CHAPTER XVIII

INTERNAL WORK, AND THE COOLING OF GASES ON FREE

EXPANSION

Joule's Experiment.-In Chap. XIII., when considering the results which might be anticipated from the Kinetic Theory of gases, the question arose, is internal work performed during the separation of the molecules which occurs during the expansion of a gas? An experiment of Joule's was then described; in this, a quantity of gas contained in a vessel under high pressure, was allowed to expand into another vessel which had been exhausted. Both vessels were, in the first experiment, surrounded by water contained in the same calorimeter.

In this experiment, no external work was performed by the gas during expansion. The gas expanding in one vessel compresses the gas contained in the other vessel. Thus, a cooling effect might be anticipated in the first vessel, and a heating effect in the second one. At the end of the experiment the gas occupied a greater volume than at first; and if, after stirring the water in the calorimeter, its final temperature is found to be unchanged, we must conclude that no appreciable amount of work has been performed during the separation of the molecules; or, in other words, that the average attraction or repulsion exerted by one molecule on another is very small.

Joule's experiment led to the result just stated. It must, however, be remembered that if a small heating or cooling, say through a fraction of a degree, does really occur when a gas expands without doing external work, the quantity of heat given out (or absorbed) will be so small that little change

will be produced in surrounding bodies. Hence, Joule's experiment must only be taken as proving that no heating or cooling effect of any considerable magnitude occurs during the free expansion of the ordinary gases.

The weak point in Joule's experiment was unquestionably the use of water to indicate, by its change of temperature, whether heat disappeared or was generated in the expanding gas. The specific heat of air at constant volume, according to Joly, is equal to 1721 therms per gram. Now a gram of air, at 。° C. and under atmospheric pressure, will occupy a volume of 773'4 c.cs. The heat required to raise the temperature of this volume of air through 1o C., if abstracted from a gram of water, would only cool the latter through 17° C. Thus, though Joule's thermometer was capable of indicating a difference of temperature of 26° F. (360° C.) it is obvious that no heating or cooling effect, unless of a considerable magnitude, could possibly have been detected by the method he employed.

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Investigation of Joule and Lord Kelvin.—Thermodynamical considerations lead to the conclusion that whereas there should be no heating or cooling of a perfect gas on free expansion, gases which do not obey Boyle's Law should exhibit a small thermal change under the same conditions. Lord Kelvin and Joule therefore determined to investigate the phenomena attending the free expansion of gases, using a method in which the temperature of the gases could be directly measured.

Before describing the actual experiment performed by Joule and Lord Kelvin, we will consider an ideal arrangement illustrating the character of the process employed.

Let us suppose that we are provided with a long cylinder CD, fitted with two air-tight and frictionless pistons, A and B, and possessing a diaphragm E pierced by a small aperture.

FIG. 164.-Gas forced through narrow orifice under pressure.

Let us suppose that the piston B is initially pressed close up to the diaphragm E, whilst a certain quantity (say I gram) of compressed gas, at a pressure 1, is introduced into the part of the cylinder between

A and E. We must further assume that the walls of the cylinder are perfect non-conductors of heat. Thus, if T is the initial temperature of the gas, any departure from this value can only be produced by the performance of work on or by the air, and not by the direct transference of heat.

Now let the motion of the piston B be opposed by a uniform force F2.

Let a be the area of either piston. Then the piston B will move outward uniformly when the pressure of the gas between E and B has attained such a value p2 that

F2.

Also let the force, tending from the first to force the piston A inwards, be denoted by F1. Then, as the gas passes through the orifice in E, so as to press the piston B forwards, the piston A will move inwards towards E at such a rate that the pressure of the air between A and E remains constant. Then Pia = F1.

