.6 \times 10^{5}\) J kg⁻¹. In words, \(3.33 \times 10^{5}\) J of heat is needed to melt 1 kg of ice at 0 °C, and \(22.6 \times 10^{5}\) J is needed to turn 1 kg of water at 100 °C into steam at 100 °C.
Read that last figure the other way around. When 1 kg of steam at 100 °C condenses back to water at 100 °C, it releases \(22.6 \times 10^{5}\) J. So steam at 100 °C carries this much more energy than boiling water at the very same temperature.
This is why a burn from steam is usually far more serious than a burn from boiling water, even though both are at 100 °C. Steam landing on the skin first condenses, dumping its large latent heat of vaporisation into the skin, and only then cools as water — delivering a double dose of heat that plain boiling water cannot.
10.9 Heat Transfer
We have already established what heat is: energy that flows from one system to another, or from one part of a system to another, because of a temperature difference. So far we have focused on how much heat flows. We now ask a different question — how does that heat actually get from one place to another?
There are three distinct ways, or modes, by which heat is transferred: conduction, convection, and radiation. All three are illustrated together in Fig. 10.13.
Figure to come
Fig. 10.13 – A hand holding a metal rod over a fire, showing heat reaching the hand by conduction along the rod, warm air rising by convection, and heat travelling directly from the flames by radiation.
Each mode works differently. In brief: conduction passes heat through a material without the material itself moving, convection carries heat by the actual movement of a heated fluid, and radiation sends heat as electromagnetic waves that need no material medium at all. The next three subsections take up each mode in turn.
10.9.1 Conduction
Hold one end of a metal rod in a flame, and before long the other end grows too hot to touch — even though only one end is in the fire. Heat has travelled along the rod from the hot end to the cold end, passing through the material without the material itself moving. This mode of heat transfer is called conduction.
Different materials conduct heat very differently. Metals conduct heat readily, which is why the metal rod’s far end heats up so fast. Gases are poor conductors of heat, and liquids fall somewhere in between solids and gases.
Setting up the problem
To describe conduction quantitatively, we measure the rate of heat flow — how much heat passes per second for a given temperature difference.
Consider a metal bar of length \(L\) and uniform cross-sectional area \(A\), with its two ends held at different temperatures, as in Fig. 10.14. We keep the ends at fixed temperatures by pressing them against large heat reservoirs at temperatures \(T_C\) and \(T_D\) (with \(T_C > T_D\)). We also imagine the sides of the bar perfectly insulated, so heat can escape only through the ends and none leaks out sideways.
Figure to come
Fig. 10.14 – A horizontal bar of length \(L\) and cross-section \(A\), its left end in contact with a hot reservoir at \(T_C\) and its right end with a cold reservoir at \(T_D\), heat flowing left to right, with the sides insulated.
After some time, the bar reaches a steady state — a condition in which the temperature at each point of the bar no longer changes with time. In this steady state, the temperature falls uniformly along the bar from \(T_C\) down to \(T_D\). The hot reservoir keeps supplying heat at a constant rate, that heat travels through the bar, and the same rate of heat is delivered to the cold reservoir at the other end.
The law of heat conduction
Experiments on this steady state show that the rate of heat flow — called the heat current \(H\) — behaves in three sensible ways. It is larger when the temperature difference \((T_C - T_D)\) is larger, larger when the cross-section \(A\) is bigger, and smaller when the bar is longer (larger \(L\)). Combining these:
\[H = KA\,\frac{T_C - T_D}{L} \tag{10.14}\]
Here \(H\) is the heat current — the heat flowing per unit time (SI unit W, i.e. J s⁻¹); \(A\) is the cross-sectional area (m²); \((T_C - T_D)\) is the temperature difference between the ends (K); \(L\) is the length of the bar (m); and \(K\) is a constant that depends on the material.
Each dependence makes physical sense. A bigger temperature difference pushes heat through faster. A wider bar offers more parallel paths for heat, so more flows. A longer bar makes the heat travel farther through resisting material, so less flows per second — much like a longer pipe slows the flow of water.
Thermal conductivity
The constant \(K\) in Eq. (10.14) is the thermal conductivity of the material. It measures how good the material is at conducting heat.
A large \(K\) means a good conductor (like a metal); a small \(K\) means a good insulator (like glass wool). Table 10.6 lists thermal conductivities for many materials. These values change only slightly with temperature and can be treated as constant over a normal temperature range.
Table 10.6 — Thermal conductivities of some materials
| Material | Thermal conductivity (J s⁻¹ m⁻¹ K⁻¹) |
|---|---|
| Metals | |
| Silver | 406 |
| Copper | 385 |
| Aluminium | 205 |
| Brass | 109 |
| Steel | 50.2 |
| Lead | 34.7 |
| Mercury | 8.3 |
| Non-metals | |
| Insulating brick | 0.15 |
| Concrete | 0.8 |
| Body fat | 0.20 |
| Felt | 0.04 |
| Glass | 0.8 |
| Ice | 1.6 |
| Glass wool | 0.04 |
| Wood | 0.12 |
| Water | 0.8 |
| Gases | |
| Air | 0.024 |
| Argon | 0.016 |
| Hydrogen | 0.14 |
Notice the enormous gap between good conductors and good insulators. Silver and copper are hundreds of times better at conducting heat than wood or glass wool.
