Chapter 11: Calorimetry
In the 1760s, in Glasgow, a Scottish chemist named Joseph Black kept returning to a question that everyone else thought was already settled. Winter snow, he noticed, did not vanish the moment the air turned mild. It sat there for days, melting slowly, while the air around it stayed warm. If warmth alone decided everything, the snow should have collapsed into water almost at once.
So Black did something careful. He put a lump of ice in a warm room and watched a thermometer stuck into it. The room poured heat into the ice hour after hour — and the thermometer refused to move. It stayed at the melting point the whole time the ice was turning into water. Only when the last of the ice was gone did the reading begin to climb.
Heat was going in, and the temperature was not rising. Black concluded that the heat had not disappeared; it had gone into changing the ice into water, hidden from the thermometer. He called it latent heat — heat that is there but does not show.
That single observation split two ideas that people had treated as one. Heat is the energy that flows. Temperature only tells you which way it flows. Measuring the heat itself, rather than the temperature, is what this chapter is about, and the name for that measurement is calorimetry.
Diagram
Fig. 11.0-1 — Joseph Black’s Observation On Melting Ice Two-panel illustration. Left panel: a beaker holding a mixture of ice pieces and water in a warm room, with a thermometer inserted into the mixture reading 0 °C, and arrows drawn inward towards the beaker labelled ‘heat entering from the surroundings’. Right panel: the same beaker after all the ice has melted, containing only water, with the thermometer now reading a higher value and the same inward arrows. Label the left panel ‘temperature does not change while ice melts’ and the right panel ‘temperature rises once all the ice has melted’.
11.1 Heat And Its Units
11.1.1 What Heat Is
Every substance around you is made of molecules, and those molecules are never still. In a solid they vibrate about fixed positions; in a liquid and a gas they move about far more freely. Because they are moving, the molecules carry kinetic energy.
Molecules also pull on one another. Each molecule exerts a force of attraction on its neighbours, and because of these attractive forces a molecule also carries potential energy. The kinetic energy depends on how fast the molecules move; the potential energy depends on how far apart they are held against those forces.
The sum of the potential energy and the kinetic energy of a molecule is called its internal energy. Add up the internal energy of every molecule in a substance and you have its thermal energy, measured in joule (J). The molecular picture behind these two ideas is drawn in Fig. 11.1-1.
Diagram
Fig. 11.1-1 — Molecules And Internal Energy Two side-by-side molecular sketches of the same solid block. Left sketch: molecules drawn as circles arranged in a regular pattern, each with a short motion arrow showing vibration about its position, labelled ‘kinetic energy due to random motion’. Right sketch: the same arrangement with double-headed force arrows drawn between neighbouring circles, labelled ‘potential energy due to attractive forces between molecules’. Below both, a single caption box labelled ‘internal energy of a molecule = kinetic energy + potential energy’.
When Two Bodies Touch
Take a hot body and an identical cold body of the same substance and the same mass. The hot one has more internal energy than the cold one. Now place them in contact, as in Fig. 11.1-2, and watch what happens.
The cold body warms up and the hot body cools down. In other words, the internal energy of the cold body increases while the internal energy of the hot body decreases. Energy has flowed from the hot body to the cold body, and it will keep flowing until both are equally warm.
That flowing energy is what we call heat.
Diagram
Fig. 11.1-2 — Flow Of Heat From A Hot Body To A Cold Body Two rectangular blocks drawn touching along one face. Label the left block ‘hot body, more internal energy’ and the right block ‘cold body, less internal energy’. Draw three thick arrows across the joining face pointing from the left block to the right block, labelled ‘heat energy’. Below the left block write ‘cools down’ and below the right block write ‘warms up’.
Notice what this definition does and does not say. It does not describe energy that a body possesses; it describes energy on the move. A body does not contain heat — it contains internal energy, and heat is the name for the part of that energy which crosses from one body to another.
Heat is a form of energy, and like every other form of energy it can be measured. The measurement of the quantity of heat is called calorimetry, and the whole of this chapter is built on it.
11.1.2 Units Of Heat
Since heat is energy, it is measured in the same unit as every other kind of energy. The SI unit of heat is the joule (symbol J).
There is an older unit that is still in wide use, and it was defined directly from an experiment with water rather than from mechanical work. It is the calorie (symbol cal).
This definition quietly assumes something that is not quite true. It assumes that the heat needed to warm \(1\\ \\text{g}\) of water through \(1^\\circ\\text{C}\) is the same whatever temperature you start from. Water does not expand evenly as it is heated, so the amount actually differs a little from one starting temperature to another. To remove the ambiguity, the calorie is defined precisely over one named degree, and is then called the \(15^\\circ\\text{C}\) calorie.
Careful measurement links the calorie to the joule:
\[1\\ \\text{cal} = 4.186\\ \\text{J} \\approx 4.2\\ \\text{J}\]
The value \(4.2\\ \\text{J}\) is the one used for calculations throughout this chapter.
The Kilo-calorie
The calorie is a small unit of heat, so for larger quantities a bigger one is convenient. That unit is the kilo-calorie (symbol kcal), where
\[1\\ \\text{kcal} = 1000\\ \\text{cal} = 4200\\ \\text{J}\\ \\text{(nearly)}\]
The kilo-calorie is the unit generally used for stating the energy value of foods.
11.2 Temperature And Its Units
11.2.1 What Temperature Is
Keep a hot body in contact with a cold body and heat flows from the hot one to the cold one. The hot body gets cooled and the cold body gets warmed. Something about the two bodies decides which way the energy goes, and that something is their temperature.
The body which imparts heat is said to be at a higher temperature than the body which receives heat. So temperature determines the direction of flow of heat for bodies in contact. It does not tell you how much energy will flow — only which way.
When a body receives heat energy, the particles constituting it start vibrating more vigorously, and so its temperature rises. This holds provided its physical state and its dimensions remain unchanged. If the body is melting or boiling instead, the heat goes elsewhere, and that important exception is taken up later in this chapter.
Same Temperature Does Not Mean Same Energy
If there is no transfer of heat between two bodies placed in contact, they are said to be at the same temperature. That is a useful test, but it is easily over-read. Bodies at the same temperature do not necessarily hold equal amounts of thermal energy.
Experiment settles the point. Give the same quantity of heat energy to different bodies and the rise in temperature turns out to be different for each of them. So the quantity of thermal energy contained in a body does not depend on its temperature alone. A mug of boiling water and a swimming pool at the same temperature are worlds apart in the energy they hold.
11.2.2 Units Of Temperature
The SI unit of temperature is the kelvin (symbol K). The other unit in everyday use is the degree celsius (symbol \(^\\circ\\text{C}\)). The two scales are compared in Fig. 11.2-1.
The two are related in a simple way. If \(t\) is the temperature in degree celsius and \(T\) the temperature in kelvin, then
\[T\\ \\text{K} = 273 + t\\ ^\\circ\\text{C}\]
or more precisely, \(T\\ \\text{K} = 273.15 + t\\ ^\\circ\\text{C}\). Thus, by adding 273 (or 273.15) to the temperature in degree celsius, we get the temperature in kelvin.
Diagram
Fig. 11.2-1 — The Celsius And Kelvin Scales Compared Two vertical scales drawn side by side and aligned horizontally at matching temperatures. Left scale headed ‘celsius (°C)’ with marks at −273, 0, 27 and 100. Right scale headed ‘kelvin (K)’ with marks at 0, 273, 300 and 373 drawn level with the corresponding celsius marks. Join the levels with faint horizontal dashed lines. Label the −273 °C / 0 K level ‘absolute zero’, the 0 °C / 273 K level ‘ice melts’ and the 100 °C / 373 K level ‘water boils at one atmosphere’. Alongside, mark the equal spacing of one degree on both scales with a small bracket labelled ‘1 °C interval = 1 K interval’.
The zero of the kelvin scale, called absolute zero or \(0\\ \\text{K}\), is the temperature at which the molecular motion ceases. It is equal to \(-273^\\circ\\text{C}\) (or \(-273.15^\\circ\\text{C}\)). Since molecules cannot move less than not at all, nothing can be colder than this, and so no negative temperature is possible on the kelvin scale.
Now look again at Fig. 11.2-1 and at what shifting by 273 actually does. It moves the zero of the scale; it does not stretch or squeeze the divisions. Since the temperatures on the two scales differ by a constant number, a degree — that is, a temperature difference — is the same on both the celsius and the kelvin scales:
\[\\Delta t\\ ^\\circ\\text{C} = \\Delta T\\ \\text{K}\]
where \(\\Delta t\) is a temperature difference in degree celsius and \(\\Delta T\) the same difference in kelvin. This is why a rise of \(20^\\circ\\text{C}\) is also a rise of \(20\\ \\text{K}\), even though \(20^\\circ\\text{C}\) itself is \(293\\ \\text{K}\).
11.2.3 Difference Between Heat And Temperature
Heat and temperature are used loosely in ordinary speech as though they meant the same thing. In physics they are different quantities, with different units and different instruments. Table 11.1 sets them against each other.
The last row is worth pausing on, because it is the one students most often get wrong. Heat adds up when bodies are brought together; temperatures do not. Mix two equal buckets of water at \(30^\\circ\\text{C}\) and you get twice the heat, but still \(30^\\circ\\text{C}\).
11.3 Factors Affecting The Heat Absorbed By A Body
Suppose you want to warm something up. How much heat energy will it take? The quantity of heat energy absorbed to increase the temperature of a body depends on three factors: the mass of the body, the increase in its temperature, and the material of which it is made.
Each of these was found by experiment, and each can be checked one at a time by holding the other two fixed. Take them in turn.
The First Factor: Mass
Objects of different mass made from the same substance absorb different amounts of heat energy to raise their temperature by the same amount.
For example, to raise the temperature of \(1\\ \\text{kg}\) of water by \(1^\\circ\\text{C}\), the heat energy absorbed is \(1\\ \\text{kcal}\), while to raise the temperature of \(2\\ \\text{kg}\) of water by \(1^\\circ\\text{C}\) it is \(2\\ \\text{kcal}\). Double the mass and you double the heat. Thus the amount of heat energy \(Q\) absorbed is directly proportional to the mass \(m\) of the object:
\[Q \\propto m\]
The Second Factor: Rise In Temperature
Objects of equal mass made from the same substance absorb different amounts of heat energy to raise their temperature by different amounts.
For example, to raise the temperature of \(1\\ \\text{kg}\) of water by \(1^\\circ\\text{C}\), the heat energy absorbed is \(1\\ \\text{kcal}\), while to raise the temperature of the same mass of water by \(2^\\circ\\text{C}\), the heat energy required is \(2\\ \\text{kcal}\). Thus the heat energy \(Q\) absorbed is directly proportional to the rise in temperature \(\\Delta t\):
\[Q \\propto \\Delta t\]
The Third Factor: The Material
Objects of the same mass but made from different substances absorb different amounts of heat energy to raise their temperature by the same amount.
