The mixing entropy formula derived in the previous problem actually applies to any ideal gas, and to some dense gases, liquids, and solids as well, For the denser systems, we have to assume that the two types of molecules are the same size and that molecules of different types interact with each other in the same way as molecules of the same type (same forces, etc.). Such a system is called an ideal mixture. Explain why, for an ideal mixture, the mixing entropy is given by

where N is the total number of molecules and NA is the number of molecules of type A. Use Stirling’s approximation to show that this expression is the same as the result of the previous problem when both N and NA are large.

Solution 38P

Suppose we start with a number of identical molecules. We can apply this argument to any system in which the molecules all have similar properties and interact with each other in the same way. The entropy of this system is which is not that easy to calculate.

Now suppose after some time, some of the molecules changes to a different type which has similar properties to the original.. The entropy will increase by the number of distinct ways we can choose to locate these molecules among the places available. The entropy due to the number of possible locations and momenta of the molecules won’t change when we replace of the molecules by a different type of molecule, since that is already accounted for by S.

We’re interested only in the extra entropy generated by introducing a second type of molecule.

The number of ways of choosing locations from a total of is just so the entropy of mixing is

Using Stirling’s approximation for large N and taking = ()as before ,we get

=

[] (we can neglect the first term as it is negligible for large N)

This expression is the same as the result of the previous problem when both N andare large.