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
ΔSmixing=kln(NAN),
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.