For example, if equal masses of water and copper are heated through \(1^\\circ\\text{C}\), the amount of heat absorbed by the water is nearly ten times the amount absorbed by the copper. The difference is not small, and it is not accidental — it is a property of the substance itself. This is what Fig. 11.3-1 is set up to show.
Diagram
Fig. 11.3-1 — Equal Masses Of Water And Copper Given The Same Heat Two identical burners of equal flame drawn side by side. On the left burner, a beaker containing 1 kg of water with a thermometer in it; on the right burner, a block of copper of mass 1 kg with a thermometer inserted into a hole in it. Label both ‘same mass, same rate of heating, same time’. Show the left thermometer reading a small rise and the right thermometer reading a much larger rise, with the labels ‘water: small rise in temperature’ and ‘copper: large rise in temperature’.
Thus the amount of heat energy absorbed depends on the nature of the substance of the object. This characteristic of the substance is expressed in terms of its specific heat capacity, written \(c\), and it is taken up in full in section 11.5.
Putting The Three Together
From the first two observations,
\[Q \\propto m \\quad \\text{and} \\quad Q \\propto \\Delta t\]
and since a body needs more heat when either the mass or the temperature rise is larger, the two proportionalities combine into one:
\[Q \\propto m,\\Delta t\]
Replacing the proportionality by an equality brings in a constant, and the third observation tells us what that constant must be — something that differs from substance to substance. Writing it as \(c\),
\[Q = c,m,\\Delta t\]
where \(Q\) is the heat energy absorbed in J, \(m\) is the mass of the body in kg, \(\\Delta t\) is the rise in temperature in \(^\\circ\\text{C}\) (or K, since the two are equal for a difference), and \(c\) is the specific heat capacity of the substance in \(\\text{J kg}^{-1},\\text{K}^{-1}\). The relation holds as long as the body is only getting warmer — not while it is melting or boiling.
Here \(c\) is the constant of proportionality, called the specific heat capacity of the substance. It is a characteristic of the substance and is different for different substances.
So the amount of heat energy contained in a body depends on its mass, its temperature and the material of the body — a statement worth holding on to, because it is exactly why temperature by itself never tells you how much energy is in something.
11.4 Heat Capacity
11.4.1 Meaning Of Heat Capacity
From everyday experience we find that different bodies require different amounts of heat energy for an equal rise in their temperatures. A spoon left in hot water warms through in seconds; a full pan on the same stove takes minutes. This property of a body is expressed in terms of its heat capacity.
Heat capacity is denoted by the symbol \(C'\). Written as a relation,
\[C' = \\frac{\\text{amount of heat energy supplied }(Q)}{\\text{rise in temperature }(\\Delta t)}\]
Read this definition carefully. It speaks of the whole body, not of the substance. Two blocks cut from the same copper bar, one twice as heavy as the other, have different heat capacities, as Fig. 11.4-1 shows — heat capacity depends on how much of the body there is.
Diagram
Fig. 11.4-1 — Heat Capacity Depends On The Whole Body Two copper blocks drawn side by side, the right one twice the size of the left. Label the left block ‘copper, mass 1 kg’ and the right block ‘copper, mass 2 kg’. Above each block draw an arrow pointing down into it labelled ‘heat supplied to raise temperature by 1 K’, with the left arrow labelled ‘390 J’ and the right arrow, drawn twice as thick, labelled ‘780 J’. Below, write ‘same substance, different heat capacity’.
The definition also works in reverse. The body gives out the same amount of heat energy, that is \(C'\), when it is cooled through \(1^\\circ\\text{C}\) (or \(1\\ \\text{K}\)). Heat capacity measures how much energy a body takes in per degree on the way up, and equally how much it hands back per degree on the way down.
If on imparting an amount of heat \(Q\) to a body its temperature rises through \(\\Delta t\\ ^\\circ\\text{C}\) (or \(\\Delta t\\ \\text{K}\)), then
\[C' = \\frac{Q}{\\Delta t}\]
where \(C'\) is the heat capacity of the body in \(\\text{J K}^{-1}\), \(Q\) is the heat energy supplied in J, and \(\\Delta t\) is the rise in temperature in K. As in section 11.3, this applies while the body is only warming — not while it is changing state.
11.4.2 Units Of Heat Capacity
Heat capacity is a quantity of heat divided by a temperature difference, so its unit is a unit of energy divided by a unit of temperature. The SI unit of heat capacity is joule per kelvin (\(\\text{J K}^{-1}\)).
It is also written as joule per degree celsius (\(\\text{J }^\\circ\\text{C}^{-1}\)), and the two are numerically identical. That is not a coincidence: a temperature difference has the same value on the celsius and the kelvin scales, as section 11.2 established, so dividing by \(1\\ \\text{K}\) and dividing by \(1^\\circ\\text{C}\) come to the same thing.
The other common units of heat capacity are \(\\text{cal }^\\circ\\text{C}^{-1}\) (or \(\\text{cal K}^{-1}\)) and \(\\text{kcal }^\\circ\\text{C}^{-1}\) (or \(\\text{kcal K}^{-1}\)). They are related as
\[1\\ \\text{kcal }^\\circ\\text{C}^{-1} = 1000\\ \\text{cal }^\\circ\\text{C}^{-1}\] \[1\\ \\text{cal K}^{-1} = 4.2\\ \\text{J K}^{-1}\]
A stated heat capacity is a promise about a single degree. If the heat capacity of an object is \(30\\ \\text{J K}^{-1}\), it means that \(30\\ \\text{J}\) of heat energy is required to raise the temperature of that object by \(1\\ \\text{K}\) (or \(1^\\circ\\text{C}\)) — and that the object will release \(30\\ \\text{J}\) for every kelvin it cools.
11.5 Specific Heat Capacity
11.5.1 Meaning Of Specific Heat Capacity
Heat capacity tells you about one particular body — this pan, that spoon. Change the size of the body and the number changes with it. To describe the substance rather than the object, we take the heat capacity of a body and share it out over its mass.
Heat capacity of a body when expressed for unit mass is called the specific heat capacity of the substance of that body. It is denoted by the symbol \(c\). Because the mass has been divided out, \(c\) is a characteristic property of the substance and is different for different substances.
\[c = \\frac{\\text{heat capacity of body }C'}{\\text{mass of the body }m}\]
where \(c\) is the specific heat capacity in \(\\text{J kg}^{-1},\\text{K}^{-1}\), \(C'\) is the heat capacity of the body in \(\\text{J K}^{-1}\), and \(m\) is the mass of the body in kg.
Section 11.4 gave \(C' = Q / \\Delta t\). Substituting that into the relation above replaces the heat capacity by the heat and the temperature rise that produced it:
\[c = \\frac{Q}{m \\times \\Delta t}\]
which is the working form of the definition.
In words, then,
\[c = \\frac{\\text{amount of heat energy supplied}}{\\text{mass} \\times \\text{rise in temperature}}\]
Rearranging this to make \(Q\) the subject gives the relation you will use in almost every calculation in this chapter:
\[Q = m \\times c \\times \\Delta t\]
where \(Q\) is the heat energy in J, \(m\) the mass in kg, \(c\) the specific heat capacity in \(\\text{J kg}^{-1},\\text{K}^{-1}\), and \(\\Delta t\) the rise in temperature in K. It holds only while the temperature is changing, and not during a change of state.
This single relation contains all three findings of section 11.3. The quantity of heat energy absorbed to increase the temperature of a body depends on the mass of the body, \(Q \\propto m\); on the increase in temperature of the body, \(Q \\propto \\Delta t\); and on the material or substance of the body, \(Q \\propto c\).
11.5.2 Units Of Specific Heat Capacity
The SI unit of specific heat capacity is joule per kilogram per kelvin (\(\\text{J kg}^{-1},\\text{K}^{-1}\)), also written as joule per kilogram per degree celsius (\(\\text{J kg}^{-1},^\\circ\\text{C}^{-1}\)). The two forms are identical in value, because a temperature difference is the same on both scales.
The other units in use are \(\\text{cal g}^{-1},^\\circ\\text{C}^{-1}\) and \(\\text{kcal kg}^{-1},^\\circ\\text{C}^{-1}\). They are related as
\[1\\ \\text{cal g}^{-1},^\\circ\\text{C}^{-1} = 1\\ \\text{kcal kg}^{-1},\\text{K}^{-1} = 4.2 \\times 10^{3}\\ \\text{J kg}^{-1},\\text{K}^{-1}\]
Notice that going from the gram unit to the kilogram unit multiplies the number by 1000, since a kilogram holds a thousand grams. So a specific heat capacity of \(0.4\\ \\text{J g}^{-1},\\text{K}^{-1}\) is the same as \(400\\ \\text{J kg}^{-1},\\text{K}^{-1}\) — the same physical property, written two ways.
A value of specific heat capacity, like a value of heat capacity, is a statement about one degree. If the specific heat capacity of copper is \(0.4\\ \\text{J g}^{-1},\\text{K}^{-1}\), it means that the heat energy required to raise the temperature of \(1\\ \\text{g}\) of copper by \(1\\ \\text{K}\) (or \(1^\\circ\\text{C}\)) is \(0.4\\ \\text{J}\).
11.5.3 Relation Between Heat Capacity And Specific Heat Capacity
The definition of specific heat capacity can be turned round. If \(c\) is the heat capacity of each kilogram, then \(m\) kilograms must have \(m\) times as much:
\[C' = m \\times c\]
where \(C'\) is the heat capacity of the body in \(\\text{J K}^{-1}\), \(m\) its mass in kg, and \(c\) the specific heat capacity of its substance in \(\\text{J kg}^{-1},\\text{K}^{-1}\). This relation is the bridge between the property of a body and the property of a substance, and it is the one to reach for whenever a question gives you one and asks for the other.
11.5.4 Distinction Between Heat Capacity And Specific Heat Capacity
The two quantities have almost the same name and are constantly confused. Table 11.2 sets them side by side.
The quickest way to keep them apart is to look at the unit. A kilogram in the unit means the quantity belongs to the substance; no kilogram means it belongs to the body in front of you.
11.5.5 Specific Heat Capacity Of Some Common Substances
The specific heat capacity of a substance is its characteristic property, and it is different for different substances. Table 11.3 gives values for some substances you will meet often.
