(i) The energy level of a quantum harmonic oscillator with frequency \nu is given by E_n=\left(n+\frac{1}{2}\right)h\nu,\quad \text{where } n=0,1,2,\ldots Calculate its partition function.
(ii) Calculate the partition function of a two level system.
State the first law of thermodynamics for a diffusively interacting system. The temperature of 10\ \mathrm{g} of air is raised by 2^\circ\mathrm{C} at constant volume. Calculate the increase in its internal energy. Given: C_v=0.172\ \mathrm{cal\ g^{-1}\ {}^\circ C^{-1}}.
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.
Einstein's 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 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.
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.
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.
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}.
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.
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.
Schematically, show the variation of density of states, D(\varepsilon) and distribution function, f(\varepsilon,T), of particles in a non-relativistic Fermi gas at high temperatures. At a temperature T, an electron occupies a state with energy 100\mathrm{meV} above the Fermi energy (\varepsilon_{\mathrm F}) with the probability of 1\%. Find the temperature T.
A system having two energy levels, -\frac{1}{2}\Delta and +\frac{1}{2}\Delta with \Delta=10\mathrm{meV} is populated by 1000 particles at a low temperature close to 100\ \mathrm{K}. Obtain the average energy per particle using classical distribution law.
The pressure on 100 g of solid copper is increased quasi-statically and isothermally at 0^\circ\mathrm{C} from 0 to 0.5 \times 10^{8}\ \mathrm{Pa}. Assuming the density and isothermal compressibility to remain at constant values of 8.96\ \mathrm{g/cm^3} and 7.16 \times 10^{-12}\ \mathrm{Pa^{-1}}, respectively, calculate the work done. Comment on the sign and magnitude of work.
At 4^{\circ}\mathrm{C} temperature, the density of water is found to be maximum. Prove that heat capacity at the constant pressure (c_p) is equal to the heat capacity at constant volume (c_v) for water at 4^{\circ}\mathrm{C}.
If the temperature variation of heat capacity is known, how do you calculate the change of entropy during an isochoric process? According to Debye's theory of specific heat of a solid, the molar heat capacity of diamond crystal at constant volume varies with temperature (T) as follows: c_v=\frac{12}{5}\pi^{4}R\left(\frac{T}{\Theta}\right)^{3} where R is the molar gas constant =8.315\ \mathrm{J/mol\ K} and \Theta=2230\ \mathrm{K} for diamond. Calculate the change in entropy of diamond of 0.36 g mass when it is heated at constant volume from 0 K to 300 K.
One mole of a gas obeys the following equation of state: \left(P + \frac{a}{v^{2}}\right)(v-b)=RT, where v is the molar volume and, a and b are constants. Show that internal energy of the gas increases as the volume increases, with the temperature remaining constant.
Calculate the critical temperature for helium, given the values for critical constants, a = 6.15 \times 10^{-5}, b = 9.95 \times 10^{-4}, where the unit of pressure is atm and the sample is kept at NTP.
Write and explain the Maxwell-Boltzmann distribution. Using this distribution, find the expressions for the most probable speed, mean speed and root-mean-square speed.
Explain Bose-Einstein distribution and obtain the same from the grand canonical ensemble.
A reversible engine converts 1/6 of the heat input into work. When the temperature of the sink is reduced by 62\ {}^{\circ}\mathrm{C}, its efficiency is doubled. Find the temperatures of source and sink.
