Statistical Physics

66 4 Ideal GasesExercise 11. Gibbs noticed that without the factor N!, the entropy of anideal gas has a strange behavior. Obtain an expression for S without thisfactor. Use this expression and show that the total entropy of two identicalsystems S(2E,2V,2N) is different from 2S(E,V,N).Exercise 12. Calculate the entropy of an ideal gas using (4.15), and comparethe result with (4.14).Exercise 13. The frequency ν of vibration of an O 2 molecule is 4.74×10 13 Hz.Evaluate the spring constant k = µω 2 of the bond between the oxygen atoms,where µ = m/2 is the reduced mass, m =2.66 × 10 −26 kg, and ω =2πν.