PYQ Bank

Energy distribution for n_i particles in classical statistical mechanics is given by n_i = g_i e^{-\alpha -\beta \varepsilon_i} where \alpha and \beta are constants. g_i is the single particle states in the i\text{ th} level. Using equipartition theorem, show that correct thermodynamic interpretation is \beta = \frac{1}{kT} (Use \int_0^\infty e^{-x} x^{1/2} \, dx = \frac{\sqrt{\pi}}{2} and \int_0^\infty e^{-\beta x} x^{3/2} \, dx = \frac{3\sqrt{\pi}}{4\beta^{5/2}})

Consider N independent particles, each of which can be in either of two energy states +\epsilon and -\epsilon. Derive an expression for the entropy of a microcanonical ensemble for this system, where $ f = \sum_i \epsilon_i / N\epsilon $ is fixed, where $ \epsilon_i $ is the energy of the i-th particle.

State second law of thermodynamics. Prove that no engine operating between two given temperatures is more efficient than a carnot engine operating between same two temperatures.

One mole of an ideal gas is compressed at constant temperature T from a volume V_1 to a volume V_2. Find the work done and heat absorbed by the gas. The gas now expands adiabatically to a volume 2V. Taking the gas to be diatomic, calculate the final temperature of the gas.

Explain the Debye model of specific heat of solids. What are its success and failures ?

State the law of equipartition of energy. Show how this law can be used to calculate specific heat of gases and hence find the ratio $ \gamma = C_p/C_v $ for diatomic and triatomic gases.

Show that at T = 0, the Fermi distribution function has a value 1 for energies less than the Fermi energy $ \epsilon_F $ and is zero above it. For a system of non-interacting electrons at T = 0, show that the ground state energy of the system of N particles is $ \frac{3}{5} N \epsilon_F $.

Derive a relation between the total number of Fermions in terms of Fermi momentum and hence obtain the expression for the total energy E of the system at absolute zero. Combine this expression with the equation of state PV = \frac{2}{3} E to show that the pressure of an ideal Fermi gas at T = 0 is proportional to 5/3 power of its number density.

Establish the relation C_p - C_v = \left[ P + \left(\frac{\partial U}{\partial V}\right)_T \right] \left(\frac{\partial V}{\partial T}\right)_P Use it to find out an expression for C_p of one mole of a gas whose internal energy is given by U = cT - \frac{a}{V} and which satisfies the equation of state \left[ P + \frac{a}{V^2} \right](V - b) = RT. Here a, b and c are constants.

Prove the law of increase of entropy. Show that for a system at fixed temperature and pressure to be in equilibrium, its Gibbs free energy should be minimum.

A system at temperature T_1 is brought in contact with a reservoir at temperature T_2 > T_1. When the system and the reservoir reach thermal equilibrium, calculate the change in entropy of the universe assuming the heat capacity of C_p of the system to be constant. Discuss whether the considered change is positive or not.

1 kg of ice at 0^\circ\text{ C} floats on 10 kg of water at 30^\circ\text{ C}, the whole system being thermally isolated. What will be the change in entropy of the sytem when thermal equilibrium is reached ? [Specific heat of water = 4.2\text{ kJ kg}^{-1}\text{ K}^{-1} and latent heat of fusion of ice = 336\text{ kJ kg}^{-1}]

State Dulong and Petit's law. How does it agree with experiment ? Discuss the limitations of the classical theory and success of the quantum theory in explaining the specific heat of solids.

Using thermodynamic principles show that the Joule-Thomson coefficient \mu can be expressed as \mu = \frac{1}{C_p} \left[ T \left( \frac{\partial V}{\partial T} \right)_p - V \right] . Calculate the value of \mu for an ideal gas and interpret your result physically.

Derive the expression for the Fermi-Dirac distribution function. Represent it graphically for T = 0 and T \neq 0.

(b) Utilize the above expression to elaborate the concept of the Bose-Einstein condensation and discuss that the phenomenon explains qualitatively the properties in the low-temperature phase of liquid ^4\text{He}.

8. (a) Starting from the expression N = \sum_k <n_k> where <n_k> is the average number of particles in the k^{\text{th}} quantum state, derive an expression for the average number of particles in the ground state of an ideal Bose gas.

(c) Show that the chemical potential of a system is an intensive quantity and is a function of temperature and pressure only.

(d) Consider the expression C_P - C_V = - T \left( \frac{\partial V}{\partial T} \right)_P^2 \left( \frac{\partial P}{\partial V} \right)_T and give reasoning regarding the values of T when C_P = C_V for water. Also, evaluate C_P - C_V for a vander Waals gas to elaborate that its value is larger for any real gas as compared to an ideal gas.

(c) Discuss the differences in the assumptions underlying Einstein and Debye theories of specific heat C_v. Give schematic plots of C_v versus reduced temperature for these theories and elucidate the differences therein. Elaborate the meaning of the “law of corresponding states” for these plots.