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A gas of N spinless Bose particles of mass m is enclosed in a volume V at a temperature T.
(i) Find an expression for the density of single-particle state D(\varepsilon) as a function of the single-particle energy \varepsilon. Sketch the result.
(ii) Write down an integral expression which implicitly determines \mu(T). Referring to your sketch in (i), determine in which direction \mu(T) moves as T is lowered.
A hypothetical engine, with an ideal gas as the working substance, operates in the cycle shown below. Show that the efficiency of the engine is \eta = 1 - \frac{1}{\gamma} \left( \frac{1 - \dfrac{P_3}{P_1}}{1 - \dfrac{V_1}{V_3}} \right) .
10\text{ g} of water at 60\text{ }^\circ\text{C} is mixed with 30\text{ g} of water at 20\text{ }^\circ\text{C}. Will the entropy of the system increase or decrease? Calculate the change.
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}}.
Discuss the principle of adiabatic demagnetization process to achieve low temperatures. Determine the fall in temperature produced by adiabatic demagnetization of a paramagnetic material at initial temperature of 3\ \mathrm{K} when the magnetic field is switched off from 10{,}000 oersted to zero. Given: heat capacity at constant magnetic field =0.2\ \mathrm{J\ g^{-1}\ K^{-1}} and Curie constant per gram mole per \mathrm{cm^3} =0.042\ \mathrm{erg\ K^{-1}\ g^{-1}\ Oe^{-2}}.
Starting from Maxwell-Boltzmann distribution for a free particle in 3-dimension, obtain the expression for root mean square (rms) speed of a particle. Calculate the rms speed of nitrogen \mathrm{N}_2 molecule at room temperature 27^\circ\mathrm{C}.
Obtain the Clausius-Clapeyron equation. Using this equation, show that for the phase boundary of the liquid and vapour phases, p--T relation can be written as p=p_0e^{-L/kT}. Here it has been assumed that the latent heat L is independent of temperature, that vapour is treated as an ideal gas and that V_{\mathrm{vapour}}=V\gg V_{\mathrm{liquid}} and that p\to p_0 as T\to\infty.
Write down the distribution law obeyed by electron gas and apply the same to derive Richardson-Dushman equation.
(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.
Deduce Clausius-Clapeyron equations based on reversible cycle. Show that the specific heat of steam is negative. What is the significance of negative specific heat?
Calculate (i) the internal energy of the electron gas per unit volume and (ii) the molar specific heat at constant volume for sodium at 100\text{ K} containing one free electron per atom. Given that the density of sodium = 0{\cdot}97 \times 10^3\text{ kg m}^{-3} and the atomic weight of sodium = 23.
There are g cells of energy \varepsilon. Show that the number n of bosons of energy \varepsilon distributed among these cells is given by n = \frac{g}{e^{(\varepsilon - \mu)/kT} - 1} What is \mu and how will you find it?
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.
Derive Maxwell's thermodynamic relations using concepts of internal energy, Helmholtz function, Gibbs' function and enthalpy.
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 ?
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 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.
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 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.
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.