Table 11.3 — Specific heat capacity of some common substances
| Substance | Specific heat capacity in \(\\text{J kg}^{-1},\\text{K}^{-1}\) | Specific heat capacity in \(\\text{cal g}^{-1},^\\circ\\text{C}^{-1}\) |
|---|---|---|
| Lead | 130 | 0.031 |
| Mercury | 139 | 0.033 |
| Brass | 380 | 0.090 |
| Zinc | 391 | 0.093 |
| Copper | 385 | 0.092 |
| Iron (cast) | 554 | 0.132 |
| Glass | 792 | 0.19 |
| Aluminium | 882 | 0.21 |
| Kerosene oil | 2100 | 0.50 |
| Ice | 2100 | 0.50 |
| Sea water | 3900 | 0.93 |
| Water | 4180 | 1.0 |
Reading The Pattern In The Table
Run your eye down the middle column. The metals cluster at the low end and the liquids at the high end, and there is a reason for that. Usually a good conductor of heat has a low specific heat capacity, while a bad conductor has a high specific heat capacity.
For example, the specific heat capacity of copper is \(0.4\\ \\text{J g}^{-1},\\text{K}^{-1}\) and that of water is \(4.2\\ \\text{J g}^{-1},\\text{K}^{-1}\). Copper is a good conductor of heat, but water is a bad (or poor) conductor of heat.
Now heat equal masses of two different substances on the same burner, so that the rate of heat supply is the same. After the same time interval, the rise in temperature for the two substances is different. This is because of their different specific heat capacities. The substance with a low specific heat capacity shows a rapid and high rise in temperature, and is thus a better conductor of heat than the substance with a high specific heat capacity, which shows a slow and small rise in temperature.
The Special Case Of Water
Water sits at the bottom of Table 11.3 with an unusually high specific heat capacity, \(4200\\ \\text{J kg}^{-1},\\text{K}^{-1}\). More precisely the value is \(4180\\ \\text{J kg}^{-1},\\text{K}^{-1}\), but for convenience it is taken as \(4200\\ \\text{J kg}^{-1},\\text{K}^{-1}\), and that is the value used in the calculations of this chapter. Section 11.8 is given over to the consequences of it, and they reach from your kitchen to the climate of a coastline.
The specific heat capacity of the same substance is different in its different phases. The specific heat capacity of water is \(4200\\ \\text{J kg}^{-1},\\text{K}^{-1}\), of ice is \(2100\\ \\text{J kg}^{-1},\\text{K}^{-1}\), and of steam is \(460\\ \\text{J kg}^{-1},\\text{K}^{-1}\). Ice, water and steam are chemically the same substance, yet each needs a different amount of heat per kilogram per kelvin — so a problem that carries a substance through a change of state must switch to the correct value at the right moment.
Among all substances, the specific heat capacity is maximum for hydrogen, which is of the order of \(14 \\times 10^{3}\\ \\text{J kg}^{-1},\\text{K}^{-1}\) — more than three times that of water.
11.6 The Calorimeter
If heat is to be measured rather than merely talked about, there must be a vessel in which the measurement can be trusted. That vessel is the calorimeter.
A calorimeter is a cylindrical vessel which is used to measure the amount of heat gained (or lost) by a body when it is mixed with another body or substance. Its construction is shown in Fig. 11.6-1, and every part of it is there to solve one particular problem.
Diagram
Fig. 11.6-1 — A Calorimeter Vertical sectional view of a calorimeter. Draw an outer wooden box (the jacket) with a cylindrical copper vessel standing centrally inside it on a small insulating support. Fill the space between the vessel and the jacket with a hatched region labelled ‘wool’. Cover the copper vessel with a wooden lid having two holes; through one hole pass a thermometer with its bulb dipping into the liquid inside the vessel, and through the other pass a stirrer that reaches into the liquid. Label, with leader lines: thermometer, stirrer, wooden lid, copper vessel, wooden jacket, wool.
Why The Vessel Is Made Of Copper
The calorimeter is made up of a thin sheet of copper, for two reasons that pull in the same direction.
First, copper is a good conductor of heat, so the vessel soon acquires the temperature of its contents. If it lagged behind, the thermometer reading would not describe the mixture as a whole.
Second, copper has a low specific heat capacity, so the amount of heat energy taken by the calorimeter itself from the contents to acquire its temperature is very small. Every joule the vessel absorbs is a joule missing from the exchange being measured, and the design keeps that theft to a minimum. Section 11.9 returns to the same argument from the side of heat capacity.
How The Three Escape Routes Are Closed
Heat can leak out of the vessel in three ways, and the calorimeter is built to block each of them separately, as set out in FC 11.1.
The outer and inner surfaces of the vessel are polished, so as to reduce the loss of heat due to radiation. A shiny surface is a poor radiator, so less energy leaves the walls as radiation.
For insulation, the vessel is placed inside a wooden jacket. The space between the calorimeter and the jacket is filled with some poor conductor such as wool or cotton, to avoid the heat loss by conduction. Wool traps air, and neither wool nor still air passes heat readily.
It is covered with a wooden lid to avoid the heat loss by convection. Without the lid, warm air above the liquid would rise and carry energy away. The lid has two holes, one for the stirrer, used to mix the contents properly, and the other for the thermometer, used to measure the temperature of its contents. Stirring matters, because a reading taken from an unmixed liquid describes only the layer around the bulb.
Flowchart
FC 11.1 — How A Calorimeter Blocks Each Route Of Heat Loss Heat tending to escape from the contents → branch 1: by conduction → through the walls and the surrounding space → blocked by the wool packing inside the wooden jacket; branch 2: by convection → through air currents above the liquid → blocked by the wooden lid; branch 3: by radiation → from the vessel surfaces → blocked by polishing the inner and outer surfaces
11.7 Principle Of Method Of Mixtures
When a hot body is mixed with a cold body, or simply kept in contact with it, heat energy passes from the hot body to the cold body, till both the bodies attain the same temperature. Section 11.1 established the direction of that flow; here we account for the amount.
Suppose no heat energy is lost to the surroundings — that is, the system is perfectly insulated. Then every joule that leaves the hot body arrives in the cold body. Nothing is created on the way and nothing goes missing. This is drawn in Fig. 11.7-1.
Diagram
Fig. 11.7-1 — Exchange Of Heat In A Mixture Two-panel figure. Left panel headed ‘before mixing’: a beaker of hot liquid labelled ‘mass m₁, specific heat capacity c₁, temperature t₁’ beside a beaker of cold liquid labelled ‘mass m₂, specific heat capacity c₂, temperature t₂’, with t₂ marked as lower than t₁. Right panel headed ‘after mixing’: one beaker containing the combined liquid, labelled ‘common final temperature t’, with a curved arrow drawn from the hot beaker of the left panel into the cold beaker labelled ‘heat energy lost by hot body = heat energy gained by cold body’. Mark the whole arrangement ‘perfectly insulated — no heat lost to the surroundings’.
This is called the principle of method of mixtures, or the principle of calorimetry. It is not a new law of nature. It is the law of conservation of energy applied to a case where the only energy changing hands is heat.
The Mathematical Statement
Let a substance A of mass \(m\_1\) and specific heat capacity \(c\_1\), at a higher temperature \(t\_1\), be mixed with another substance B of mass \(m\_2\) and specific heat capacity \(c\_2\), at a lower temperature \(t\_2\), so that \(t\_2 < t\_1\). Let the final temperature of the mixture after the heat exchange become \(t\).
Since A cools and B warms, and both end at \(t\),
\[\\text{fall in temperature of A} = t\_1 - t, \\qquad \\text{rise in temperature of B} = t - t\_2\]
Both of these are written so as to come out positive, which is why the order of the terms differs in the two brackets. Now apply \(Q = m c, \\Delta t\) to each substance in turn:
\[\\text{heat energy lost by A} = m\_1 c\_1 (t\_1 - t)\] \[\\text{heat energy gained by B} = m\_2 c\_2 (t - t\_2)\]
If no heat energy is lost to the surroundings, then by the principle of method of mixtures these two are equal:
\[m\_1 c\_1 (t\_1 - t) = m\_2 c\_2 (t - t\_2)\]
where \(m\_1\) and \(m\_2\) are masses in kg, \(c\_1\) and \(c\_2\) are specific heat capacities in \(\\text{J kg}^{-1},\\text{K}^{-1}\), and \(t\_1\), \(t\_2\) and \(t\) are temperatures in \(^\\circ\\text{C}\) (or, equally, in K, since only differences appear). The relation holds only while both substances stay in the same state throughout — no melting, no boiling — and only when the loss of heat to the surroundings can be neglected.
The above expression can be used to calculate whichever one of these quantities is unknown. Note also that the masses may be left in gram on both sides, provided the specific heat capacities are then in \(\\text{J g}^{-1},\\text{K}^{-1}\), since the units must match across the equation.
11.8 Consequences Of The High Specific Heat Capacity Of Water
Water has an unusually high specific heat capacity, \(4200\\ \\text{J kg}^{-1},\\text{K}^{-1}\). A single number in a table can look like a small matter. It is not. It shapes the climate of coastlines, the design of engines, the survival of a winter crop and the steadiness of your own body temperature. Some of its consequences are set out below.
The Climate Near The Sea Shore Is Moderate
The specific heat capacity of water is very high, \(1000\\ \\text{cal kg}^{-1},^\\circ\\text{C}^{-1}\) or \(4200\\ \\text{J kg}^{-1},\\text{K}^{-1}\). It is about five times as high as that of sand.
Hence the heat energy required for the same rise in temperature by a certain mass of water is nearly five times more than that required by the same mass of sand. Similarly, a certain mass of water imparts nearly five times more heat energy than that given by the same mass of sand for the same fall in temperature. As such, sand — or earth — gets heated or cooled more rapidly as compared to water under similar conditions, that is, under the same exposure to the sun.
Thus, near the sea shore, there develops a large difference in temperature between the land and sea surfaces, due to which convection air currents are set up. The cold air blows from the land towards the sea during the night, which is the land breeze, and during the day cold air blows from the sea towards the land, which is the sea breeze. These are drawn in Fig. 11.8-1, and the chain of cause and effect behind the daytime one is set out in FC 11.2. These breezes make the climate near the sea shore moderate.
Diagram
Fig. 11.8-1 — Land Breeze And Sea Breeze Two panels, each showing a coastline in section with land on the left and sea on the right, and the sky above. Left panel headed ‘daytime — sea breeze’: show the sun above the land, label the land ‘heats up quickly’ and the sea ‘heats up slowly’, draw a rising arrow over the land, a descending arrow over the sea, and a bold horizontal arrow at ground level pointing from the sea towards the land labelled ‘sea breeze’. Right panel headed ‘night — land breeze’: no sun, label the land ‘cools quickly’ and the sea ‘stays warm’, draw a rising arrow over the sea, a descending arrow over the land, and a bold horizontal arrow at ground level pointing from the land towards the sea labelled ‘land breeze’.
