Solve the given problems, detailed solutions is needed.

Problem 1: Show all derivation

(a) The Boltzman equation for entropy is: S = kBlnΩ        

It has a generalized form: S = -kB∑p_jln_jp_j  (1) for a system that has a probability of p_j in its jth state. The sum is over all the possible states: j = 0, 1, 2 ... k. Please show that S = kBlnΩ is an extreme case of its generalized form under certain condition. Explain the condition.

(b) Consider one classical particle inside a 2-dimensional circular container of radius R. The particle is equally probable anywhere in the container. Use Equation (1) to show that: S = kBln(πR2)

(c) Consider R as the thermodynamic variable that has a range from 0 to infinity and can change to achieve thermal equilibrium. Using the entropy from (b) and the internal energy (d, ε₁, ε₂ are positive constants):

U(R) = - ε₁(R/d) + ε₂(R/d)²

Show that most probably value of R in the limit of kT >> ε₁, ε₂ and kT << ε₁, ε₂.

(d) Assume ε₁ = 0, find the expression for U using the most probably value of R.

Problem 2: Show all derivation

Two systems have same heat capacity C (ignore volume for this problem). One is at temperature T + t and the other is at temperature T - t.

(a) Two systems are put in thermal contact. Find out the final temperature and the change of total entropy.

(b) Operate a reversible thermodynamics engine between the two systems until the two reach equilibrium. Find out the final temperature of the system.

(c) In (b), what is the change of entropy (for the sum of two systems) and work performed by the reversible thermodynamics engine? What is the change of entropy to the environment?

(d) Starting from the equilibrium at the end of (b), put the system in contact with a reversible heat reservoir until the final temperature is T. Please find out the change of entropy for the sum system and the universe. Explain the results.

Problem 3: Show all derivation

The Helmhotz free energy of a system is:

F(T, V) = 1/2kV₀(V - V₀)² + 3NRTln(1 - [1 + α(V - V₀)]e^θ/T)

Every parameter (other than T, V) in the expression is constant.

(a) Find out the compressibility and the coefficient of thermal expansion.

(b) Find out the entropy and internal energy of the system.

(c) Calculate (∂T/∂V)_U and (∂T/∂V)_S.

(d) Find out Cv and Cp and their limits to T >> θ and T << θ.