N particles obeying Classical Statistics are distributed among three states having energies \varepsilon_1 = 0, \varepsilon_2 = k_{\mathrm B}T and \varepsilon_3 = 2k_{\mathrm B}T, where k_{\mathrm B} is Boltzmann constant. If the total equilibrium energy of the system is 1000k_{\mathrm B}T, calculate the value of N.
The coefficient of viscosity of helium at 27^\circ\mathrm{C} is 2 \times 10^{-5}\ \mathrm{kg\,m^{-1}\,s^{-1}}. Calculate
(i) the average speed and
(ii) the diameter of a helium molecule, if it is assumed that the gas obeys Maxwell-Boltzmann distribution. Given Boltzmann constant k_{\mathrm B} = 1.38 \times 10^{-23}\ \mathrm{J\,K^{-1}} and mass of helium atom = 6.67 \times 10^{-27}\ \mathrm{kg}.
In Leh, temperature of ice on a cold winter night is measured as -20^\circ\mathrm{C}. Calculate the change in entropy when 1\ \mathrm{kg} of ice is converted into steam at 100^\circ\mathrm{C}. Given specific heat capacity of ice is 500\ \mathrm{cal\,kg^{-1}\,K^{-1}}, latent heat of ice is 3.36\times10^5\ \mathrm{J\,kg^{-1}}, latent heat of steam is 2.26\times10^6\ \mathrm{J\,kg^{-1}} and J=4.2\ \mathrm{J\,cal^{-1}}.
Consider non-equilibrium situation for a system in which the population inversion has been achieved. Explain that such a system can be treated as if it has negative absolute temperature.
Show that both Fermi-Dirac and Bose-Einstein distributions reduce under certain condition in a form which gives the total number of particles as N = A\int_0^\infty \sqrt{\varepsilon}e^{-\beta\varepsilon}\,d\varepsilon where A is a constant and \beta = 1/k_{\mathrm B}T. Further show that this expression is just the same as that obtained from the Maxwellian speed distribution.
The Einstein theory of specific heat of solids gives the expression C_V=\frac{3Nk_Bx^2e^x}{(e^x-1)^2} where x=\frac{\theta_E}{T} with \theta_E as the Einstein temperature.
(i) Mention Einstein's assumptions in deriving it. Also obtain low- and high-temperature limiting expressions for it.
(ii) Give schematic plot of \dfrac{C_V}{3N k_B} versus \dfrac{T}{\theta_E} and comment on the validity of expressions in (i) in comparison with experiments.
Establish the relation \left(\frac{\partial T}{\partial V}\right)_P=-\left(\frac{\partial P}{\partial S}\right)_T and then derive \left(\frac{\partial C_P}{\partial P}\right)_T=-T\left(\frac{\partial^2 V}{\partial T^2}\right)_P. Hence show that the heat capacity C_P of an ideal gas is independent of pressure P.
Show that the Helmholtz free energy of a system never increases in any isothermal-isochoric transformation.
Find out the expressions for van der Waals constants a and b.
Calculate the values of van der Waals constants a and b for oxygen with T_c=154.2\,\mathrm{K}, P_c=49.7 atmosphere and R=80\,\mathrm{cm^3\,atmosphere/K}.
Write down the expressions for Bose-Einstein and Fermi-Dirac distribution functions, and show how Fermi-Dirac distribution leads to the explanation of Pauli's exclusion principle.
What led van der Waals to modify the ideal gas equation? Using the concepts of critical temperature T_c, pressure P_c and volume V_c, show that the critical constant for a real gas is 8/3.
Calculate the change in pressure for a change in freezing point of water equal to -0.91^\circ\mathrm{C}. Given, the increase of specific volume when 1 gm of water freezes into ice is 0.091 cc/gm and latent heat of fusion of ice is 80 cal/gm.
Derive an expression for the thermal efficiency of a reversible heat engine operating on the Diesel cycle with an ideal gas of constant heat capacity as the working medium.
Calculate the number of different arrangements of 10 indistinguishable particles in 15 cells of equal a priori probability, considering that one cell contains only one particle.
Consider the following statement: ``The Fermi energy of a given material is the energy of that quantum state which has the probability equal to \frac{1}{2} of being occupied by the conduction electrons.'' Is the above statement correct? Give reasons for your answer.
1 kmol of an ideal gas is compressed isothermally at 400 K from 100 kPa to 1000 kPa in a piston and cylinder arrangement. Calculate the entropy change of the gas, the entropy change of the surroundings and the total entropy change resulting from the process if the process is mechanically reversible and the surroundings consist of a heat reservoir at 400 K.
Consider one mole of an ideal gas whose pressure changes with volume as P=\alpha V, where \alpha is a constant. If it is expanded such that its volume increases m times, find the change in internal energy, work done by the gas and heat capacity of the gas.
Define entropy. How is it related to disorder? Hence, derive the Boltzmann relation S = k \log \Omega, where \Omega is the probability and k is the Boltzmann constant. Show that for any type of process, involving a closed system \Delta S \ge \frac{\Delta Q}{T} where the equality sign applies for internally reversible processes and the inequality for internally irreversible processes.
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}})