Flowchart
FC 11.2 — Why A Sea Breeze Blows During The Day Sun heats land and sea equally → land has the lower specific heat capacity, so its temperature rises faster → air above the land becomes warmer and rises → air pressure near the land surface falls → cooler air from over the sea moves in to take its place → this inward flow is the sea breeze → the incoming cool air keeps the shore from getting too hot
Hot Water Bottles Are Used For Fomentation
Fomentation means applying warmth to a sore or stiff part of the body. The reason a hot water bottle is used is that water does not cool quickly, because of its high specific heat capacity. It therefore provides more heat energy for fomentation, and over a longer period, than the same mass of most other substances would.
Water Is Used As An Effective Coolant
By allowing water to flow in pipes around the heated parts of a machine, heat energy from such parts is removed. Water in pipes can extract more heat from the surroundings without much rise in its own temperature, because of its high specific heat capacity. This is why radiators in cars and in generators use water for cooling.
In Cold Countries, Water Is Used As A Heat Reservoir
In cold countries, water is used as a heat reservoir for bottles of wine and juice, to prevent them from freezing. The reason is that water, due to its high specific heat capacity, can impart a large amount of heat before reaching its own freezing point. Hence bottles kept in water remain warm and they do not freeze even when there is a considerable fall in the surrounding temperature.
Farmers Fill Their Fields With Water To Protect Crops From Frost
On a cold winter night, when the atmospheric temperature falls below \(0^\\circ\\text{C}\), water in the fine capillaries of plants freezes, so the veins burst due to the increase in volume of water on freezing. As a result the plants die and the crop gets destroyed.
In order to save the crop on such nights, farmers fill their fields with water. Water has a high specific heat capacity, so it gives out a great deal of heat for every degree it falls, and it does not allow the temperature in the surrounding area of the plants to fall up to \(0^\\circ\\text{C}\).
Plants And Animals Have A High Content Of Water In Their Bodies
All plants and animals have nearly 80 to 90 per cent of water in their bodies. This water, because of its high specific heat capacity, helps to maintain nearly the same temperature of their body in all seasons. A body largely made of water is a body that resists sudden change in temperature.
11.9 Some Examples Of High And Low Heat Capacity
Section 11.8 was about the specific heat capacity of a substance. This section is about the heat capacity of a particular object, which the designer controls by choosing both the substance and the mass. Sometimes a large heat capacity is wanted, and sometimes a very small one.
The Base Of A Cooking Pan Is Made Thick
By making the base of a cooking pan thick, its mass is increased, and since \(C' = m \\times c\), its heat capacity becomes large. Because of this it gets heated slowly, and it then imparts sufficient heat energy at a slow, steady rate to the food for its proper cooking. After cooking, the thick base also keeps the food warm for a longer period, since it has a large store of energy to give back as it cools. The two designs are compared in Fig. 11.9-1.
Diagram
Fig. 11.9-1 — Thick And Thin Pan Bases Two cooking pans drawn in vertical cross-section, side by side, of the same width and holding the same food. Left pan: a thick base, labelled ‘thick base — large mass, large heat capacity’, with a flame beneath and a caption ‘heats slowly, gives heat to the food at a steady rate, stays warm long after the flame is off’. Right pan: a thin base, labelled ‘thin base — small mass, small heat capacity’, with an identical flame beneath and a caption ‘heats quickly, food is liable to burn, cools quickly’.
The Base Of An Electric Press Is Made Thick And Heavy
By doing so, the heat capacity of the base of the electric press becomes large, and it remains hot for a long duration even after the current is switched off. That is exactly what pressing clothes needs — a steady supply of heat rather than a quick burst.
It is also why a press must never be left face-down on a cloth or a table after being switched off. The base is still holding a large quantity of heat energy and will go on giving it out for many minutes; it belongs on its heel-rest or a heat-proof stand until it is cool.
The Calorimeter Is Made Of A Thin Sheet Of Copper
Here the requirement is the opposite one. The specific heat capacity of copper is low, and by making the vessel thin its mass, and hence its heat capacity, becomes low as well. It therefore takes a negligible amount of heat from the contents in order to attain their temperature, and the heat we set out to measure is not quietly swallowed by the measuring vessel. The full design was described in section 11.6.
11.10 Change Of Phase
There are three states, or phases, of matter, namely solid, liquid and gas. A substance can exist in all the three phases under different conditions of temperature and pressure.
Water is the everyday example. Ice, which is a solid, at \(0^\\circ\\text{C}\) when heated becomes water, a liquid, at \(0^\\circ\\text{C}\); and that water on further heating changes to steam, a gas, at \(100^\\circ\\text{C}\). Thus, at one atmospheric pressure, water is found in all the three phases at different temperatures.
The process of change from one state to another at a constant temperature is called the change of phase. It is brought about by the exchange of heat — heat must go in, or come out, for the change to happen at all. Fig. 11.10-1 collects the different processes of change of phase and the names they carry.
Diagram
Fig. 11.10-1 — Processes Of Change Of Phase Three labelled boxes in a horizontal row: SOLID on the left, LIQUID in the middle, GAS on the right. Between SOLID and LIQUID draw two curved arrows: the upper one pointing from SOLID to LIQUID labelled ‘melting’, the lower one pointing from LIQUID to SOLID labelled ‘freezing’. Between LIQUID and GAS draw two curved arrows: the upper one pointing from LIQUID to GAS labelled ‘vaporisation’, the lower one pointing from GAS to LIQUID labelled ‘condensation’. Above all three boxes draw a long curved arrow from SOLID directly to GAS labelled ‘sublimation’, and below them a long curved arrow from GAS directly to SOLID labelled ‘solidification’. Mark the three upper-route arrows (melting, vaporisation, sublimation) with a small ‘heat absorbed’ tag and the three lower-route arrows (freezing, condensation, solidification) with a small ‘heat liberated’ tag.
The change from solid to liquid phase is known as melting, while the reverse change from liquid to solid is called freezing.
The change from liquid to vapour is known as vaporisation, while the reverse change from gas, or vapour, to liquid is called condensation, also known as liquefaction.
A substance need not pass through the liquid phase at all. The direct change from solid to vapour is called sublimation, and the reverse change from vapour to solid is called solidification.
Read Fig. 11.10-1 once more as a picture of energy rather than of names. Every arrow on the upper route, going from a more tightly held phase to a freer one, needs heat to be absorbed. Every arrow on the lower route, going the other way, liberates heat. That symmetry runs through the rest of this chapter.
11.11 Melting And Freezing
11.11.1 Melting, Freezing And Melting Point
The reverse change from liquid to solid phase, with the liberation of heat at a constant temperature, is called freezing, and the temperature at which a liquid freezes to a solid is called its freezing point.
Thus heat energy is absorbed during melting and it is rejected during freezing, and in both cases at a constant temperature. Fig. 11.11-1 shows the pair of changes for ice and water.
Diagram
Fig. 11.11-1 — Melting And Freezing Of Ice At 0 °C Two labelled boxes side by side: ‘0 °C ice’ on the left and ‘0 °C water’ on the right. Draw an arrow from the left box to the right box above the pair, labelled on two lines ‘melting’ and ‘absorption of heat’. Draw a second arrow from the right box back to the left box below the pair, labelled on two lines ‘freezing’ and ‘liberation of heat’. Mark clearly that the temperature stays 0 °C throughout both changes.
Two facts follow, and both are used constantly in calculations. First, for a pure substance the melting point and the freezing point are identical — ice melts at \(0^\\circ\\text{C}\) and water freezes at \(0^\\circ\\text{C}\). Second, for a given mass of a substance, the amount of heat energy absorbed during melting is the same as that liberated during freezing.
11.11.2 Heating Curve Of Ice During Melting
To draw the heating curve of ice, we perform the experiment described below. The apparatus is shown in Fig. 11.11-2.
The Experiment
Take a boiling test tube and fill it half with ice chips. Insert a thermometer gently into the ice, taking care that its bulb does not touch the walls of the test tube — if it did, it would read the glass rather than the ice. Note the temperature of the ice. It will be \(0^\\circ\\text{C}\).
Heat the bottom of the test tube slowly over a flame, or by immersing the test tube in a beaker containing hot water, and note the temperature after every half minute. Continue till the temperature of the water formed after the melting of the ice increases to \(30^\\circ\\text{C}\). Heating of this kind is done in a laboratory and under a teacher’s supervision.
Diagram
Fig. 11.11-2 — Apparatus For The Heating Curve Of Ice A boiling test tube held upright in a clamp on a stand, filled to about half its depth with small ice chips. A thermometer is inserted into the ice through the mouth of the tube, its bulb well inside the ice and clear of the walls; label this ‘bulb must not touch the walls’. Below the test tube, a burner with a small flame; alongside, a stopwatch. Label the test tube ‘ice chips’, the thermometer ‘thermometer’, and the flame ‘heat supplied slowly at a steady rate’.
Plot a graph for temperature against time, taking temperature on the Y-axis and time on the X-axis. The graph so obtained is shown in Fig. 11.11-3. It is called the heating curve for ice.
Diagram
Fig. 11.11-3 — Heating Curve For Ice Initially At 0 °C A line graph. X-axis labelled ‘time (s)’ with marks at 0, 30, 60, 90, 120, 150, 180, 210. Y-axis labelled ‘temperature (°C)’ with marks at 0, 10, 20, 30. Draw a horizontal line at 0 °C from point A at time 0 to point B at 120 s, labelled along its length ‘melting ice + water’. From B draw a straight line rising steeply to point C at 30 °C at about 210 s, labelled alongside ‘water’. Label the points A, B and C, and mark the level of the horizontal part clearly as 0 °C.
From the graph it is clear that the temperature of the ice remains constant, equal to \(0^\\circ\\text{C}\), in the part AB, till the whole of the ice melts. The constant temperature at which the ice melts, that is \(0^\\circ\\text{C}\), is the melting point of ice. The heat supplied during this time is being used in melting the ice. After this, the temperature of the water formed by the melted ice begins to rise from \(0^\\circ\\text{C}\) in the part BC.
11.11.3 Change In Volume On Melting
Most of the substances, like lead and wax, expand on melting, but some substances, like ice and bismuth, contract on melting.
For example, \(1\\ \\text{g}\) of solid wax of volume \(1.161\\ \\text{cm}^3\) on melting at \(64^\\circ\\text{C}\) becomes \(1\\ \\text{g}\) of molten wax of volume \(1.166\\ \\text{cm}^3\) — the same mass now takes up slightly more room. On the other hand, \(1\\ \\text{g}\) of ice of volume \(1.091\\ \\text{cm}^3\) at \(0^\\circ\\text{C}\) on melting becomes \(1\\ \\text{g}\) of water of volume \(1\\ \\text{cm}^3\) at \(0^\\circ\\text{C}\), a clear shrinking.