10.9.2 Convection
In conduction, heat moves through a material while the material itself stays put. Convection works in a completely different way: here the heated matter itself moves, carrying its heat along with it.
Because convection needs matter to flow from place to place, it cannot occur in solids, whose particles are locked in position. Only fluids, which can flow, allow it. Convection comes in two kinds: natural and forced.
Natural convection
In natural convection, gravity does the work. Suppose a fluid is heated from below. The hot portion at the bottom expands, and — as we saw with thermal expansion — expanding makes it less dense than the fluid around it.
A less dense fluid surrounded by denser fluid experiences an upward buoyant force, the same effect that makes a cork rise in water. So the warm, light fluid rises, and cooler, denser fluid sinks down to take its place. This cooler fluid then gets heated in turn, rises, and is again replaced. A continuous circulation, or convection current, is set up, as illustrated in Fig. 10.9.2a.
Figure to come
Fig. 10.9.2a – A vessel of water heated from below, showing looping convection currents: warm water rising in the middle and cooler water sinking along the sides.
This is clearly different from conduction — here whole parcels of fluid physically move, transporting their heat with them.
Forced convection
Sometimes we do not wait for gravity — we push the fluid along ourselves. In forced convection, the fluid is made to move by a pump, fan, or other mechanical means.
Everyday examples include forced-air heating systems in homes (a fan blows warm air around) and the cooling system of an automobile engine (a pump circulates coolant to carry heat away from the engine).
Sea breeze and land breeze
Natural convection explains a familiar coastal experience. Recall from the discussion of specific heat that land heats up and cools down faster than a large body of water, partly because water has a much higher specific heat capacity and partly because currents spread the absorbed heat through its great volume.
During the day, the ground warms faster than the sea. Air touching the warm ground is heated by conduction, expands, becomes less dense, and rises. Cooler air from over the sea moves in to fill the space — a sea breeze blowing from sea to land. The risen air cools, descends over the sea, and a convection cycle is set up that carries heat away from the land, as shown in Fig. 10.17.
Figure to come
Fig. 10.17 – Two panels of convection cycles: (day) land warmer than water, warm air rising over land and a sea breeze blowing inland; (night) water warmer than land, warm air rising over the sea and a land breeze blowing out to sea.
At night the situation reverses. The land loses its heat faster than the water, so now the sea surface is warmer than the land. The air rises over the warmer sea instead, and the breeze blows the other way — from land to sea, a land breeze. This is exactly why the wind at a beach so often changes direction after the sun goes down.
Trade winds
Natural convection also operates on a planetary scale. The steady surface wind that blows from the north-east toward the equator — the trade wind — is a giant convection current.
The equatorial regions receive far more solar heating than the poles. Air near the hot equatorial surface warms, expands, and rises, moving toward the poles high up, while cooler air streams back along the surface toward the equator. On a non-rotating Earth this simple cycle would carry surface air straight from the poles to the equator.
But the Earth’s rotation modifies the flow. Because of the planet’s spin, air near the equator moves eastward at about 1600 km/h, while air near the poles has almost no eastward speed. As a result, the rising equatorial air does not travel all the way to the poles; it descends at about 30° N latitude and returns to the equator. This deflected surface flow is the trade wind.
10.9.3 Radiation
The first two modes of heat transfer share a limitation. Conduction needs a solid to pass heat along, and convection needs a fluid to carry it. Both require some material medium — so neither can move heat across empty space.
Yet heat clearly does cross empty space. The Earth is warmed by the Sun across millions of kilometres of vacuum. Closer to home, we feel the warmth of a fire almost at once — far too quickly for convection, which takes time to set up, and even though air is a poor conductor. Some third mechanism must be at work, one that needs no medium at all.
That mechanism is radiation, and the energy it carries is called radiant energy.
Why radiation needs no medium
To see why radiation is different, we need the idea of an electromagnetic wave. In such a wave, an electric field and a magnetic field oscillate together in space and time. You will study electromagnetic waves in detail later; for now, the key facts are enough.
Like any wave, electromagnetic waves can have different wavelengths. Crucially, they can travel through vacuum, and they all move at the same enormous speed — the speed of light:
\[c = 3 \times 10^{8}\ \text{m s}^{-1}\]
Because these waves carry energy on their own oscillating fields, they need nothing to travel through. This is exactly why radiation requires no medium and why it is so fast — heat reaches us from the Sun across empty space at the speed of light.
Thermal radiation
Every object, whether solid, liquid, or gas, emits radiant energy simply because of its temperature. The radiation given off by a body on account of its temperature — such as the glow of red-hot iron or the light of a lamp filament — is called thermal radiation.
