166 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



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 



V V 



or (ra^ -f m 2 a 2 ) log 2. 



xr P v 



Now m-r and wa 



Therefore the increase of entropy may be represented by the 

 expression 



(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 circum- 

 stances 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 thermo- 

 dynamics relate. 



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