Ice is the odd one out, and that oddity has a consequence you have already met in section 11.8: because water expands as it freezes, it bursts the fine capillaries of a plant on a frosty night.
11.11.4 Effect Of Pressure On The Melting Point
Whether pressure raises or lowers the melting point depends on which way the volume changes, so the rule follows the two cases of section 11.11.3.
The melting point of the substances which contract on melting, like ice, decreases by the increase in pressure. For example, the melting point of ice decreases by \(0.0072^\\circ\\text{C}\) for every one atmosphere rise in pressure.
On the other hand, the melting point of the substances such as wax and lead, which expand on melting, increases by the increase in pressure.
11.11.5 Effect Of Impurities On The Melting Point
The melting point of a substance decreases by the presence of impurities in it. The melting point of ice decreases from \(0^\\circ\\text{C}\) to \(-22^\\circ\\text{C}\) on mixing salt in it in proper proportion.
This fact is utilised in making the freezing mixture by adding salt to ice. The freezing mixture is used in preparing kulfis.
11.12 Vaporisation And Boiling
11.12.1 Vaporisation, Condensation And Boiling Point
Similarly, the change from vapour to liquid phase on liberation of heat at a constant temperature is called condensation, or liquefaction, and the particular temperature at which the condensation occurs is called the condensation point of the vapour.
As with melting and freezing, the two temperatures are one and the same for a pure substance: the boiling point and the condensation point are identical. And the energy balances too. Heat energy is absorbed at a constant temperature during vaporisation, while the same amount of heat energy is liberated during condensation at that temperature, for the same mass of substance. Fig. 11.12-1 shows the pair for water and steam.
Diagram
Fig. 11.12-1 — Boiling And Condensation Of Water Two labelled boxes side by side, the left reading ‘100 °C water’ and the right reading ‘100 °C steam’. Draw an arrow from the left box to the right box above the line, labelled on top ‘boiling’ and below ‘absorption of heat’. Draw a second arrow from the right box back to the left box below the line, labelled below ‘condensation’ and above ‘liberation of heat’. Note beside the figure that the temperature stays at 100 °C throughout both changes at one atmospheric pressure.
11.12.2 Heating Curve For Water
To obtain the heating curve of water, we perform the experiment below, using the arrangement of Fig. 11.12-2.
The Experiment
Take some water at room temperature, say \(20^\\circ\\text{C}\), in a flask and suspend a thermometer in it.
Heat the flask by keeping it over a bunsen burner and note its temperature after every half minute, till the water starts boiling and the flask contains vapours. At this stage, bubbles are formed throughout the water. This indicates that boiling occurs throughout the volume of the water, and not merely at its surface.
Plot a graph for temperature against time, taking the temperature on the Y-axis and time on the X-axis. The graph is shown in Fig. 11.12-3. It is called the heating curve for water.
Diagram
Fig. 11.12-2 — Apparatus For The Heating Curve Of Water A round-bottomed flask, about half filled with water, standing on a wire gauze over a tripod stand with a bunsen burner beneath. A thermometer is suspended through the mouth of the flask with its bulb inside the water and clear of the glass. Show small bubbles rising through the whole body of the water and vapour escaping from the mouth of the flask. Label: flask, water, thermometer, bubbles formed throughout the liquid, wire gauze, tripod stand, bunsen burner.
Diagram
Fig. 11.12-3 — Heating Curve For Water Line graph. X-axis labelled ‘time (s)’ marked at intervals of 30 from 0 to 420; Y-axis labelled ‘temperature (°C)’ marked at 20, 40, 60, 80, 100, 120, 140. From point A at 20 °C on the Y-axis, draw a straight line rising steadily to point B at 100 °C at about 150 s, and label this stretch ‘liquid’. From B draw a horizontal line at 100 °C running to point C near 420 s, and label this stretch ‘boiling — liquid + vapour’. Mark and letter the points A, B and C.
11.12.3 Change In Volume On Boiling
All liquids expand on boiling, and the expansion is not a small one. For example, \(1\\ \\text{cm}^3\) of water at \(100^\\circ\\text{C}\) becomes \(1760\\ \\text{cm}^3\) of steam at \(100^\\circ\\text{C}\).
That figure is worth pausing over. The same mass of substance now occupies more than seventeen hundred times the space, which is why steam trapped in a closed vessel builds up such large pressures, and why any vessel in which water is boiled under pressure must have a designed route for the steam to escape.
11.12.4 Effect Of Pressure On The Boiling Point
The boiling point of pure water at one atmospheric pressure, which is \(760\\ \\text{mm}\) of Hg, is \(100^\\circ\\text{C}\). Change the pressure and that number changes with it.
Inside A Pressure Cooker
In a pressure cooker, steam is not allowed to escape freely. The vapour pressure on the water inside the pressure cooker becomes nearly \(1.75\) times the atmospheric pressure. This increases the boiling point of water, so water boils in a cooker at about \(120^\\circ\\text{C}\) to \(125^\\circ\\text{C}\).
Thus cooking of vegetables, pulses and the like becomes much faster in it, since they get sufficient heat energy before the water boils, and the water is now at a higher temperature. Further, the cooked material inside, being at a higher temperature, remains warm over a long period.
A cooker only works because the pressure is contained, so the weight on its vent and its safety valve are not accessories but part of the physics. A blocked vent would let the pressure, and with it the boiling point, climb past what the vessel is built for.
At High Altitudes
At high altitudes, such as hills and mountains, the atmospheric pressure is low, that is, less than one atmospheric pressure. Therefore at these places water boils at a temperature lower than \(100^\\circ\\text{C}\).
The water then does not provide the required heat energy to its contents for cooking before it starts boiling. Thus cooking at high altitudes, if done in the open, becomes difficult and takes a much longer time. The pot is boiling merrily and the food is still not cooked, because boiling is not the same thing as being hot enough.
11.12.5 Effect Of Impurities On The Boiling Point
The boiling point of a liquid increases by the addition of impurities to it. Note that this is the opposite of what impurities do to the melting point, which they lower.
If common salt is added to water, it boils at a temperature higher than \(100^\\circ\\text{C}\). Hence we add salt to the water while cooking pulses, vegetables and so on, so that the water provides sufficient heat energy to its contents before boiling starts. The cooking is now at a higher temperature and thus becomes easier and faster.
11.13 Latent Heat And Specific Latent Heat
11.13.1 Latent Heat
We have read that during the change of phase of a substance, which takes place at a constant temperature, a considerable amount of heat energy is absorbed or liberated.
Heat energy is absorbed by a solid during melting, and an equal amount of heat energy is liberated by the liquid during freezing, without showing any change in temperature. Similarly, heat energy is absorbed by a liquid during vaporisation, and an equal amount of heat energy is liberated by the vapour during condensation, without showing any change in temperature.
Since the heat energy absorbed or liberated in a change of phase is not externally manifested by any rise or fall in temperature, it is called latent heat. The word means hidden, and it is well chosen: the energy is entirely real, but no thermometer will show it.
Latent heat, when expressed for unit mass of substance, is called the specific latent heat. It is a characteristic property of the substance, depending on its composition, and is different for different substances.
11.13.2 Specific Latent Heat And Its Units
Specific latent heat is denoted by the symbol \(L\). In words,
\[L = \\frac{\\text{heat absorbed (or liberated) for the change of phase}}{\\text{mass}}\]
If an amount of heat energy \(Q\) is absorbed or liberated by a mass \(m\) of a substance during its change of phase at a constant temperature, then
\[L = \\frac{Q}{m}\]
where \(L\) is the specific latent heat in \(\\text{J kg}^{-1}\), \(Q\) is the heat energy in J, and \(m\) is the mass of the substance in kg. Note what is absent from this relation: there is no temperature term, because the temperature does not change during the process at all.
Rearranged to give the heat, the relation reads
\[Q = m \\times L\]
where \(Q\) is the heat energy absorbed or liberated in J, \(m\) the mass in kg, and \(L\) the specific latent heat in \(\\text{J kg}^{-1}\). This is the companion of \(Q = m c \\Delta t\), and the two are used at different stages of the same problem: \(Q = mc\\Delta t\) while the temperature is changing, \(Q = mL\) while the phase is changing.
Units Of Specific Latent Heat
The SI unit of specific latent heat is \(\\text{J kg}^{-1}\). Other common units are \(\\text{cal g}^{-1}\) and \(\\text{kcal kg}^{-1}\). They are related as
\[1\\ \\text{kcal kg}^{-1} = 1\\ \\text{cal g}^{-1}\] \[1\\ \\text{cal g}^{-1} = 4.2\\ \\text{J g}^{-1} = 4.2 \\times 10^{3}\\ \\text{J kg}^{-1}\]
11.13.3 Specific Latent Heat Of Fusion Of Ice
On the other hand, the specific latent heat of freezing of ice is the heat energy released when a unit mass of water at \(0^\\circ\\text{C}\) freezes to ice at \(0^\\circ\\text{C}\) without any change in temperature. For a pure substance the specific latent heat of freezing is the same as the specific latent heat of fusion — the same energy, going in one way and coming out the other.
For ice, the specific latent heat of fusion is
\[L\_{\\text{ice}} = 336000\\ \\text{J kg}^{-1} = 80\\ \\text{cal g}^{-1}\]
Read that value three ways, because ICSE questions ask for each of them.
It means that \(1\\ \\text{kg}\) of ice at \(0^\\circ\\text{C}\) absorbs \(336000\\ \\text{J}\) of heat energy to convert into water at \(0^\\circ\\text{C}\); equivalently, \(1\\ \\text{g}\) of ice at \(0^\\circ\\text{C}\) absorbs \(80\\ \\text{cal}\) of heat energy to convert into water at \(0^\\circ\\text{C}\).
In other words, \(1\\ \\text{kg}\) of water at \(0^\\circ\\text{C}\) liberates \(336000\\ \\text{J}\) of heat energy to convert into ice at \(0^\\circ\\text{C}\), or \(1\\ \\text{g}\) of water at \(0^\\circ\\text{C}\) liberates \(80\\ \\text{cal}\) of heat energy to convert into ice at \(0^\\circ\\text{C}\).
And put as a comparison of two things at the same temperature: \(1\\ \\text{g}\) of water at \(0^\\circ\\text{C}\) has \(336\\ \\text{J}\), or \(80\\ \\text{cal}\), more heat energy than \(1\\ \\text{g}\) of ice at \(0^\\circ\\text{C}\). Same temperature, different energy — which is section 11.2’s warning made concrete.
