Page:Scientific Papers of Josiah Willard Gibbs.djvu/202

that the quantities of the gases are such that each occupies initially one half of the total volume. If we denote this volume by $V$, the increase of entropy will be

$m_{1}a_{1}\log V+m_{2}a_{2}\log V-m_{1}a_{1}\log {\frac {V}{2}}-m_{2}a_{2}\log {\frac {V}{2}},$

or⁠$(m_{1}a_{1}+m_{2}a_{2})\log 2.$

Now⁠$(m_{1}a_{1}={\frac {pV}{2t}},$⁠and⁠$m_{2}a_{2}={\frac {pV}{2t}}\cdot$

Therefore the increase of entropy may be represented by the expression

${\frac {pV}{t}}\log 2.$

(297)

It is noticeable that the value of this expression does not depend upon the kinds of gas which are concerned, if the quantities are such as has been supposed, except that the gases which are mixed must be of different kinds. If we should bring into contact two masses of the same kind of gas, they would also mix, but there would be no increase of entropy. But in regard to the relation which this case bears to the preceding, we must bear in mind the following considerations. When we say that when two different gases mix by diffusion, as we have supposed, the energy of the whole remains constant, and the entropy receives a certain increase, we mean that the gases could be separated and brought to the same volume and temperature which they had at first by means of certain changes in external bodies, for example, by the passage of a certain amount of heat from a warmer to a colder body. But when we say that when two gas-masses of the same kind are mixed under similar circumstances there is no change of energy or entropy, we do not mean that the gases which have been mixed can be separated without change to external bodies. On the contrary, the separation of the gases is entirely impossible. We call the energy and entropy of the gas-masses when mixed the same as when they were unmixed, because we do not recognize any difference in the substance of the two masses. So when gases of different kinds are mixed, if we ask what changes in external bodies are necessary to bring the system to its original state, we do not mean a state in which each particle shall occupy more or less exactly the same position as at some previous epoch, but only a state which shall be undistinguishable from the previous one in its sensible properties. It is to states of systems thus incompletely defined that the problems of thermodynamics relate.

But if such considerations explain why the mixture of gas-masses