When thermal radiation lands on another body, part of it is reflected and part is absorbed. How much a body absorbs depends strongly on its colour. Experiment shows that black surfaces absorb — and also emit — radiant energy far better than light-coloured surfaces do.
Figure to come
Fig. 10.9.3a – The Sun radiating heat as electromagnetic waves across empty space to the Earth, illustrating heat transfer by radiation through a vacuum.
The Dewar flask (thermos)
If radiation, conduction, and convection are the only ways heat moves, then a device that blocks all three can keep things hot or cold for a long time. This is exactly what a Dewar flask, or thermos bottle, does.
A Dewar flask is built to defeat every mode of heat transfer at once, as shown in Fig. 10.9.3b. It is a double-walled glass vessel whose inner and outer walls are coated with silver. The silver surfaces reflect thermal radiation — the inner wall reflects the contents’ radiation back inward, and the outer wall reflects incoming radiation back outward, cutting radiation losses. The space between the two walls is evacuated (emptied of air), which removes the medium needed for conduction and convection, cutting those losses too. Finally, the flask rests on an insulating support such as cork, to block conduction through its base.
Figure to come
Fig. 10.9.3b – Cross-section of a Dewar flask: double glass walls with silvered surfaces, an evacuated space between them, hot or cold contents inside, and a cork support at the base, with arrows showing reflected radiation.
So far we have treated radiation only in general terms — that hot bodies emit it and dark bodies emit it best. The next subsection looks more closely at what wavelengths thermal radiation contains and how that changes with temperature.
10.9.4 Blackbody Radiation
So far we have spoken of thermal radiation without asking a basic question: what wavelengths does it contain? The answer is that thermal radiation is never a single wavelength. At any temperature, a hot body emits a continuous spectrum — a whole range of wavelengths, from short to long, all at once.
But the energy is not spread evenly across those wavelengths. Some wavelengths carry more energy than others. Figure 10.18 shows the experimental curves of radiation energy (per unit area, per unit wavelength) emitted by a blackbody, plotted against wavelength, for several different temperatures.
A blackbody here means an idealised object that absorbs all the radiation falling on it and, correspondingly, is the best possible emitter at every wavelength. Its radiation curve depends only on temperature, which makes it the natural reference for studying thermal radiation.
Figure to come
Fig. 10.18 – Curves of radiation energy per unit area per unit wavelength versus wavelength for a blackbody at different temperatures (lamp filament 2000 K, arc 3000 K, sunlight 6000 K), the peak shifting toward shorter wavelengths as temperature rises, with the visible-light band marked.
Wien’s displacement law
Look carefully at the curves in Fig. 10.18. Each has a peak — a wavelength \(\lambda_m\) at which the emitted energy is maximum. As the temperature rises, this peak wavelength \(\lambda_m\) shifts toward shorter wavelengths. The relation between them is Wien’s Displacement Law:
\[\lambda_m T = \text{constant} \tag{10.15}\]
Here \(\lambda_m\) is the wavelength of maximum energy emission (m) and \(T\) is the absolute temperature of the body (K). The constant, called Wien’s constant, has the value \(2.9 \times 10^{-3}\) m K. The law is named after the physicist Wilhelm Wien.
Because \(\lambda_m\) and \(T\) multiply to a fixed number, a hotter body has a smaller peak wavelength. This single fact explains the changing colour of heated iron.
A deep feature of the blackbody curves in Fig. 10.18 is that they are universal: they depend only on the temperature, not on the size, shape, or material of the blackbody. Explaining these universal curves proved impossible with the physics of the 1800s.
Stefan-Boltzmann law
Wien’s law tells us where the radiation peaks. A second law tells us how much total energy a body radiates. Since radiation needs no medium, energy can be transferred this way across vast distances, even through vacuum.
The total electromagnetic energy radiated per second by a body depends on its surface area, its ability to radiate, and — most strongly — on its absolute temperature. For a perfect radiator (a blackbody), the energy emitted per unit time, \(H\), is:
\[H = A\sigma T^4 \tag{10.16}\]
Here \(H\) is the energy radiated per unit time, i.e. power (W); \(A\) is the surface area of the body (m²); \(T\) is its absolute temperature (K); and \(\sigma\) is the Stefan-Boltzmann constant, with value \(\sigma = 5.67 \times 10^{-8}\) W m⁻² K⁻⁴.
Emissivity: real bodies
Most real bodies are not perfect radiators — they emit only a fraction of the ideal amount given by Eq. (10.16). A substance like lamp black comes close to the perfect limit, but ordinary surfaces fall short. To account for this, we introduce a dimensionless fraction \(e\) called the emissivity, and write:
\[H = Ae\sigma T^4 \tag{10.17}\]
For a tungsten lamp, for example, \(e\) is about 0.4. So a tungsten lamp at 3000 K with a surface area of 0.3 cm² radiates at the rate:
\[H = 0.3 \times 10^{-4} \times 0.4 \times 5.67 \times 10^{-8} \times (3000)^4 = 60\ \text{W}\]
Note how the tiny area (0.3 cm² = \(0.3 \times 10^{-4}\) m²) is combined with the huge \((3000)^4\) to give a familiar-sized power for a bulb.
