Q.5) A system has two quantum states, of energy 0, Ε.
a) Find the probabilities P1 and P2 when the average energy of the system Ε = 0.4Ε.
b) Calculate the entropy.
Q.6) Two independent systems 1 and 2 in thermal contact at a common temperature T is equal to the product of the partition functions of the separate systems Show that the partition function Z1+2 and the Helmholtz free energy F1+2 in terms of the partition functions Z1, Z2 and free energies F1, F2 are given by:
a) Z1+2 = Z1Z2 b) F1+2 = F1 + F2
Q.7) Consider a systems of N>>1 identical, distinguishable and independent particles that can be placed in three energy levels of energies 0, Ε and 2Ε, respectively. Only the level of energy Ε is degenerate, of degeneracy g=2. This system is in equilibrium with a heat reservoir at temperature T.
a) Obtain the partition function of the system.
b) What is the probability of finding each particle in each energy level?
c) Calculate the average energy <E>, the specific heat at constant volume, Cv, and the entropy S, of the system.
d) Define the high and low temperature limits. Give the mean energy and the entropy of the system in these limits. Justify qualitatively these results.