Problems
Problem 1: Consider a monatomic ideal gas of N molecules confined in a volume V having total energy E in thermodynamic equilibrium. Calculate the number of microstates of the system considering
Calculate the entropy S of the system as a function of (E,N,V) considering both the formulae and calculate the temperature of the system..
Problem 2: Consider a system with two compartments with an impenetrable partition. Both the compartments of equal volume V are filled with the same monatomic ideal gas of N molecules and total energy E each. The whole system is in thermodynamic equilibrium. Calculate the change in entropy 
after removing the partition once without dividing Ω by 
and then dividing Ω by . Check that 
in the first case and =0 in the second case. (Gibb's paradox is resolved only if the gas molecules are assumed to be indistinguishable).
Problem 3: Consider a system of N localized spin-1/2 magnetic ions of magnetic moment μ in an external magnetic field H having total energy E. Calculate the entropy of the system 
where 
is total number of accessible states with 
up spins and 
down spins configurations. Check that
Treating E to be continuous, plot 
versus 
. Show that this system can have negative absolute temperature for the positive energy region. Why negative absolute temperature is possible here but not for a gas in a box?
Problem 4: Consider a monatomic ideal gas of N particles enclosed in a volume V . The system is in thermodynamic equilibrium with a heat bath at temperature T . Calculate the canonical partition function Z, obtain internal energy E , Helmholtz free energy F and entropy S . Verify the equation of state PV=NkBT .
Problem 5: Consider N localized one dimensional classical Harmonic oscillators of frequency ω in thermal equilibrium at temperature T. Obtain the canonical partition function 
. Calculate the internal energy E of the system. Check that the energy obtained is satisfying the equipartition theorem, 
thermal energy per square term in the Hamiltonian.
Problem 6: Consider N localized one dimensional quantum Harmonic oscillators of frequency ω in thermal equilibrium at temperature T. Obtain the canonical partition function 
. Show that the internal energy E of the system is given by
(Note that E is not satisfying equipartition theorem.) Check that 
as 
and 
as 
as expected.