Emitting and absorbing at once
A real body does not only emit — it also receives radiation from its surroundings. If a body at temperature \(T\) sits in surroundings at temperature \(T_s\), it emits and absorbs at the same time, and what matters is the net loss.
For a perfect radiator, the net rate of loss of radiant energy is:
\[H = \sigma A (T^4 - T_s^4)\]
For a real body with emissivity \(e\), this becomes:
\[H = e\sigma A (T^4 - T_s^4) \tag{10.18}\]
Here \(T\) is the body’s absolute temperature and \(T_s\) is the absolute temperature of the surroundings (both in K). The subtraction of \(T_s^4\) accounts for the radiation the body absorbs from its surroundings.
As an example, consider the heat radiated by a human body. Take the body’s surface area as about 1.9 m² and the room temperature as 22 °C (295 K). The skin temperature may be about 28 °C (301 K), and the emissivity of skin is about 0.97 for the relevant radiation. The net rate of heat loss is:
\[H = 5.67 \times 10^{-8} \times 1.9 \times 0.97 \times \{(301)^4 - (295)^4\} = 66.4\ \text{W}\]
This is more than half the rate of energy the body produces at rest (about 120 W) — a large loss.
10.10 Newton’s Law of Cooling
A cup of hot tea or a glass of warm milk left on a table always cools down, and eventually reaches the temperature of the room around it. This is nothing new — it is the heat transfer of the earlier sections at work. What we now want to know is how fast a body cools, and what that rate depends on.
An experiment on cooling
We can study cooling with a simple activity. Take about 300 mL of water in a calorimeter fitted with a stirrer and a two-holed lid. Push the stirrer through one hole and a thermometer through the other, with the thermometer bulb well inside the water. The thermometer’s first reading, \(T_1\), is the temperature of the surroundings.
Now heat the water until it is about 40 °C above room temperature, then remove the heat source. Start a stopwatch and, stirring gently, note the water’s temperature \(T_2\) at fixed intervals — say every minute — until it is only about 5 °C above the surroundings.
Plot the temperature excess \(\Delta T = T_2 - T_1\) (how far the water is above the surroundings) on the y-axis against time \(t\) on the x-axis. The result is the curve in Fig. 10.19.
Figure to come
Fig. 10.19 – A cooling curve showing the temperature excess \(\Delta T = T_2 - T_1\) of hot water falling with time \(t\): steep at first, then flattening as \(\Delta T\) becomes small.
The curve reveals a clear pattern. The cooling is fastest at the start, when the water is much hotter than the room, and it slows down as the water’s temperature falls closer to that of the surroundings. In short, the rate of cooling depends on how big the temperature difference is.
Newton’s law of cooling
A hot body loses heat to its surroundings mainly as radiation, and the rate of that loss grows with the temperature difference. Isaac Newton was the first to study this relationship in a systematic way.
Newton’s law states that the rate of loss of heat, \(-\dfrac{dQ}{dt}\), of a body is directly proportional to the temperature difference \(\Delta T = (T_2 - T_1)\) between the body and its surroundings. This holds only for a small temperature difference. The loss also depends on the nature and area of the exposed surface. In symbols:
\[-\frac{dQ}{dt} = k\,(T_2 - T_1) \tag{10.19}\]
Here \(-\dfrac{dQ}{dt}\) is the rate of heat loss (W); \(T_2\) is the body’s temperature and \(T_1\) the temperature of the surroundings (K or °C); and \(k\) is a positive constant that depends on the area and nature of the body’s surface. The negative sign shows that the body is losing heat as time passes.
From the law to a cooling formula
We can turn Newton’s law into a formula for temperature versus time. Suppose the body has mass \(m\) and specific heat capacity \(s\), and is at temperature \(T_2\), with the surroundings at \(T_1\). If its temperature falls by a small amount \(dT_2\) in a small time \(dt\), the heat lost is:
\[dQ = ms\,dT_2\]
So the rate of loss of heat is:
\[\frac{dQ}{dt} = ms\,\frac{dT_2}{dt} \tag{10.20}\]
Combining Eqs. (10.19) and (10.20) — that is, setting the two expressions for the rate of heat loss equal (with the sign showing loss):
\[-ms\,\frac{dT_2}{dt} = k\,(T_2 - T_1)\]
Rearranging to gather the temperature terms on one side and time on the other:
\[\frac{dT_2}{T_2 - T_1} = -\frac{k}{ms}\,dt = -K\,dt \tag{10.21}\]
where we have written \(K = \dfrac{k}{ms}\) as a single combined constant.
