If we consider only heat and mechanical phenomena and do not include electrical occurrences, the irreversible processes may be grouped in four classes:
(a) The body whose changes of state are considered is in contact with one or more bodies whose temperatures differ by a finite amount from its own. There is here flow of heat from hot to cold and the process is an irreversible one.
(b) When the body experiences friction which develops heat it is not possible to effect completely the opposite operation.
(c) The third group includes those changes of state in which a body expands without at the same time developing an amount of external energy which is exactly equal to the work of its elastic forces. For example this occurs when the pressure which a body has to overcome is essentially (that is, finitely) less than the body's own internal tension. In such a case it is not possible to bring said body back to its initial state by a completely opposite procedure. Examples of this group are: steam escaping from a high-pressure boiler, compressed air flowing into a vacuum tank and a spring suddenly released from its state of high tension.
(d) Suppose two gases existing at the same pressure and temperature are on opposite sides of a partition; when the partition is quickly removed the two gases will diffuse or mix. These gases will not unmix of themselves and the diffusion process is an irreversible one and is somewhat like the process considered under (c).
The foregoing facts and propositions have in the main already been stated in this presentation and it will be profitable to make comparisons with the definition of irreversible and reversible events given on p. 30 and with the examples on pp. 31, 32.
HEAT CONDUCTION
The group under head (a) represents the irreversible processes which perhaps occur most often, namely, the direct passage of heat, by conduction or radiation, from a hot body to a cold body, here say from a hot gas to a cold gas. The former loses in heat energy what the latter gains. As radiation phenomena have very special features of their own and for the present may be said to be outside of our selected province, we will confine our attention to heat conduction alone. Moreover, for our present purpose, we will suppose said flow or change to take place without alteration of volume of either the hot or the cold gas. Then will the hot gas experience a drop in temperature and the cold one a rise in temperature. We have already treated such isometric changes and know that the number of complexions is thereby diminished in the originally hotter body and increased in the originally colder one. If this increment is greater than the accompanying decrement, then the final outcome of this direct passage from hot to cold is an increase in the total number of complexions of the two gases. There will then, by our precise definition, be a corresponding increase in the total entropy of the two systems. It is foreign to our present purpose to prove in an independent, purely mechanical way, that such excess does finally exist and will here content ourselves with the well-known and simple thermodynamic expression for this excess, where is the heat energy thus directly transferred from the hot to the cold body, the absolute temperature of the hot body and that of the cold body.
THE WORK OF FRICTION IS CONVERTED INTO HEAT
The group under head (b) contains a class of events which usually attends, in one form or another, most natural phenomena.
We will here consider an interesting (but perhaps too special) case, namely, the experiment performed by W. THOMSON and JOULE on the flow of gas through a porous plug. The plug obstructed the uniform, non-conducting, passage through which the gas was forced without sensibly changing its velocity of flow: (See LORENZ' Technische Wärmelehre, p. 275). It can easily be shown (in L., p. 274) that with an ideally perfect gas, As a matter of fact, there was an actual though slight drop in temperature found to exist with the most perfect gases available. Evidently the process was a throttling one, reducing the larger initial pressure to the smaller final one, which reduction was of course accompanied by a corresponding increase in volume.
Assuming that an ideally perfect gas was employed in the experiment, and that the final state for our consideration is that corresponding to its attainment of thermal equilibrium, we see that because of the unchanged temperature there is here no loss of internal energy, for the work consumed by the friction of the porous plug is all returned to the gas by the heat developed by said friction. Moreover, the + and - external work in this experiment also balance. Now although there has been no loss of energy there has been a growth of entropy corresponding to the evident increase in the number of complexions. This increase is exactly equal to that found for reversible isothermal change of state when accompanied by an increase in volume, and the discussion is therefore not repeated here.
One phase of the above process is the conversion of mechanical work into heat through the medium of friction.
