How do you define density of states? Show that the density of states with wave vector less than \vec{k} in a three-dimensional cubic box of volume V can be given by D(\omega)=\frac{V}{2\pi^2}k^2\left(\frac{dk}{d\omega}\right) in the frequency spectrum between \omega and \omega+d\omega. Here, assume that the number of modes per unit range of k is L/(2\pi), L being the length of each side of the cubic box.

An electron is confined in the ground state of a one-dimensional harmonic oscillator such that \Delta x = 10^{-10}~\text{m}. Assuming \langle T \rangle = \langle V \rangle, find the energy in eV required to excite it to the first excited state.

Write down the Hamiltonian operator for a linear harmonic oscillator. Show that the energy eigenvalue of the same can be given by E_n=\left(n+\frac{1}{2}\right)\hbar\omega_0 at energy state n with \omega_0 being the natural frequency of vibration of the linear oscillator. Prove that n=0 energy state has a wave function of typical Gaussian form.

For a free quantum mechanical particle under the influence of a one-dimensional potential, show that the energy is quantized in discrete fashion. How do these energy values differ from those of a linear harmonic oscillator?

$. Hence find the uncertainty product (\Delta x)(\Delta H).

Show that the phase velocity v_p for a particle with rest mass m_0 is always greater than the velocity of light and that v_p is a function of wavelength.

Consider a quantum -- mechanical gas of non-interacting spin zero bosons, each of mass m which are free to move within volume V. (i) Find the energy and heat capacity in the very low temperature region. Discuss why it is appropriate at low temperatures to put chemical potential equal to zero. (ii) Show how the calculation is modified for a photon (mass = 0) gas. Prove that the energy is proportional to \text{T}^4.

Derive Maxwell's thermodynamic relations using concepts of internal energy, Helmholtz function, Gibbs' function and enthalpy.

The equation of state of a dilute gas at very high temperature is described by \frac{\text{PV}}{\text{kT}} = 1 + \frac{\text{B(T)}}{\text{V}}, where V is the volume per particle and B(T) is a negative quantity. One can conclude that this is a property of a Van der Waal's gas. Explain why it is a property of Van der Waal's gas.

Obtain an expression for the specific heat capacity of a solid on the basis of Einstein's theory. How far do the results from this theory agree with experimental data ?

A gas has only two particles, a and b. With the help of a diagram, show that how these two particles can be arranged in the three quantum series 1, 2, 3 using (i) Maxwell-Boltzmann, (ii) Fermi-Dirac, and (iii) Bose-Einstein statistics.

Obtain Clausius -- Clapeyron equation which applies to any first-order change of phase or any transition that occurs at constant temperature and pressure. Use Maxwell's thermodynamic relation for deriving the equation.

What are the conditions for the change in temperature of a van der Waals gas passing through a porous plug? Prove that the ideal gas passing through the porous plug does not show any change in temperature.

What is Carnot's theorem? Prove that Carnot's reversible engine is the most efficient one and no other engine can be more efficient than Carnot's engine.

Derive the expression for the average energy of a quantum oscillation of frequency \nu. Assume Fermi-Dirac distribution and E-E_F>2, where E_F is the Fermi level.

Einsteins molar specific heat capacity of a solid is given by C_V = 3R\left(\frac{\theta_E}{T}\right)^2\frac{e^{\theta_E/T}}{\left(e^{\theta_E/T}-1\right)^2}, where \theta_E=\frac{\hbar\omega}{k_B}. Obtain the expressions for the cases:

(i) when T \gg \theta_E

(ii) when T \ll \theta_E What is the discrepancy of Einstein model to explain the variation of specific heat capacities of solids with the temperature? The molar specific heat capacity of a solid at constant volume is 2.77\ \mathrm{J\,K^{-1}} at 36.8\ \mathrm{K}. Determine the Debye temperature of the solid.

What is Gibbs phase rule? Find the values of degrees of freedom when

(i) only the liquid \mathrm{CO_2} is in equilibrium with the gaseous \mathrm{CO_2}.

(ii) water is in the vapour-liquid saturation region.

(iii) water is in a single-phase region.

(iv) water is at the triple point.

Explain the effect of pressure on the melting and boiling points of a substance using Clapeyron's latent heat equation. Calculate under what pressure, water will boil at 120^\circ\mathrm{C}, if the change in specific volume when 1 gram of water is converted into steam is 1676\ \mathrm{cm^3}. Latent heat of steam =540\ \mathrm{cal/g}, 1 atmospheric pressure =10^6\ \mathrm{dynes/cm^2}.

If the partition function for a perfect gas is given by Z=\frac{V}{h^3}(2\pi m k T)^{3/2}, calculate (i) average kinetic energy per molecule and (ii) specific heat of the gas.

Derive an expression for the entropy change in the expansion of a gas from volume \text{V}_i to volume \text{V}_f. Use a statistical definition of entropy for derivation.