Table 11.4 — Specific latent heat of fusion of some common substances
| Substance | Latent heat of fusion in \(\\text{cal g}^{-1}\) | Latent heat of fusion in \(\\text{J kg}^{-1}\) |
|---|---|---|
| Mercury | 3 | \(12.5 \\times 10^{3}\) |
| Sulphur | 9 | \(37.7 \\times 10^{3}\) |
| Silver | 21 | \(88.0 \\times 10^{3}\) |
| Paraffin wax | 35 | \(146 \\times 10^{3}\) |
| Copper | 43 | \(180 \\times 10^{3}\) |
| Ice | 80 | \(336 \\times 10^{3}\) |
Ice stands at the top of this table by a wide margin, and section 11.14 is about what follows from that.
11.13.4 Explanation Of Latent Heat Of Melting On The Kinetic Model
Where does the hidden energy actually go? The kinetic model of matter answers this, and the answer is drawn in Fig. 11.13-1.
According to the kinetic model, the molecules in a solid vibrate about their mean positions. The total energy of a molecule is the sum of its kinetic energy, due to its motion, which depends on the temperature; and its potential energy, which depends on the force of attraction between the molecules and the separation between them.
When a solid changes into a liquid without any change in temperature, the average kinetic energy of the molecules does not change — that is exactly what the constant temperature tells us. But the separation between the molecules, on an average, increases.
Some energy is required for increasing this separation against the attractive forces between the molecules, that is, for the increase in the potential energy of the molecules. Since the increase in volume is negligible, the heat energy used in volume expansion is ignored.
Thus, the heat energy supplied during melting is utilised only in increasing the potential energy of the molecules, and is called the latent heat of melting. When \(1\\ \\text{g}\) of ice at \(0^\\circ\\text{C}\) changes to water at \(0^\\circ\\text{C}\), nearly \(336\\ \\text{J}\) of energy is used in increasing the potential energy due to the increase in intermolecular separation, so the latent heat of ice is \(336\\ \\text{J g}^{-1}\).
Diagram
Fig. 11.13-1 — Latent Heat On The Kinetic Model Two molecular sketches side by side with a wide arrow between them labelled ‘heat supplied at constant temperature’. Left sketch headed ‘solid at its melting point’: circles packed in a close regular arrangement, each with a small vibration arrow of a given length, and springs or attraction lines drawn between neighbours. Right sketch headed ‘liquid at the same temperature’: the same number of circles, more loosely and irregularly arranged with visibly larger gaps, each with a vibration arrow of the SAME length as before. Caption boxes: under the left, ‘molecules close together, low potential energy’; under the right, ‘molecules further apart, higher potential energy’; and across the arrow, ‘average kinetic energy unchanged, so the temperature does not change’.
11.14 Natural Consequences Of The High Specific Latent Heat Of Fusion Of Ice
The specific latent heat of fusion of ice, \(336000\\ \\text{J kg}^{-1}\), is the largest in Table 11.4 by a long way. Every kilogram of ice that melts must be handed that much energy, and every kilogram of water that freezes must give it up. Several familiar features of the natural world follow directly.
Snow On Mountains Does Not Melt All At Once
The reason is that ice has a high specific latent heat of fusion, equal to \(336000\\ \\text{J kg}^{-1}\). It is due to this fact that snow changes into water slowly, as it gets heat energy from the sun.
If the latent heat of ice had been low, all the snow would have melted in a short time on getting heat from the sun. There would then have been floods in the rivers, and no water after that. The slowness is not a nuisance; it is what turns a snowfield into a river that runs all summer.
In Cold Countries, Water In Lakes And Ponds Does Not Freeze All At Once
The reason is that the specific latent heat of fusion of ice is sufficiently high, \(336\\ \\text{J g}^{-1}\). The water in lakes and ponds has to liberate a large quantity of heat to the surroundings, which are below \(0^\\circ\\text{C}\), before it can freeze — and giving up that much energy takes a long time.
The layer of ice formed over the water surface then helps as well. Being a poor conductor of heat, it prevents the loss of heat from the water of the lake, so the water below does not freeze all at once. This property is very important for the survival of aquatic life, as Fig. 11.14-1 shows.
Diagram
Fig. 11.14-1 — A Frozen Lake In Winter Vertical cross-section of a lake. At the top, a layer of ice spanning the surface, labelled ‘layer of ice — a poor conductor of heat’. Above it, the air with an arrow pointing upward out of the ice labelled ‘slow loss of heat to air below 0 °C’. Below the ice, a body of liquid water labelled ‘water still liquid’, with a few fish and water plants drawn in it. At the bottom, the lake bed. Add a side label: ‘the ice layer slows further heat loss, so the lake does not freeze right through’.
Drinks Get Cooled More Quickly By Ice Than By Ice-cold Water At 0 °C
This is because \(1\\ \\text{g}\) of ice at \(0^\\circ\\text{C}\) takes \(336\\ \\text{J}\) of heat energy from the drink in order to melt into water at \(0^\\circ\\text{C}\). Thus the drink gives up an additional \(336\\ \\text{J}\) of heat energy to \(1\\ \\text{g}\) of ice at \(0^\\circ\\text{C}\) than it does to \(1\\ \\text{g}\) of ice-cold water at \(0^\\circ\\text{C}\).
Therefore the cooling produced by \(1\\ \\text{g}\) of ice at \(0^\\circ\\text{C}\) is much more than that produced by \(1\\ \\text{g}\) of water at \(0^\\circ\\text{C}\), even though both start at exactly the same temperature.
When Ice In A Frozen Lake Starts Melting, Its Surroundings Become Very Cold
The reason is that quite a large amount of heat energy is required for melting the ice of the frozen lake, and this heat is absorbed from the surrounding atmosphere. As a result, the temperature of the surroundings falls and it becomes very cold.
It Is Generally More Cold After A Hail-storm Than During Or Before It
The reason is that after the hail-storm, the ice starts absorbing the heat energy required for its melting from the surroundings. So the temperature of the surroundings falls further down and we feel more cold.
Notice that the last two entries are the same physics as the third one, seen from the outside. Melting ice does not merely fail to warm its surroundings; it actively robs them of heat.
11.15 Summary
The internal energy of a molecule is the sum of its kinetic energy, due to random motion, and its potential energy, due to the attractive forces between molecules. The total internal energy of all the molecules of a substance is its thermal energy.
Heat is that form of energy which flows from a hot body to a cold body when they are kept in contact. It is energy in transit; a body contains internal energy, not heat. The measurement of the quantity of heat is called calorimetry.
The SI unit of heat is the joule (J). One calorie is the quantity of heat energy required to raise the temperature of \(1\\ \\text{g}\) of water from \(14.5^\\circ\\text{C}\) to \(15.5^\\circ\\text{C}\), and \(1\\ \\text{cal} = 4.186\\ \\text{J} \\approx 4.2\\ \\text{J}\). One kilo-calorie is the same for \(1\\ \\text{kg}\) of water, and \(1\\ \\text{kcal} = 1000\\ \\text{cal} = 4200\\ \\text{J}\) nearly.
Temperature is a parameter which tells the thermal state of a body — its degree of hotness or coldness — and determines the direction of flow of heat when two bodies at different temperatures are placed in contact. Its SI unit is the kelvin (K).
The celsius and kelvin scales are related by \[T\\ \\text{K} = 273 + t\\ ^\\circ\\text{C}\] where \(T\) is the temperature in K and \(t\) the temperature in \(^\\circ\\text{C}\). Absolute zero, \(0\\ \\text{K}\), equals \(-273^\\circ\\text{C}\), and no negative temperature is possible on the kelvin scale.
A temperature difference has the same value on both scales: \[\\Delta t\\ ^\\circ\\text{C} = \\Delta T\\ \\text{K}\]
Two bodies at the same temperature may hold very different quantities of thermal energy, since thermal energy depends on the mass and the substance as well as on the temperature.
The heat energy absorbed to raise the temperature of a body depends on three factors: its mass, its rise in temperature, and the material of which it is made. \[Q = m \\times c \\times \\Delta t\] where \(Q\) is the heat energy in J, \(m\) the mass in kg, \(c\) the specific heat capacity in \(\\text{J kg}^{-1},\\text{K}^{-1}\), and \(\\Delta t\) the rise in temperature in K. It applies only while the temperature is changing.
The heat capacity of a body is the amount of heat energy required to raise its temperature by \(1\\ \\text{K}\): \[C' = \\frac{Q}{\\Delta t}\] where \(C'\) is the heat capacity in \(\\text{J K}^{-1}\). Its other units are \(\\text{cal K}^{-1}\) and \(\\text{kcal K}^{-1}\), with \(1\\ \\text{cal K}^{-1} = 4.2\\ \\text{J K}^{-1}\).
The specific heat capacity of a substance is the heat energy required to raise the temperature of unit mass of it by \(1\\ \\text{K}\): \[c = \\frac{Q}{m,\\Delta t}\] where \(c\) is in \(\\text{J kg}^{-1},\\text{K}^{-1}\). Its other unit is \(\\text{cal g}^{-1},^\\circ\\text{C}^{-1}\), with \(1\\ \\text{cal g}^{-1},^\\circ\\text{C}^{-1} = 4.2 \\times 10^{3}\\ \\text{J kg}^{-1},\\text{K}^{-1}\).
Heat capacity belongs to a body and depends on its mass; specific heat capacity belongs to a substance and does not. They are related by \[C' = m \\times c\]
A good conductor of heat usually has a low specific heat capacity, and a bad conductor a high one. The substance with a low specific heat capacity shows a rapid and high rise in temperature for the same heat supplied.
Water has an unusually high specific heat capacity, \(4200\\ \\text{J kg}^{-1},\\text{K}^{-1}\) (more precisely \(4180\\ \\text{J kg}^{-1},\\text{K}^{-1}\)). The value differs for the phases of the same substance: ice \(2100\\ \\text{J kg}^{-1},\\text{K}^{-1}\) and steam \(460\\ \\text{J kg}^{-1},\\text{K}^{-1}\).
Consequences of the high specific heat capacity of water include the moderate climate near a sea shore, the use of hot water bottles for fomentation, water as a coolant in radiators, water as a heat reservoir to prevent bottles from freezing, the flooding of fields to protect crops from frost, and the steady body temperature of plants and animals.
A calorimeter is a cylindrical vessel used to measure heat gained or lost when a body is mixed with another. It is made of a thin sheet of copper, because copper is a good conductor and has a low specific heat capacity, and a thin sheet has a small mass. Its surfaces are polished to reduce loss by radiation, it sits in a wooden jacket packed with wool to reduce loss by conduction, and a wooden lid reduces loss by convection.