INCREASE OF VOLUME WITHOUT PERFORMANCE OF EXTERNAL WORK BY ELASTIC FORCES OF THE GAS
This case of an irreversible process comes into group c. We will consider here JOULE'S well-known experiment with the air tanks, in which the compressed air, initially stored in the one tank, was allowed to discharge into the other tank which, at the start, contained only a vacuum. At the end of the experiment, when thermal equilibrium obtained, the temperature in the two, now connected, tanks was the same as originally existed in the compressed-air tank. Here of course it is assumed that the air exchanged no heat whatever with the outside.
As the final state, like the initial state, is in thermal equilibrium, and possesses the same temperature, we can ascertain the total change in the number of complexions as we did when discussing isothermal and reversible changes and because of the accompanying increase in the volume of the air, find that here as there the number of complexions has increased and that therefore the entropy of the air has increased in this case.
We might rest satisfied with this conclusion, but additional light will be shed on entropy significance if we consider more in detail the intermediate stages of this evidently irreversible process. The rush of air from the full to the empty tank produces whirls and eddies of a finite character and it is only when these have subsided, by the conversion of the visible or sensible kinetic energy of their particles into heat, that thermal equilibrium obtains. But at each intermediate stage (while still visibly whirling and eddying) the gas possesses entropy, even while in the turbulent condition. This is clear from our present physical definition of entropy, namely, the logarithm of the number of complexions of the state, for it is evident that even in this turbulent state it possesses a certain number of complexions, however difficult mathematically it may actually be to find this number. Boltzmann found an expression for any condition; PLANCK gave it the form of Eq. (18), p. 63, where (in the C.G.S. system) is a universal constant, function is the law of distribution of the particles and their velocity elements and is a sort of fictitious elementary region in a six-dimensional space. From its derivation and definition the value given for entropy in Eq. (33) depends only on the state of the body at the instant in question and does not at all depend on its history preceding this instant.
The difference between the value of for the final state (say, as given for a gas by Eq. 20) and the value of as given by Eq. (33) for the instant, constitutes the driving motive which urges the gas toward thermal equilibrium. A similar difference or driving motive is the underlying impelling cause of all natural phenomena.
OF THE DIFFUSION OF GASES
This case of an irreversible process comes under group d. Concerning this phenomenon J. W. GIBBS established the following proposition:
"The entropy of a mixture of gases is the sum of the entropies which the individual gases would have, if each at the same temperature occupied a volume equal to the total volume of the mixture."
That the total entropy will be larger as a result of the mixing detailed under d, p. 73, may be inferred from the following consideration: When two gases are thus brought together, it is more probable that in any part of the total space available for this mixture there will be found both kinds of molecules than only one kind of these molecules.
But this irreversible process can be explained in a more distinctly physical way. The two gases are originally at the same pressure and temperature; they mix without other changes occurring in surrounding bodies; the mixture (when diffusion is completed) is at the same pressure and temperature as the original gaseous constituents. Considering each gas by itself, what has happened as the result of diffusion is that each gas in its final state occupies a larger volume than in its initial, unmixed, state. The presence of the other gas in the mixture in no wise changes this fact. Of course this increment in volume is accompanied by a corresponding decrement in its pressure, without change in temperature. A sort of isothermal change of state has taken place in the passage from one condition of thermal equilibrium to the other. We have already seen that then the number of complexions of the gas increases and consequently also its entropy. The sum of the increments of the number of complexions separately experienced by the two diffusing gases constitutes an increase in the total number of complexions over and above the total number of complexions existing in both gases before diffusion. There is of course a corresponding increase in entropy due to such diffusion.
All these irreversible processes are passages from less stable to more stable conditions, from less probable to more probable states, or summarizing:
There is in Nature a constant tendency to equalize temperature differences, to convert work into heat, to increase disgregation and to promote diffusion.
This tendency has also been described as the tendency in Nature to pass from concentrated to distributed conditions of energy.
The four irreversible processes just discussed are all spontaneous ones, i.e., they occur without the help of agencies external to the bodies directly engaged in the transformations.
It is evident that the foregoing statements are really identical, expressing the same thought in different ways.