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The calorimetric unit of heat. When two bodies at different temperatures are put into thermal contact with each other, they will change in temperature until each is the same as the other. We suppose that there is some kind of exchange between the two bodies. The exchange is not of matter; it is of energy. We can go on to make the assumption that what happens is a transfer of energy from the hotter to the cooler body. An increase in the temperature of a body means an increase in its energy, and a decrease in its temperature means a decrease in its energy. By the principle of the conservation of energy, the energy loss of the hotter body is the energy gain of the cooler one (see fig. 1.1 ).


If we keep the two initial temperatures fixed but use several different bodies, then we obtain different filial common temperatures. We say that the bodies have different thermal capacities (C). In the simple case of two bodies initially at temperatures and , where > :

C ( - ) = C ( - ) [1.1]

where C, C are the two thermal capacities, and is the final temperature. This is to suppose that if an input of heat energy Q to a body increases its temperature by an amount , then:

Q = C [1.2]

The thermal capacity of a body is not strictly speaking a constant, so it should always be named for a specific temperature range. For a given substance at constant density, the larger a body of that substance is, the greater is its thermal capacity. In fact, as experiment shows, the thermal capacity is directly proportional to the mass of the given substance. We can therefore write:

C = m.c [1.3]

where m is the mass and c is the specific heat of the substance, or the thermal capacity per unit mass.

By combining together equations [1.2] and [1.3], we can form a definition of a unit of heat. The calorie is defined as the amount of heat required to raise the temperature of one gram of water from 14.5 ° C to 15.5°C. In Study Unit 2 we shall see how quantities of heat can be expressed in mechanical units, such as ergs or joules. Here, it is sufficient to have some concept of the practical basis of defining a unit of heat in calorimetry.

Connection between heat input and rise in temperature. The thermal capacity of a given mass is not independent of the temperature; though for certain gases, for example, it is almost constant over a very wide range of temperature. If we made a graph of temperature rise against heat input for a given substance, in most cases we would obtain an irregular line such as that shown in fig. 1.2.

The thermal capacity for the range to is given by (see figure) Q / ( - ). But the thermal capacity at temperature is given by ( dQ / d ) ; that is, the slope or gradient of the line at that temperature.

If we were making measurements on a liquid then, as the temperature increased, the graph eventually would show a region in which heat enters the mass but produces no rise in temperature (see fig. 1.3).

This region corresponds to the boiling of the liquid. The heat energy all goes into the change of structure of the substance (which turns it from the liquid to the vapour phase) until there is only vapour, when the temperature will again begin to rise. On cooling, the heat taken up in this change of structure is released when the vapour condenses-which again takes place at constant temperature. The quantity of heat required to bring about a change in structure of unit mass (usually one gram) of a substance is called the latent heat of that substance for the change in question e.g. vaporization, condensation, etc. In Study Unit 6 we will investigate the nature of a change of phase, such as that from liquid to vapour or from vapour to liquid. There we will use the concept of entropy. In Study Unit 3 we will explain how an input of heat must always increase this entropy, but not necessarily increase the temperature.

What would have happened if we had made these measurements at different pressures? Both the temperature of boiling and the magnitude of the latent heat would be different. Further, in the case of the vapour, the measured specific heat would be different depending on whether we kept the volume or the pressure constant. The reason for this is that, in changing its volume, the vapour would do work and expend some of the energy entering it as heat. Such properties are discussed in Study Units 2 and 5.

In giving a latent heat we have to indicate the pressure (or the corresponding temperature) at which it was measured, and in giving a specific heat we have to indicate the conditions of our measurement (usually constant volume or constant pressure). Now, how in practice can these quantities be measured in terms of our calorimetric unit of heat?

Measurement of specific heat. The first step is to compare the thermal capacities per unit mass-i.e. the specific heats of different substances. if we mix two substances which are initially at different temperatures and measure their final equilibrium temperature, we can find the ratio of the specific heats c and c of the two substances. We write down the equation:

mc + mc = O [1.4]

m is the mass of one substance and m the mass of the other. is the change in temperature of the first substance and the corresponding change in temperature of the second substance (either or must be negative). The equation assumes that no thermal energy is either gained or lost by the mixture. Then:

c / c = -((m) / (m))

Now if we make one ofthe substances water, we can (under certain conditions) put c numerically equal to one since the specific heat of water between 14.5°C and l5.5°C is defined as one calorie per gram ner centigrade degree. Then:

c = -((m) / (m))

This method of mixtures can be extended to involve any number of substances. Usually, we mix two substances together in a calorimeter, so that there are in all three substances involved. By a series of experiments we can then measure the specific heat of any substance in terms of calories per unit mass per degree rise. But at no point do we actually measure quantities of energy. A good question to think about at this stage is how it can be possible to measure specific heats for temperatures much higher or much lower than 15°C (for example, the specific heat of iron between 500°C and 600°C).