Now integrate both sides. The left side integrates to a natural logarithm (since the integral of \(\frac{1}{x}\) is \(\log_e x\)), and the right side to \(-Kt\) plus a constant of integration \(c\):
\[\log_e (T_2 - T_1) = -Kt + c \tag{10.22}\]
Taking the exponential of both sides removes the logarithm and gives temperature explicitly as a function of time:
\[T_2 = T_1 + C'\,e^{-Kt} \qquad \text{where } C' = e^{c} \tag{10.23}\]
Here \(C'\) is just a constant (fixed by the starting temperature). Equation (10.23) shows that the temperature excess above the surroundings dies away exponentially with time — which is exactly why the cooling curve in Fig. 10.19 is steep at first and then flattens. This equation lets us calculate the time a body takes to cool through a chosen temperature range.
For small temperature differences, this combined rate of cooling — arising from conduction, convection, and radiation together — is proportional to the temperature difference. This makes Newton’s law a useful approximation in many everyday situations.
Verifying the law
Newton’s law can be checked experimentally with the set-up in Fig. 10.20(a). A double-walled vessel (V) holds water between its walls, and a copper calorimeter (C) filled with hot water sits inside it. One thermometer reads the temperature \(T_2\) of the water in the calorimeter, and another reads the temperature \(T_1\) of the water between the double walls.
Figure to come
Fig. 10.20 – (a) The verification apparatus: a copper calorimeter of hot water inside a double-walled water vessel, with thermometers reading \(T_2\) (calorimeter) and \(T_1\) (outer walls); (b) a graph of \(\log_e(T_2 - T_1)\) versus time \(t\), a straight line with negative slope.
The calorimeter’s temperature is recorded at equal time intervals, and a graph is plotted of \(\log_e(T_2 - T_1)\) against time \(t\). As shown in Fig. 10.20(b), this graph turns out to be a straight line with a negative slope — exactly what Eq. (10.22) predicts, confirming the law.
10.11 Summary
Heat is a form of energy that flows between a body and its surrounding medium by virtue of the temperature difference between them. The degree of hotness of the body is quantitatively represented by temperature.
A temperature-measuring device (thermometer) makes use of some measurable property (called a thermometric property) that changes with temperature. Different thermometers lead to different temperature scales. To construct a temperature scale, two fixed points are chosen and assigned some arbitrary values of temperature. The two numbers fix the origin of the scale and the size of its unit.
The Celsius temperature (\(t_C\)) and the Fahrenheit temperature (\(t_F\)) are related by:
\[t_F = \frac{9}{5}\,t_C + 32\]
- The ideal gas equation connecting pressure (\(P\)), volume (\(V\)), and absolute temperature (\(T\)) is:
\[PV = \mu R T\]
where \(\mu\) is the number of moles and \(R\) is the universal gas constant.
- In the absolute temperature scale, the zero of the scale corresponds to the temperature where every substance in nature has the least possible molecular activity. The Kelvin absolute temperature scale (\(T\)) has the same unit size as the Celsius scale (\(T_C\)), but differs in the origin:
\[T_C = T - 273.15\]
- The coefficient of linear expansion (\(\alpha_l\)) and the coefficient of volume expansion (\(\alpha_v\)) are defined by the relations:
\[\frac{\Delta l}{l} = \alpha_l\,\Delta T \qquad\qquad \frac{\Delta V}{V} = \alpha_v\,\Delta T\]
where \(\Delta l\) and \(\Delta V\) denote the change in length \(l\) and volume \(V\) for a change of temperature \(\Delta T\). The relation between them is:
\[\alpha_v = 3\,\alpha_l\]
- The specific heat capacity of a substance is defined by:
\[s = \frac{1}{m}\frac{\Delta Q}{\Delta T}\]
where \(m\) is the mass of the substance and \(\Delta Q\) is the heat required to change its temperature by \(\Delta T\). The molar specific heat capacity of a substance is defined by:
\[C = \frac{1}{\mu}\frac{\Delta Q}{\Delta T}\]
where \(\mu\) is the number of moles of the substance.
The latent heat of fusion (\(L_f\)) is the heat per unit mass required to change a substance from solid into liquid at the same temperature and pressure. The latent heat of vaporisation (\(L_v\)) is the heat per unit mass required to change a substance from liquid to the vapour state without change in temperature and pressure.
The three modes of heat transfer are conduction, convection, and radiation.
In conduction, heat is transferred between neighbouring parts of a body through molecular collisions, without any flow of matter. For a bar of length \(L\) and uniform cross-section \(A\), with its ends maintained at temperatures \(T_C\) and \(T_D\), the rate of flow of heat \(H\) is:
\[H = K A\,\frac{T_C - T_D}{L}\]
where \(K\) is the thermal conductivity of the material of the bar.
- Newton’s Law of Cooling says that the rate of cooling of a body is proportional to the excess temperature of the body over the surroundings:
\[\frac{dQ}{dt} = -k\,(T_2 - T_1)\]
where \(T_1\) is the temperature of the surrounding medium and \(T_2\) is the temperature of the body.