By the principle of method of mixtures, also called the principle of calorimetry, heat energy lost by the hot body equals heat energy gained by the cold body, provided no heat is lost to the surroundings. It follows from the law of conservation of energy. \[m\_1 c\_1 (t\_1 - t) = m\_2 c\_2 (t - t\_2)\] where \(m\_1, c\_1, t\_1\) describe the hot body, \(m\_2, c\_2, t\_2\) the cold body, and \(t\) is the final temperature. If the vessel also warms up, its heat capacity is added to that of the cold body: \[m\_1 c\_1 (t\_1 - t) = (m\_2 c\_2 + Mc)(t - t\_2)\]
A large heat capacity is wanted where heat must be stored and released slowly — the thick base of a cooking pan, the thick heavy base of an electric press. A small heat capacity is wanted where the vessel must not take heat from its contents — the thin copper calorimeter.
The change from one state to another at a constant temperature is called the change of phase, and it is brought about by the exchange of heat. Solid to liquid is melting and liquid to solid is freezing; liquid to vapour is vaporisation and vapour to liquid is condensation; solid to vapour is sublimation and vapour to solid is solidification.
The melting point is the constant temperature at which a solid changes to liquid. For a pure substance the melting point and the freezing point are identical, and the heat absorbed on melting a given mass equals the heat liberated on freezing it.
On the heating curve for ice, the horizontal part at \(0^\\circ\\text{C}\) shows the ice melting at constant temperature, and the sloping part after it shows the water formed rising in temperature.
Most substances, such as lead and wax, expand on melting; some, such as ice and bismuth, contract on melting. \(1\\ \\text{g}\) of ice of volume \(1.091\\ \\text{cm}^3\) at \(0^\\circ\\text{C}\) becomes \(1\\ \\text{g}\) of water of volume \(1\\ \\text{cm}^3\) at \(0^\\circ\\text{C}\).
An increase in pressure decreases the melting point of substances that contract on melting — that of ice falls by \(0.0072^\\circ\\text{C}\) per atmosphere — and increases the melting point of substances that expand on melting.
Impurities lower the melting point. The melting point of ice falls from \(0^\\circ\\text{C}\) to \(-22^\\circ\\text{C}\) when salt is mixed in it in proper proportion, which is the basis of the freezing mixture used to prepare kulfis.
The boiling point is the constant temperature at which a liquid vaporises, and for a pure substance it equals the condensation point. On the heating curve for water, the sloping part shows the liquid warming and the horizontal part at \(100^\\circ\\text{C}\) shows it boiling while heat continues to be absorbed.
All liquids expand on boiling: \(1\\ \\text{cm}^3\) of water at \(100^\\circ\\text{C}\) becomes \(1760\\ \\text{cm}^3\) of steam at \(100^\\circ\\text{C}\).
The boiling point of a liquid increases with an increase in pressure and decreases with a decrease in pressure. Pure water boils at \(100^\\circ\\text{C}\) at \(760\\ \\text{mm}\) of Hg; in a pressure cooker, where the pressure is nearly \(1.75\) times atmospheric, it boils at about \(120^\\circ\\text{C}\) to \(125^\\circ\\text{C}\), and at high altitudes it boils below \(100^\\circ\\text{C}\), which makes cooking in the open slow and difficult.
Impurities raise the boiling point, which is why salt is added to water used for cooking pulses and vegetables.
Heat absorbed or liberated during a change of phase produces no change in temperature, and is called latent heat. Expressed for unit mass it is the specific latent heat: \[L = \\frac{Q}{m}, \\qquad Q = m \\times L\] where \(L\) is the specific latent heat in \(\\text{J kg}^{-1}\), \(Q\) the heat energy in J and \(m\) the mass in kg. Its other units are \(\\text{cal g}^{-1}\) and \(\\text{kcal kg}^{-1}\), with \(1\\ \\text{cal g}^{-1} = 4.2 \\times 10^{3}\\ \\text{J kg}^{-1}\).
The specific latent heat of fusion of ice is the heat energy required to melt unit mass of ice at \(0^\\circ\\text{C}\) to water at \(0^\\circ\\text{C}\) without any change in temperature: \[L\_{\\text{ice}} = 336000\\ \\text{J kg}^{-1} = 80\\ \\text{cal g}^{-1}\] Hence \(1\\ \\text{g}\) of water at \(0^\\circ\\text{C}\) has \(336\\ \\text{J}\) more heat energy than \(1\\ \\text{g}\) of ice at \(0^\\circ\\text{C}\).
On the kinetic model, melting at constant temperature leaves the average kinetic energy of the molecules unchanged but increases their average separation. The latent heat supplied is used in increasing the potential energy of the molecules against their mutual attraction.
Consequences of the high specific latent heat of fusion of ice include the slow melting of mountain snow, lakes and ponds in cold countries not freezing all at once, ice cooling a drink far more than ice-cold water does, the chilling of the surroundings when a frozen lake melts, and the cold felt after a hail-storm.
Flowchart
FC 11.3 — Heating A Substance From Solid To Vapour Solid below its melting point, temperature rising, Q = m c Δt → melting point reached → change of phase at constant temperature, Q = m L → liquid below its boiling point, temperature rising, Q = m c Δt → boiling point reached → change of phase at constant temperature, Q = m L → vapour, temperature rising again, Q = m c Δt
11.16 Key Terms
Boiling point — the constant temperature at which a liquid changes to vapour on absorbing heat; measured in K or \(^\\circ\\text{C}\).
Calorie — the heat energy required to raise the temperature of \(1\\ \\text{g}\) of water from \(14.5^\\circ\\text{C}\) to \(15.5^\\circ\\text{C}\); \(1\\ \\text{cal} = 4.186\\ \\text{J} \\approx 4.2\\ \\text{J}\).
Calorimeter — a cylindrical vessel of thin polished copper, used to measure the heat gained or lost when one body is mixed with another.
Calorimetry — the measurement of the quantity of heat.
Change of phase — the change of a substance from one state to another at a constant temperature, brought about by the exchange of heat.
Condensation — the change from vapour to liquid on the liberation of heat at a constant temperature; also called liquefaction.
Condensation point — the constant temperature at which a vapour changes to liquid; for a pure substance it equals the boiling point. Measured in K or \(^\\circ\\text{C}\).
Degree celsius — a unit of temperature, symbol \(^\\circ\\text{C}\), related to the kelvin by \(T\\ \\text{K} = 273 + t\\ ^\\circ\\text{C}\).
Freezing — the change from liquid to solid on the liberation of heat at a constant temperature.
Freezing point — the constant temperature at which a liquid changes to solid; for a pure substance it equals the melting point. Measured in K or \(^\\circ\\text{C}\).
Heat — the form of energy which flows from a hot body to a cold body when they are kept in contact; SI unit joule (J).
Heat capacity — the amount of heat energy required to raise the temperature of a whole body by \(1\\ \\text{K}\); SI unit \(\\text{J K}^{-1}\).
Internal energy — the sum of the kinetic energy and the potential energy of a molecule; measured in joule (J).
Joule — the SI unit of heat and of every other form of energy; symbol J.
Kelvin — the SI unit of temperature; symbol K, with \(0\\ \\text{K}\) at \(-273^\\circ\\text{C}\).
Kilo-calorie — the heat energy required to raise the temperature of \(1\\ \\text{kg}\) of water from \(14.5^\\circ\\text{C}\) to \(15.5^\\circ\\text{C}\); \(1\\ \\text{kcal} = 1000\\ \\text{cal} = 4200\\ \\text{J}\) nearly.
Latent heat — the heat absorbed or liberated during a change of phase, which produces no change in temperature; measured in joule (J).
Melting — the change from solid to liquid by the absorption of heat at a constant temperature.
Melting point — the constant temperature at which a solid changes to liquid; measured in K or \(^\\circ\\text{C}\).
Principle of calorimetry — another name for the principle of method of mixtures.
Principle of method of mixtures — the statement that the heat energy lost by the hot body equals the heat energy gained by the cold body, provided no heat is lost to the surroundings.
Solidification — the direct change from vapour to solid, the reverse of sublimation.
Specific heat capacity — the heat energy required to raise the temperature of unit mass of a substance by \(1\\ \\text{K}\); SI unit \(\\text{J kg}^{-1},\\text{K}^{-1}\).
Specific latent heat — the heat energy absorbed or liberated by unit mass of a substance during a change of phase at a constant temperature; SI unit \(\\text{J kg}^{-1}\).
Sublimation — the direct change from solid to vapour on absorbing heat, without passing through the liquid phase.
Temperature — the parameter which tells the degree of hotness or coldness of a body and determines the direction of flow of heat between bodies in contact; SI unit kelvin (K).
Thermal energy — the total internal energy of all the molecules of a substance; measured in joule (J).
Vaporisation — the change from liquid to vapour by the absorption of heat at a constant temperature.
11.17 Points To Ponder
Suppose water had a specific heat capacity as low as that of sand. Which would change more — the climate of a coastal city, or the design of a car’s cooling system? Work out what would happen in each case before deciding.
Along the flat part of a heating curve, energy goes in for minutes together and the thermometer does not move. Is that energy lost? If it is still in the substance, describe a way in which you could get it back out again.
Ice contracts on melting, while most solids expand. Go back through this chapter and count how many of the effects described would come out differently if ice behaved like wax instead. Which single one would matter most to living things?
A calorimeter is designed to have the smallest possible heat capacity, while a cooking pan is designed to have a large one. Could a single material ever be the best choice for both? Say precisely which quantity the designer is controlling in each case, and how.
In a method of mixtures experiment, some heat always escapes to the room, however careful the insulation. If a student ignores that escape, will the specific heat capacity they calculate for a hot metal come out larger or smaller than the true value? Reason it through from the equation rather than guessing.
Two bodies at the same temperature can hold very different amounts of energy, and a thermometer cannot tell them apart. What, then, is a thermometer actually measuring — and why is that still the right quantity to use for deciding which way heat will flow?
