Topic Start | Study Unit | Previous Page | Next Page

Presentation

Fig. 5.1 shows the basic framework of our thinking. Our problem is to bring what we know of the behaviour of molecules into correspondence with what we know of the behaviour of systems.

From the molecular side, we begin with what appear to be the simplest notions concerning what molecules are and how they behave. We project ourselves into a time when nothing was known of the internal structure of molecules and the complexity of their internal behaviour. Instead, we picture them simply as bundles of matter whose behaviour can be completely described in terms of motions. In this way, we can bring the concept of kinetic energy into our description of molecules. If we treat the energy of molecules as being only kinetic, then it can be completely described in terms of masses and velocities.

On the macroscopic side, we know something about the internal relationships which exist between the different parameters by which we measure the state of a system. We know, for example, that what we call the pressure is directly proportional to what we call the temperature of a given system in equilibrium. We also know that, within certain limits, the specific heat at constant volume of a gas can be taken as constant.

Now, if we take it that these internal relationships between the parameters of a system result from the interactions between molecules and their environment, then it would also seem possible to construct a model of a gas in terms of ideas of how molecules behave. Such a model, which represents the totality of a large number of molecules, should form an analogue of the system; that is to say, the behaviour of the totality of molecules should be equivalent to the behaviour of the system as a whole.

It should be understood that there are two mutually complementing parts to our analogue. First, we have the concept of what a single molecule is like, and how it behaves. Second, we have a picture of how such molecules aggregate together. A change in one of these parts of the analogue must inevitably lead to a corresponding change in the other part.

By supposing that the internal energy of a gas is entirely equivalent to the sum of the kinetic energies of its constituent molecules, we already presuppose that everything can be explained in terms of simple mechanical interactions between bundles of matter. But when we come to construct our analogue out of a representation of a totality of such molecules, we will see that we have to introduce means of interaction between these molecules which cannot be explained in simple kinetic terms. In fact, it is just these non-kinetic interactions which enable there to be that coalescence between the molecules of a gas which constitute it as a system.

Behaviour of the system as a whole.
We must begin by understanding what it is we have to analogue. By investigating the behaviour of gases at low pressures, we find a common pattern of behaviour:

PV = constant x T

When 1 gram molecule of any gas is investigated, again at low pressure:

PV = RT (ideal gas equation of state) [5.1]

This is the equation of state of the ideal gas and R is a constant applicable to all gases. The actual behaviour of gases does not, of course, follow this pattern exactly, but the approximation is good provided we keep well away from the transition to the liquid phase. The three variables P, V and T are parameters of state of the system as a whole. If any two of them are fixed, then the state of the gas is determined. In a definite state, the gas will have a definite internal energy, so the internal energy should be determined by any two of the parameters of state. For example, U can be represented as a function of V and T: U = U(V,T). Then:

We can find out about the variation of U with variation in V or T by various experiments. We take, as an example, Joule expansion where the gas expands into a vacuum. For a gas to which the ideal gas equation strictly applies, there is no change of temperature due to this expansion. We look at the equation for changes in U in terms of V and T, and put T and U equal to zero. U is zero because no work is done against the vacuum and no heat ransfer is observed: since no energy enters or leaves the system by heat or by work the internal energy must remain constant. Then:

Therefore the variation of U with volume is zero, and hence U depends only on T: the internal energy is a function of temperature alone.

The ideal gas.
The system for which the equation PV = RT holds exactly is called an ideal gas. Because of the simplicity of the formula, important deductions about the behaviour of ideal gases can be made using mathematical techniques and the First Law of Thermodynamics. We found (by using one experiment) that:

is the rate of change of the internal energy with temperature at constant volume. At constant volume, no work can be involved. We can now use the First Law in this way:

Q = U + W and W = O

i.e. increases in internal energy are due to heat transfer alone.

The expression is the specific heat at constant volume c.
Thus U = cT
Further experiments show that c, is a constant, independent of temperature.
So U = U + cT, where U is a constant. We will set this constant U at zero. Then:
U = cT [5.2]

Mathematical deductions.
The First Law gives us an equation for the conservation of energy. The equation U = cv T gives us an expression for the internal energy in terms of temperature alone. The equation PV = RT enables us to express any one parameter in terms of the others. The First Law is expressed:

Q = U + PV

(where signifies a very small amount).

If we divide through by T we have (provided that the pressure is constant):

We can also deduce the adiabatic equation of state.

If we have a mass of ideal gas which is thermally insulated, what combinations of P, V and T are possible? Without going into detail:

Putting the two together enables us to write:

The molecular picture.
We have to represent molecules, and the microscopic level of behaviour, in such a way that we can eventually form an analogue of the behaviour of the system. One point must be made at the outset. In order to represent (i.e. put into a descriptive language we can understand) the behaviour of molecules, we have to use the terminology and concepts with which we are familiar. But these terms and concepts have been built up mainly to describe macroscopic behaviour. We set out below the representation of a molecule made in the elementary kinetic theory of the ideal gas, and indicate something of its limitations.

