Thermal energy of a solid is given by the relation E = \int_0^{ u_m} \frac{h\nu g(\nu)d\nu}{e^{h\nu/k_B T} - 1} where $ u_m = k_B \theta_D/h$, \theta_D being the Debye temperature. Given g(\nu) = 6N h^2 \nu(k_B \theta_D)^2, deduce the expression for E for T \ll \theta_D and discuss the temperature variation of specific heat for T \ll \theta_D. \left( \int_0^\infty \frac{x^3 dx}{e^x - 1} = 2.404 \right)

Calculate the work done in compressing adiabatically 10^{-3}\text{ kg} of air initially at STP to one-half its original volume. (Give density of air at \text{STP}=1.293\text{ kg/m}^3 and \gamma = 1.4.

Write a short note on Thermodynamic potentials.

How much faster does a cup of coffee cool when it is at 100^\circ\text{ C} than when it is at 30^\circ\text{ C}, if room temperature is 20^\circ\text{ C}?

Obtain the Clausius-Clapeyron equation. \frac{dp}{dt} = \frac{L}{T(v_2 - v_1)} where the symbols have their usual meanings

Use the Maxwell relation \left(\frac{\partial s}{\partial v}\right)_T = \left(\frac{\partial p}{\partial T}\right)_v to show that the internal energy, U, of a thermodynamic system depends on the volume according to the expression \left(\frac{\partial U}{\partial v}\right)_T = T\left(\frac{\partial p}{\partial T}\right)_v - p where the symbols have their usual meanings. Use this result to derive Stefan's law for a black-body radiator.

Explain, using the kinetic modal, why peaks in the curves showing Maxwell distribution of molecular speeds move towards higher speed at higher temperature.

1 kg of ice at 0^\circ\text{ C} is melted and converted to water at 0^\circ\text{ C}. Compute the change in entropy.

Write a short note on Negative temperature.

Find the temperature at which root mean square velocity of nitrogen molecules in earth’s atmosphere equals the velocity of escape from the earth’s gravitational field. Mass of N_2 atom = 23.24 \times 10^{24}\text{ g}. Mean radius of earth = 6370\text{ km}.

Calculate the mean free path of helium atoms at NTP, the coefficient of viscosity being 190 \times 10^{-7}\text{ kg m}^{-1}\text{ s}^{-1}. 1\text{ atmospheric pressure} = 0.076 \times 136 \times 10^3 \times 9.81\text{ Nm}^{-2}.

A gas possesses a Maxwellian velocity distribution function. Show that the fraction of molecules in a given volume that possess a velocity (+v_x) in one direction only and whose magnitude is greater than some selected value v_o is \int_{v_o}^{\infty} f(V_x) dv_x = \frac{1}{2} \left[ 1 - \frac{1}{\sqrt{\pi}} \text{erf} \left( \frac{1/4 m v_o^2}{KT} \right)^{1/2} \right] Symbols have their usual meanings.

A volume of one gm-mole of an ideal gas expands isothermally to four times its initial volume. Calculate the change in its entropy in terms of gas constant.

Write a short note on Production of low temperature using adiabatic demagnetisation.

A Carnot’s engine is made to work between 0^\circ\text{C} and -200^\circ\text{C}. Calculate us efficiency. Derive the expression you use of calculation.

Explain thermoionic emission. Discuss Richardson's derivation of the thermionic equation and show how the velocity distribution of the emitted electrons corroborates Fermi-Dirac distribution, Mention a few modern thermoionic emitters.

Find the density of hydrogen gas at temperature of 25^\circ\text{C} and pressure of 750\text{ mm Hg}.

Mention some experiment which establishes that electrons in metals obey Fermi-Dirac statistics and justify your answer in some detail.

For a Maxwell-Boltzmann gas derive the expression for (i) the most probable speed (ii) the average speed and (iii) the root mean square speed and find their ratio (taking the most probable speed as unity).

Write a note on Production of temperature below 1 K.