11.18 Objective Questions
Multiple choice
- The SI unit of specific heat capacity is: (1 mark)
- \(\\text{J K}^{-1}\)
- \(\\text{J kg}^{-1},\\text{K}^{-1}\)
- \(\\text{J kg}^{-1}\)
- \(\\text{J}\)
- A rise in temperature of \(25^\\circ\\text{C}\) is the same as a rise of: (1 mark)
- \(25\\ \\text{K}\)
- \(248\\ \\text{K}\)
- \(273\\ \\text{K}\)
- \(298\\ \\text{K}\)
- Which of the following contracts on melting? (1 mark)
- wax
- lead
- ice
- copper
- The specific latent heat of fusion of ice is: (1 mark)
- \(336\\ \\text{J kg}^{-1}\)
- \(336\\ \\text{J g}^{-1}\)
- \(80\\ \\text{J g}^{-1}\)
- \(4200\\ \\text{J kg}^{-1}\)
- Water in a pressure cooker boils at about: (1 mark)
- \(90^\\circ\\text{C}\)
- \(100^\\circ\\text{C}\)
- \(120^\\circ\\text{C}\) to \(125^\\circ\\text{C}\)
- \(160^\\circ\\text{C}\)
- A body of mass \(m\) is made of a substance of specific heat capacity \(c\). Its heat capacity is: (1 mark)
- \(\\dfrac{m}{c}\)
- \(\\dfrac{c}{m}\)
- \(m c\)
- \(m c \\Delta t\)
- Adding common salt to water does which of the following? (1 mark)
- raises its boiling point and raises its melting point
- raises its boiling point and lowers its melting point
- lowers its boiling point and raises its melting point
- lowers its boiling point and lowers its melting point
Assertion–reason
For each of the following, choose from: a. both A and R are true and R is the correct explanation of A b. both A and R are true but R is not the correct explanation of A c. A is false but R is true d. A is true but R is false
Assertion (A): The base of an electric press is made thick and heavy. Reason (R): A body of large heat capacity remains hot for a long time even after the supply of heat is stopped. (1 mark)
Assertion (A): A piece of ice at \(0^\\circ\\text{C}\) cools a drink more than the same mass of water at \(0^\\circ\\text{C}\). Reason (R): The specific heat capacity of water is greater than that of ice. (1 mark)
Assertion (A): The temperature of a substance does not change while it is boiling, even though heat is being supplied continuously. Reason (R): The heat supplied during boiling increases the potential energy of the molecules and not their average kinetic energy. (1 mark)
Match the following
- Match each quantity in Column A with its SI unit in Column B. (4 marks)
| Column A | Column B |
|---|---|
| a. Heat capacity | i. \(\\text{J kg}^{-1}\) |
| b. Specific heat capacity | ii. \(\\text{K}\) |
| c. Specific latent heat | iii. \(\\text{J K}^{-1}\) |
| d. Temperature | iv. \(\\text{J kg}^{-1},\\text{K}^{-1}\) |
Fill in the blanks
- Complete the following: (4 marks)
- Heat flows from a body at a ………… temperature to a body at a ………… temperature.
- The melting point of ice ………… with an increase in pressure.
- The boiling point of water ………… when common salt is dissolved in it.
- During a change of phase, the temperature of a substance ………… .
True or false
- State whether each statement is true or false: (4 marks)
- A hot body contains a large quantity of heat.
- The specific heat capacity of a substance depends on the mass of the body.
- One gram of water at \(0^\\circ\\text{C}\) has more energy than one gram of ice at \(0^\\circ\\text{C}\).
- A temperature difference of one degree celsius equals a temperature difference of one kelvin.
Name the following
- Name the following: (4 marks)
- The common liquid which has an unusually high specific heat capacity.
- The direct change of a substance from the vapour phase to the solid phase.
- The vessel used in the laboratory to measure a quantity of heat.
- The law on which the principle of method of mixtures is based.
Give one word or phrase for
- Give one word or phrase for each of the following: (3 marks)
- The heat absorbed or given out during a change of phase, which causes no change in temperature.
- The constant temperature at which a solid changes into a liquid.
- The measurement of the quantity of heat.
State the SI unit of
- State the SI unit of each of the following: (4 marks)
- heat
- temperature
- heat capacity
- specific latent heat
Odd one out
- Pick the odd one out in each row and give a reason for your choice. (2 marks)
- ice, bismuth, wax
- \(\\text{J kg}^{-1},\\text{K}^{-1}\), \(\\text{J K}^{-1}\), \(\\text{J kg}^{-1}\), \(\\text{J s}^{-1}\)
11.19 Check Your Concepts
Define specific heat capacity and state its SI unit. (2 marks)
State the principle of method of mixtures, and name the law of physics on which it is based. (2 marks)
What does it mean to say that the heat capacity of a body is \(250\\ \\text{J K}^{-1}\)? (2 marks)
Give a reason: hot water bottles, rather than bottles filled with some other liquid, are used for fomentation. (2 marks)
A liquid P has a higher specific heat capacity than a liquid Q. Which of the two would you choose as a coolant in a machine, and why? (2 marks)
Differentiate between heat and temperature, stating three points of difference and the basis of each. (3 marks)
Differentiate between heat capacity and specific heat capacity, stating the basis of difference in each case. (3 marks)
Why does the temperature of a substance remain unchanged during its change of phase, although heat is being supplied to it throughout? (3 marks)
Explain why cooking food in an open vessel takes much longer on a high mountain than at sea level. (3 marks)
State the effect of (i) an increase in pressure, and (ii) the presence of impurities, on the melting point of ice. Give one everyday use of the second effect. (3 marks)
Why is a calorimeter made of a thin sheet of copper? Give two separate reasons, one for the choice of the material and one for the choice of the thickness. (4 marks)
Explain, on the basis of the kinetic model, why heat supplied during melting produces no rise in temperature. In your answer, state what happens to the average kinetic energy and to the average separation of the molecules. (4 marks)
Discuss the role of the high specific heat capacity of water in making the climate near a sea shore moderate. Include in your answer how the land breeze and the sea breeze are set up. (5 marks)
A mass \(m\_1\) of a substance of specific heat capacity \(c\_1\) at a temperature \(t\_1\) is mixed with a mass \(m\_2\) of another substance of specific heat capacity \(c\_2\) at a lower temperature \(t\_2\). Deduce an expression for the final temperature \(t\) of the mixture, and state the assumption you have made. (5 marks)
11.20 Diagrams, Graphs & Constructions
Draw a labelled diagram of a calorimeter as used in the laboratory. Mark on it the thermometer, the stirrer, the lid and the insulating packing, and state which route of heat loss each of the last two reduces. (4 marks)
Fig. 11.11-3 shows the heating curve for ice initially at \(0^\\circ\\text{C}\). State what the part AB represents, what the part BC represents, and what the temperature along AB is called. Give a reason why AB is horizontal. (4 marks)
Fig. 11.12-3 shows the heating curve for water heated from room temperature. Identify the boiling point of the water from the graph, and explain why the line becomes parallel to the time axis at that temperature. (3 marks)
Sketch a temperature–time graph for \(1\\ \\text{kg}\) of ice initially at \(-10^\\circ\\text{C}\) heated at a constant rate until it has all become steam at \(100^\\circ\\text{C}\). Label the axes with their units, and mark on the sketch the two horizontal parts and the phase present along each sloping part. (5 marks)
A sample of molten naphthalene at \(90^\\circ\\text{C}\) is allowed to cool in a room at \(25^\\circ\\text{C}\). Its melting point is \(80^\\circ\\text{C}\). Sketch the expected temperature–time cooling curve, and mark clearly the part of the curve during which the naphthalene is freezing. (4 marks)
Draw a labelled diagram of the apparatus you would set up to obtain the heating curve of ice, and state one precaution to be taken while placing the thermometer. (3 marks)
11.21 Practice With Numericals
Calculate the heat energy required to raise the temperature of \(250\\ \\text{g}\) of copper from \(25^\\circ\\text{C}\) to \(75^\\circ\\text{C}\). Specific heat capacity of copper \(= 385\\ \\text{J kg}^{-1},\\text{K}^{-1}\). (2 marks)
A body has a heat capacity of \(240\\ \\text{J K}^{-1}\). How much heat energy does it give out while cooling from \(60^\\circ\\text{C}\) to \(20^\\circ\\text{C}\)? (2 marks)
\(2400\\ \\text{J}\) of heat energy raises the temperature of \(1.5\\ \\text{kg}\) of a metal by \(4\\ \\text{K}\). Find the specific heat capacity of the metal and the heat capacity of the piece. (3 marks)
Calculate the heat energy required to melt \(800\\ \\text{g}\) of ice at \(0^\\circ\\text{C}\) into water at \(0^\\circ\\text{C}\). Specific latent heat of fusion of ice \(= 336\\ \\text{J g}^{-1}\). (2 marks)
\(120\\ \\text{g}\) of water at \(90^\\circ\\text{C}\) is mixed with \(280\\ \\text{g}\) of water at \(15^\\circ\\text{C}\) in a vessel of negligible heat capacity. Find the final temperature of the mixture. (3 marks)
A \(500\\ \\text{W}\) heater is used to warm \(2\\ \\text{kg}\) of water from \(20^\\circ\\text{C}\) to \(50^\\circ\\text{C}\). Assuming all the heat supplied goes to the water, find the time taken. Specific heat capacity of water \(= 4200\\ \\text{J kg}^{-1},\\text{K}^{-1}\). (3 marks)
How much heat energy is needed to convert \(100\\ \\text{g}\) of ice at \(0^\\circ\\text{C}\) into water at \(40^\\circ\\text{C}\)? Specific latent heat of fusion of ice \(= 336\\ \\text{J g}^{-1}\), specific heat capacity of water \(= 4.2\\ \\text{J g}^{-1},\\text{K}^{-1}\). (3 marks)
A heater melts \(40\\ \\text{g}\) of ice at \(0^\\circ\\text{C}\) into water at \(0^\\circ\\text{C}\) in \(2\\ \\text{minutes}\). Taking the specific latent heat of fusion of ice as \(336\\ \\text{J g}^{-1}\), calculate the power of the heater, assuming no heat is lost. (3 marks)
What mass of ice at \(0^\\circ\\text{C}\) must be added to \(500\\ \\text{g}\) of water at \(30^\\circ\\text{C}\) to bring the water down to \(0^\\circ\\text{C}\), all the ice just melting? Specific heat capacity of water \(= 4.2\\ \\text{J g}^{-1},\\text{K}^{-1}\), specific latent heat of fusion of ice \(= 336\\ \\text{J g}^{-1}\). State the assumption used. (4 marks)
A metal block of mass \(200\\ \\text{g}\) at \(100^\\circ\\text{C}\) is dropped into \(250\\ \\text{g}\) of water at \(20^\\circ\\text{C}\) contained in a calorimeter of heat capacity \(150\\ \\text{J K}^{-1}\). The final temperature recorded is \(25^\\circ\\text{C}\). Calculate the specific heat capacity of the metal. Specific heat capacity of water \(= 4.2\\ \\text{J g}^{-1},\\text{K}^{-1}\). (4 marks)
Calculate the total heat energy required to convert \(2\\ \\text{kg}\) of ice at \(-10^\\circ\\text{C}\) into water at \(20^\\circ\\text{C}\). Specific heat capacity of ice \(= 2100\\ \\text{J kg}^{-1},\\text{K}^{-1}\), specific latent heat of fusion of ice \(= 336000\\ \\text{J kg}^{-1}\), specific heat capacity of water \(= 4200\\ \\text{J kg}^{-1},\\text{K}^{-1}\). (5 marks)