10.12 Points to Ponder
The relation connecting the Kelvin temperature (\(T\)) and the Celsius temperature \(t_C\), \[T = t_C + 273.15\] and the assignment \(T = 273.16\) K for the triple point of water, are exact relations (by choice). With this choice, the Celsius temperature of the melting point of water and the boiling point of water (both at 1 atm pressure) are very close to, but not exactly equal to, 0 °C and 100 °C respectively. In the original Celsius scale, these latter fixed points were exactly at 0 °C and 100 °C (by choice), but now the triple point of water is the preferred fixed point, because it has a unique temperature.
A liquid in equilibrium with its vapour has the same pressure and temperature throughout the system; the two phases in equilibrium differ in their molar volume (i.e. density). This is true for a system with any number of phases in equilibrium.
Heat transfer always involves a temperature difference between two systems, or between two parts of the same system. Any energy transfer that does not involve a temperature difference in some way is not heat.
Convection involves the flow of matter within a fluid due to unequal temperatures of its parts. A hot bar placed under a running tap loses heat by conduction between the surface of the bar and the water, and not by convection within the water.
10.13 Table of Physical Quantities
| Quantity | Symbol | Dimensions | Unit | Remark |
|---|---|---|---|---|
| Amount of substance | \(\mu\) | [mol] | mol | |
| Celsius temperature | \(t_C\) | [K] | °C | |
| Kelvin absolute temperature | \(T\) | [K] | K | \(t_C = T - 273.15\) |
| Coefficient of linear expansion | \(\alpha_l\) | [K⁻¹] | K⁻¹ | |
| Coefficient of volume expansion | \(\alpha_v\) | [K⁻¹] | K⁻¹ | \(\alpha_v = 3\,\alpha_l\) |
| Heat supplied to a system | \(\Delta Q\) | [M L² T⁻²] | J | \(Q\) is not a state variable |
| Specific heat capacity | \(s\) | [L² T⁻² K⁻¹] | J kg⁻¹ K⁻¹ | |
| Thermal conductivity | \(K\) | [M L T⁻³ K⁻¹] | J s⁻¹ m⁻¹ K⁻¹ | \(H = -KA\dfrac{dT}{dx}\) |
10.14 NCERT Questions
The triple points of neon and carbon dioxide are 24.57 K and 216.55 K respectively. Express these temperatures on the Celsius and Fahrenheit scales.
Two absolute scales A and B have triple points of water defined to be 200 A and 350 B. What is the relation between \(T_A\) and \(T_B\)?
The electrical resistance in ohms of a certain thermometer varies with temperature according to the approximate law: \[R = R_o\,[1 + \alpha\,(T - T_o)]\] The resistance is 101.6 Ω at the triple-point of water 273.16 K, and 165.5 Ω at the normal melting point of lead (600.5 K). What is the temperature when the resistance is 123.4 Ω?
Answer the following:
- The triple-point of water is a standard fixed point in modern thermometry. Why? What is wrong in taking the melting point of ice and the boiling point of water as standard fixed points (as was originally done in the Celsius scale)?
- There were two fixed points in the original Celsius scale as mentioned above which were assigned the number 0 °C and 100 °C respectively. On the absolute scale, one of the fixed points is the triple-point of water, which on the Kelvin absolute scale is assigned the number 273.16 K. What is the other fixed point on this (Kelvin) scale?
- The absolute temperature (Kelvin scale) \(T\) is related to the temperature \(t_c\) on the Celsius scale by \(t_c = T - 273.15\). Why do we have 273.15 in this relation, and not 273.16?
- What is the temperature of the triple-point of water on an absolute scale whose unit interval size is equal to that of the Fahrenheit scale?
Two ideal gas thermometers A and B use oxygen and hydrogen respectively. The following observations are made:
| Temperature | Pressure thermometer A | Pressure thermometer B |
|---|---|---|
| Triple-point of water | \(1.250 \times 10^5\) Pa | \(0.200 \times 10^5\) Pa |
| Normal melting point of sulphur | \(1.797 \times 10^5\) Pa | \(0.287 \times 10^5\) Pa |
a. What is the absolute temperature of the normal melting point of sulphur as read by thermometers A and B?
b. What do you think is the reason behind the slight difference in answers of thermometers A and B? (The thermometers are not faulty.) What further procedure is needed in the experiment to reduce the discrepancy between the two readings?A steel tape 1 m long is correctly calibrated for a temperature of 27.0 °C. The length of a steel rod measured by this tape is found to be 63.0 cm on a hot day when the temperature is 45.0 °C. What is the actual length of the steel rod on that day? What is the length of the same steel rod on a day when the temperature is 27.0 °C? Coefficient of linear expansion of steel = \(1.20 \times 10^{-5}\) K⁻¹.