Let us suppose that the gas initially contained in AE occupied a volume v1. Also let x, denote the distance between the piston A and the diaphragm E at the commencement of the experiment. Then

When the piston A has moved up to the diaphragm E, the whole of the air will have been forced from the compartment AE into the compartment EB of the cylinder. Let v2 be- the volume occupied by the air in EB, and let x, be the distance through which the piston B has meanwhile moved, from its initial position immediately against the diaphragm E. Then

No heat can enter or leave the cylinder, so that any change in the energy of the contained gas must be due to the performance of work on or by it.

Let E be the internal energy possessed by the gas when in AE, and let E2 be the internal energy possessed by it in EB.

The work performed by external agency on the gas is equal to the product of the force F1 into the distance x through which it has acted.

The work performed by the gas is equal to the product of the force resisting the motion of the piston B (i.e., F2) into the distance x2 through which that force has been overcome.

... Work performed by the gas

The difference, E2 E1 between the final and initial energies of the gas, must be equal to the difference between the work performed on the gas, and that performed by it.

Now, the energy possessed by the molecules of the gas may be partly kinetic and partly potential. The kinetic energy we have learnt to associate with heat; the potential energy will depend on the relative mean positions of the molecules, supposing that attractive or repulsive forces are exerted between them.

We can now consider the following cases :

I. Þ1v1 = p2v2 (i.e., the gas obeys Boyle's Law).

In this case E1 = E2. If there is any attractive force exerted between neighbouring molecules, this force must have been overcome during the expansion of the gas, and consequently the potential energy of the molecules must be greater in the final than in the initial condition.

But the total energy has the same value in both cases. Therefore the kinetic energy of the gas is smaller in the final than in the initial condition. In other words, the gas will be cooled during the process described above.

If repulsive forces are exerted between neighbouring molecules, the potential energy of the molecules will be diminished during expansion, and consequently their kinetic energy will increase. In other words, the gas will be heated during the process described above.

II. Þ11< Þ2V1⁄2. In this case the product of the pressure and volume of the gas decreases as the pressure is increased. This condition generally holds during the initial stages of the compression of a gas. Hydrogen is, however, an exception (see p. 204). Since

that is, the energy possessed by the gas is less in the final than in the initial condition.

Therefore, if no forces are exerted between neighbouring molecules, a slight cooling effect will be produced.

If molecular attractions are exerted, a still greater cooling will result.

If molecular repulsions are exerted, the cooling due to the divergence from Boyle's Law may be partially or entirely compensated for, or a heating effect may be produced.

III. Þ11 >Þ22. In this case the product of the pressure and volume of the gas increases as the pressure is increased. Regnault and Amagat found this to be the case with hydrogen, and Amagat showed that it is also the case with most gases, when subjected to very high pressures.

Since

E2 E1

P11 - P2P2 = E2 - E1

= some positive value = + § (say) E2 = E1 +

A heating effect will be produced if no molecular forces are exerted. This heating effect will be enhanced if molecular repulsions exist. If molecular attractions exist, a smaller heating effect or even a cooling effect, may be produced.

Modifications of the above Ideal Arrangement.Returning to Fig 164, it may easily be understood that in the neighbourhood of the orifice in E the gas will form eddies. But any motion of finite portions of the gas will entail a diminution in the energy of molecular motion. Thus, near the orifice in F the gas will be colder than at points further removed, where the gas has been brought to rest by internal friction. Consequently, in performing an experiment such as that just described, care must be taken that the temperature of the gas is measured at a point where eddies have ceased to exist.

Further, the piston B may be dispensed with. It serves to divide the gas from the surrounding atmosphere; but if it is removed, the atmosphere will be forced back in an essentially similar manner; the pressure på will then be equal to the atmospheric pressure.

Care must be taken that the jet of air issuing from the orifice in E does not produce any sound, as this would involve a loss of energy.

Instead of forcing the piston A along the cylinder, the end C of the latter may be connected to a pump, provided the action of the latter is such that the pressure of the gas, at all points to the left of the diaphragm E, is maintained constant. The essential point in the arrangement is that no change shall take place in the compressed gas till it passes through the orifice in E. Thus, the gas between E and the pump must be maintained at a constant temperature and pressure, and must serve to transmit