A further point is that equations such as [1.4] invoke the principle of the conservation of energy. This is tantamount to assuming that the calorimeters we use have perfect thermal insulation against interaction with the environment. But how is this realized in practice?

We can take a simple example of a calorimeter system: Regnault's apparatus as shown in fig. 1.4.

The surface of the calorimeter is silvered to reduce radiant transfer of heat. It is supported on three glass points set in wood to reduce conduction. Cotton wool is placed around the top of the calorimeter to prevent convection currents being set up. The calorimeter is provided with a lid to prevent evaporation, and enclosed by a constant temperature water-bath so that the environmental temperature can be made steady and easily measurable.

Even with all these precautions, heat loss cannot be reduced to zero. If we plot the temperature of the calorimeter against time, we will find a curve such as that shown in fig. 1.5 (the full curve).

The horizontal line denotes the temperature of the enclosure (it will not be perfectly horizontal). The dotted curve is the plot we suppose we would have if there were no heat exchange between the calorimeter and the enclosure. (Why does it come to the right of the solid curve in the region A?)

How do we compensate for the temperature drop due to heat losses? (i.e. the difference between and ' in the diagram). Part of the solution is indicated in the figure: begins below , so that in the first half of the duration of the experiment the calorimeter takes in heat, and in the second half it loses heat. But the two halves do not involve equal quantities of heat. Compensation for the disparity is made by calculation, using measurements on the graph of temperature against time.

There is quite a different way of avoiding heat losses, and thereby having the kind of isolation needed for measurements involving quantities of heat. The point is that there is a heat loss only if there is a temperature difference. We take up this clue later on.

Measurement of latent heat. The measurement of the latent heat of water can be done by allowing steam to pass into a water calorimeter, where it condenses, giving up its latent heat, and cools to the temperature of the calorimetric contents. Difficulties arise from the composition of the steam and from the incomplete assimilation of the steam into the calorimeter. Boiling always takes place with some degree of "violence": the temperature is slightly irregular and droplets of water are carried with the steam. However, saturated water vapour at 100°C can be produced, and this corresponds exactly to the steam produced by an ideal, nonviolent change of phase.

A fairly precise method was developed by Berthelot. Splashing is avoided by using the raised tube T (see fig. 1.6).

The calorimeter is shielded against the heating system. The steam does not mix with the water of the calorimeter but condenses in the glass tube G. This makes the mass of condensed steam easier to weigh. The open end of the tube A can be connected to a pump, so that the pressure on the boiling water can be altered and hence the ternperature of boiling changed. Thus the latent heat at different temperatures can be measured. The equation used is

C is the thermal capacity of the calorimeter and is its original temperature. m is the mass of the steam condensed and is its "boiling" temperature. L is the latent heat of vaporization and the final temperature attained. We take the specific heat of water to be unity.

It would be much simpler if we could measure the heat input required to change the phase of a given mass of liquid without becoming involved in changes of temperature. In that case, the experiment would have to take place at the temperature of the change of phase involved. In situations where there is a transfer of heat with a constant temperature distribution, i.e. the same place has the same temperature at all times, there will be a heat loss only if there are temperature differences with respect to the immediate surroundings.

Measurement of heat transferred. We have talked about measurements of quantities of heat in terms of the calorimetric unit of the calorie. If this unit is used, then the experimental methods must at some stage involve calorimetry. But the calorimeter has the defect of inaccuracy because of the inevitable heat losses involved. In certain experiments, calorimetric methods have to be used when we cannot control the heat transfer involved.

A more serious drawback to calorimetric methods based on the calorie is that, strictly speaking, we are unable to make accurate measurements in regions of temperature other than the l45°C to 15.5°C range for which the calorie is defined. lt would be very convenient if we could use sources of heat for which we could measure the heat produced directly. Such a source is an electrical resistance in a circuit. From our knowledge of electricity, we know that energy is being supplied to the resistance in order to maintain the current it carries. Now we ask: where does this energy go? (In other words, we invoke the principle of the conservation of energy.)

We have a multitude of evidence that such an electrical resistance produces results identical to those we associate with heat. We can therefore say that the electrical energy supplied to the resistance is converted into heat, and that the resistance is then a source of heat.. When the resistance is at a stable temperature (why is this stipulation necessary?) all of the energy supplied to it is lost as heat. In other words, we make the equation: mechanical energy lost = thermal energy gained. So the rate of heat production is Ri units of energy per second, where R is the resistance in ohms and i the current in amperes. in this instance then, we measure the quantities of heat involved directly and in terms of mechanical units of energy-not calories.

In Study Unit 2 we investigate the grounds on which we relate together quantities of thermal energy (e.g. "heat") and quantities of mechanical energy (e.g. work in an electrical circuit). The theme of the relation between thermal and mechanical phenomena runs through all the Study Units.