A large steel wheel is to be fitted on to a shaft of the same material. At 27 °C, the outer diameter of the shaft is 8.70 cm and the diameter of the central hole in the wheel is 8.69 cm. The shaft is cooled using ‘dry ice’. At what temperature of the shaft does the wheel slip on the shaft? Assume the coefficient of linear expansion of the steel to be constant over the required temperature range: \(\alpha_{steel} = 1.20 \times 10^{-5}\) K⁻¹.
A hole is drilled in a copper sheet. The diameter of the hole is 4.24 cm at 27.0 °C. What is the change in the diameter of the hole when the sheet is heated to 227 °C? Coefficient of linear expansion of copper = \(1.70 \times 10^{-5}\) K⁻¹.
A brass wire 1.8 m long at 27 °C is held taut with little tension between two rigid supports. If the wire is cooled to a temperature of −39 °C, what is the tension developed in the wire, if its diameter is 2.0 mm? Coefficient of linear expansion of brass = \(2.0 \times 10^{-5}\) K⁻¹; Young’s modulus of brass = \(0.91 \times 10^{11}\) Pa.
A brass rod of length 50 cm and diameter 3.0 mm is joined to a steel rod of the same length and diameter. What is the change in length of the combined rod at 250 °C, if the original lengths are at 40.0 °C? Is there a ‘thermal stress’ developed at the junction? The ends of the rod are free to expand. (Coefficient of linear expansion of brass = \(2.0 \times 10^{-5}\) K⁻¹, steel = \(1.2 \times 10^{-5}\) K⁻¹.)
The coefficient of volume expansion of glycerine is \(49 \times 10^{-5}\) K⁻¹. What is the fractional change in its density for a 30 °C rise in temperature?
A 10 kW drilling machine is used to drill a bore in a small aluminium block of mass 8.0 kg. How much is the rise in temperature of the block in 2.5 minutes, assuming 50% of the power is used up in heating the machine itself or lost to the surroundings? Specific heat of aluminium = 0.91 J g⁻¹ K⁻¹.
A copper block of mass 2.5 kg is heated in a furnace to a temperature of 500 °C and then placed on a large ice block. What is the maximum amount of ice that can melt? (Specific heat of copper = 0.39 J g⁻¹ K⁻¹; heat of fusion of water = 335 J g⁻¹.)
In an experiment on the specific heat of a metal, a 0.20 kg block of the metal at 150 °C is dropped in a copper calorimeter (of water equivalent 0.025 kg) containing 150 cm³ of water at 27 °C. The final temperature is 40 °C. Compute the specific heat of the metal. If heat losses to the surroundings are not negligible, is your answer greater or smaller than the actual value for the specific heat of the metal?
Given below are observations on molar specific heats at room temperature of some common gases.
| Gas | Molar specific heat (\(C_v\)) (cal mol⁻¹ K⁻¹) |
|---|---|
| Hydrogen | 4.87 |
| Nitrogen | 4.97 |
| Oxygen | 5.02 |
| Nitric oxide | 4.99 |
| Carbon monoxide | 5.01 |
| Chlorine | 6.17 |
The measured molar specific heats of these gases are markedly different from those for monatomic gases. Typically, the molar specific heat of a monatomic gas is 2.92 cal/mol K. Explain this difference. What can you infer from the somewhat larger (than the rest) value for chlorine?
A child running a temperature of 101 °F is given an antipyrin (i.e. a medicine that lowers fever) which causes an increase in the rate of evaporation of sweat from his body. If the fever is brought down to 98 °F in 20 minutes, what is the average rate of extra evaporation caused by the drug? Assume the evaporation mechanism to be the only way by which heat is lost. The mass of the child is 30 kg. The specific heat of the human body is approximately the same as that of water, and the latent heat of evaporation of water at that temperature is about 580 cal g⁻¹.
A ‘thermacole’ icebox is a cheap and efficient method for storing small quantities of cooked food in summer in particular. A cubical icebox of side 30 cm has a thickness of 5.0 cm. If 4.0 kg of ice is put in the box, estimate the amount of ice remaining after 6 h. The outside temperature is 45 °C, and the coefficient of thermal conductivity of thermacole is 0.01 J s⁻¹ m⁻¹ K⁻¹. [Heat of fusion of water = \(335 \times 10^3\) J kg⁻¹.]
A brass boiler has a base area of 0.15 m² and thickness 1.0 cm. It boils water at the rate of 6.0 kg/min when placed on a gas stove. Estimate the temperature of the part of the flame in contact with the boiler. Thermal conductivity of brass = 109 J s⁻¹ m⁻¹ K⁻¹; heat of vaporisation of water = \(2256 \times 10^3\) J kg⁻¹.
Explain why:
- a body with large reflectivity is a poor emitter;
- a brass tumbler feels much colder than a wooden tray on a chilly day;
- an optical pyrometer (for measuring high temperatures) calibrated for an ideal black body radiation gives too low a value for the temperature of a red hot iron piece in the open, but gives a correct value for the temperature when the same piece is in the furnace;
- the earth without its atmosphere would be inhospitably cold;
- heating systems based on circulation of steam are more efficient in warming a building than those based on circulation of hot water.