(i) It is a point mass; but an actual mass must occupy a finite volume.
(ii) It has only kinetic energy of motion; but a molecule also has an internal structure (protons, neutrons, electrons, etc.) which means that it has its own internal energy. Further, it can rotate or vibrate and hence have rotational and vibrational modes of energy.
(iii) It makes perfectly elastic and instantaneous collisions with the walls of the enclosure which contains it; but the walls are composed of molecules too, and a molecule does not actually strike another molecule. Instead, strong short repulsive forces come into play.

We consider an ideal gas of volume V, temperature T, pressure P, and internal energy U, and assume that it consists of a very large number, N, of such molecules. For the internal energy, we can write:

i.e. the sum of the kinetic energies of the molecules.

By some means - for example, vibrations of molecules in the walls of the enclosure - the N molecules have an equal chance of sharing in the total energy. Over a long enough time-span, then, the average energy of each molecule is the same. We can consider, for this time-span (much less than a second), N average energy molecules. We write the average energy:

For convenience, we are going to base our calculations on mythical "average energy molecules" instead of doing the averaging at the end, because the main ideas are easier to see this way. We are justified in this because, in the elementary kinetic theory, each molecule is treated as an isolated particle, and over a period of time it can be considered as an average energy particle in terms of its cumulative effects on system behaviour.

The number of impacts made on unit area of the surface of the enclosure in one second is to be identified with the pressure. First we need to represent the motion of the molecules. If we are considering N "average energy molecules" each will "have" a velocity but the directions of motion will be random. One way of representing the random directions is to have all these "average molecules" passing through the centre of a sphere along tracks such as those shown in fig. 5.2. These "average molecules" are abstractions from our representation which enable us to picture the distribution of energy with respect to direction. Inside a sphere, the random motion of molecules will have just the same distribution of energy with respect to direction as that shown in fig. 5.2. There must be N tracks each associated with kinetic energy in order to have total energy N = U. We use this representation as the basis of our calculations.

Use fig. 5.2. as a reference. In one second, one half of those "molecules" which are within a distance of magnitude will strike the surface; i.e. the number of "molecular hits" per second will be:

Each "molecular hit" gives an impulse of . The total impulse in one second on unit area is the force exerted per unit area-which is the pressure:

Using [5.3] this becomes:

From our knowledge of system behaviour we have that U = c y T [5.2]. We can then suggest that

and we find that this agrees well with experiment. More important, our analogue has brought us to the point where we can see a connection between , the average kinetic energy of a molecule, and T the absolute temperature of the system as a whole.

In order to develop our analogue further, we can either go back and criticize our representations and assumptions, or we can look more clearly at the system behaviour of actual gases. We take the second of these alternatives, leaving the first to the Investigation.

The macro-behaviour of actual gases.
Some examples of the pattern of behaviour of actual gases are shown in fig. 5.3. Note that as P 0 the values of PV for all gases (one gram molecule) converge to the ideal gas value.

Equations of state for actual gases are complicated. They can be expressed in the general form:

where the numbers A, B, etc. are functions of temperature (they are called the "virial coefficients"). A well known instance of this kind of equation is Van der Waals' equation of state:

This equation works pretty well when compared with results such as those shown in fig. 5.4 (for carbon dioxide), even for the liquid phase. It does not work when there are the two phases of liquid and vapour together. The (isothermal) lines in fig. 5.4. show that only when the temperature is below T - that of the critical point - can there be a liquid. But the graph also suggests that there is a continuity between the gaseous, vapour and liquid conditions.

We must be able to integrate this fact into our analogue.

Let us also consider how a gas is brought into the liquid condition. One ol the key methods uses Joule-Kelvin throttling - see fig. 5.5.

The action is as follows: gas is pushed from a high pressure region to a low pressure region in a continuous flow, and the system is insulated so that there is no heat transfer. From the First Law, we can therefore deduce that U + PV is a constant. If the gas were ideal, the temperature would not change; hut for any actual gas:

P 1 V 1 P 2 V 2 and therefore T 1 T 2

Whether the gas will heat or cool depends on the conditions and the actual gas involved. Experimental study of a gas yields an inversion curve (see fig. 5.6),

which is a series of points (joined together in a smooth curve) joining the maxima of lines with constant values of U + PV. Decrease of pressure from any of these points then means a fall in temperature, and so the gas will cool if throttled under these conditions. In the throttling processes we go from high pressure to low pressure, i.e. from right to left on the graph. Only if the initial temperature is below the inversion temperature (see fig. 5.6) will there be cooling.

If then, we can pre-cool a gas, by adiabatic expansion, or surrounding it with boiling oxygen, etc, to below its inversion temperature, repeated Joule-Kelvin throttling will cool the gas below its critical temperature and it will liquefy (perhaps under pressure).

Re-interpreting the analogue.
We managed to get quite a good way by equating the internal energy of an ideal gas with the sum of the kinetic energies of its molecules. We did this at the expense of realism in our representation of molecules, and we also found that actual gases exhibited behaviour quite inexplicable in terms of our analogue, which is not surprising, since it was made to fit ideal gas behaviour. By looking critically at the assumptions we make in the kinetic theory, we can make some guesses about actual molecular behaviour. It is then possible to investigate the energies which bring coherence into the behaviour of systems.