1. We can calculate other mechanical thermodynamic properties Mj Stat. Mech. Postulate: If you can calculate a mechanical property Xi consistent with the macroscopic parameters, then, <Xi> =macroscopic thermodynamic X Using the postulate Now that we know the probability of finding the quantum state with Ej at a given N,V

2. Internal Energy

3. Going back to the calculation of mechanical properties… Pressure With these tools in hand , we can now combine mechanical properties with thermodynamics and extract information about nonmechanical functions (like S or T)

4. A A A A A B B B B B heat bath(T) Consider 2 ensembles A and B that become in thermal contact, with no change in volume  dV = 0 ; dA = 0 dB = 0 Derivation of b Initially W({a,b}) = W({a*}) x W({b*}) Ensemble A  bA Ensemble B  bB What are the b values at the new equilibrium?

5. Derivation of b II

6. dlnW=bdE

7. The sum of the changes in energy in each system yields the infinitesimal change in the total energy

8. Derivation of b Approaching equilibrium, there is an increase in the value of EA, heat is transferred from B to A. bA>bB At equilibrium bA =bB any change in W({a}) will be canceled by a similar change in W({b}) 1st law: equilibrium means TA =TB, and approaching equilibrium heat will be transferred from B to A when TA<TB

9. and since and we obtain heatrev exchange workrev E and p relations

10. From thermodynamics, we already know that or Gibbs Entropy Entropy

11. and A(N,V,T)

12. All relationships

13. in the limit of T0 So far we worked with energy states, but we can easily change to levels, by counting how many times each “state” is repeated  degeneracy= W(N,V) States vs levels in the limit of T  Q(N,V,T)  constant = number of states