A body cools from 80 °C to 50 °C in 5 minutes. Calculate the time it takes to cool from 60 °C to 30 °C. The temperature of the surroundings is 20 °C.
10.15 Check Your Concepts
Two identical iron blocks are heated to the same temperature of 90 °C, but one is far more massive than the other. Are they at the same temperature? Do they contain the same amount of heat? Use this to explain the difference between “heat” and “temperature.”
A mercury-in-glass thermometer and an alcohol-in-glass thermometer are calibrated to agree exactly at the ice point and the steam point, yet they disagree at temperatures in between. A gas thermometer, however, gives the same reading whatever gas is used. Explain the reason for this difference.
Long steel railway tracks and large bridges are always built with small gaps or interlocking expansion joints between their sections. Explain, using the idea of thermal expansion, why these gaps are necessary and what could happen if they were absent.
Water has its maximum density at 4 °C and shows anomalous expansion between 0 °C and 4 °C. Explain how this unusual property allows fish and aquatic plants to survive in a pond during a harsh winter.
A gas has two molar specific heats, \(C_p\) and \(C_v\), whereas a solid is described adequately by a single specific heat. Explain the physical reason for this difference, and state which of \(C_p\) and \(C_v\) is larger for a gas, with justification.
State the principle of calorimetry. What essential condition must be satisfied by the system for this principle to hold, and how is a calorimeter designed to meet it?
Explain, in terms of the effect of pressure on boiling point, (a) why water boils below 100 °C at high altitudes and cooking is slow, and (b) why food cooks faster inside a pressure cooker.
Assertion: A burn caused by steam at 100 °C is usually more severe than a burn caused by boiling water at 100 °C. Reason: When steam condenses on the skin, it releases its latent heat of vaporisation before cooling as water. State whether the assertion and reason are correct, and whether the reason correctly explains the assertion.
Some cooking pots are made with a copper coating on the base. Using the idea of thermal conductivity, explain why copper is chosen for this purpose and what advantage it gives during cooking.
A thermos (Dewar) flask keeps its contents hot or cold for a long time. Identify the three modes of heat transfer, and explain how the design of the flask reduces the heat loss due to each mode.
From Newton’s law of cooling, the cooling curve of a hot body is steep at first and then flattens as the body approaches the surrounding temperature. Explain this shape. Also explain why a graph of \(\log_e(T_2 - T_1)\) against time is a straight line.
10.16 Practice with Numericals
A patient’s temperature is measured as 40 °C. Express this temperature (a) on the Fahrenheit scale and (b) on the Kelvin scale.
A metal rod is 1.5 m long at 20 °C. Find its length when its temperature is raised to 120 °C. (Coefficient of linear expansion of the metal = \(1.2 \times 10^{-5}\) K⁻¹.)
The coefficient of linear expansion of a certain metal is \(1.8 \times 10^{-5}\) K⁻¹. Using the relation between the linear and volume expansion coefficients, find its coefficient of volume expansion.
How much heat is required to raise the temperature of 3.0 kg of water from 20 °C to 90 °C? (Specific heat capacity of water = 4186 J kg⁻¹ K⁻¹.)
A 0.15 kg block of metal at 100 °C is dropped into 0.20 kg of water at 25 °C contained in a calorimeter of negligible heat capacity. If the final steady temperature is 30 °C, find the specific heat capacity of the metal. (Specific heat capacity of water = 4186 J kg⁻¹ K⁻¹.)
Calculate the total heat required to convert 0.50 kg of ice at 0 °C first into water at 0 °C and then into water at 100 °C. (Latent heat of fusion of ice = \(3.33 \times 10^{5}\) J kg⁻¹; specific heat capacity of water = 4186 J kg⁻¹ K⁻¹.)
One face of a slab of area 0.50 m² and thickness 4.0 cm is maintained at 30 °C and the other face at 10 °C. If the thermal conductivity of the slab material is 0.80 J s⁻¹ m⁻¹ K⁻¹, find the rate of heat flow through the slab in the steady state.
The radiation emitted by a certain star is most intense at a wavelength of \(4.0 \times 10^{-7}\) m. Using Wien’s displacement law (Wien’s constant = \(2.9 \times 10^{-3}\) m K), estimate the surface temperature of the star.
A blackbody of surface area \(1.0 \times 10^{-3}\) m² is maintained at an absolute temperature of 400 K. Using the Stefan-Boltzmann law (\(\sigma = 5.67 \times 10^{-8}\) W m⁻² K⁻⁴), calculate the power radiated by the body.
A body cools from 70 °C to 60 °C in 4 minutes when placed in surroundings at 20 °C. Using Newton’s law of cooling in its approximate (average-temperature) form, estimate the time it will take to cool from 50 °C to 40 °C in the same surroundings.