Calculate the Fermi energy of aluminium at absolute zero. The density of aluminium is 2\cdot 7 \times 10^3\text{ kg m}^{-3} and its atomic weight is 26\cdot 98\text{ kg (k mol)}^{-1}. Show that the electron gas in aluminium is strongly degenerate.
One mole of gas obeys van der Waals equation of state. If its molar internal energy is given by u=cT-a/V (in which V is the molar volume, a is one of the constants in the equation of state and c is a constant), calculate the molar heat capacities C_v and C_p.
What do you understand by negative temperature? Write and explain various restrictions on a system for the concept of negative temperature to be meaningful.
In the case of a gas obeying the equation of state \frac{\text{Pv}}{\text{RT}} = 1 + \frac{\beta}{\text{v}}, where \beta is a function of T only, find the expression of the heat capacity at constant volume.
A gas of interacting atoms has an equation of state and heat capacity at constant volume given by the expressions p(T,V)=aT^{1/2}+bT^3+cV^{-2} C_v(T,V)=dT^{1/2}+eT^2V+fT^{1/2} where a through f are constants which are independent of T and V. Find the differential of the internal energy dU(T,V) in terms of dT and dV.
Two Carnot engines \text{C}_1 and \text{C}_2 operate in series, Engine \text{C}_1 absorbs heat at T and rejects heat to a sink at temperature 300\text{ K}. Engine \text{C}_2 absorbs \frac{1}{4}\text{th} of the heat rejected by engine \text{C}_1 and rejects heat to the sink at 200\text{ K}. If the work done in both the cases is the same, find the temperature T.
Estimate the temperature at which the root mean square speed of nitrogen molecules exceeds their most probable speed by 100\text{ ms}^{-1}.
Calculate the critical constants for \mathrm{CO_2} for which the Van der Waals constants are given by a=0.0072 and b=0.002. Also calculate the Boyle's temperature of \mathrm{CO_2}. The unit of pressure is atmosphere and the unit of volume is that of a gm-mole of the gas at NTP.
Explain the characteristics of the following thermodynamic processes for a perfect gas:
(i) Isothermal process
(ii) Adiabatic process
(iii) Isobaric process
(iv) Isochoric process Obtain the expression for the work done by the gas during the above processes.
Using the expression for internal energy U = 3N \frac{\hbar\omega}{e^{\hbar\omega/k_B T}-1}, show that Einstein specific heat capacity is given by; C = 3R \left(\frac{\hbar\omega}{k_B T}\right)^2 \frac{e^{\hbar\omega/k_B T}}{\left(e^{\hbar\omega/k_B T}-1\right)^2} Also show that Einstein specific heat capacity given above is proportional to e^{-\hbar\omega/k_B T} at very low temperature.
The melting point of tin is 232^{\circ}\mathrm{C}, its latent heat of fusion is 14\ \mathrm{cal/g} and the specific heat of solid and molten tin are 0.055 and 0.064\ \mathrm{cal/g}\,{}^{\circ}\mathrm{C} respectively. Calculate the change in entropy when 1.0\mathrm{gm} of tin is heated from 100^{\circ}\mathrm{C} to 300^{\circ}\mathrm{C}.
How many nitrogen molecules must strike a 1\text{ cm}^2 surface each second to exert a pressure of 1\text{ atmosphere} ? (Assume the molecules are all moving at same speed, corresponding to a temperature of 300\text{ K}, and at an angle of 45^\circ to the wall). [Molecular mass of \text{N}_2 is 28\text{ u}]
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.
Calculate the probability of an electron occupying an energy level 0.02\,\mathrm{eV} above the Fermi level at T=300\,\mathrm{K}
Find the pressure at which water would boil at 150^\circ\text{C} if the change in specific volume when one gm of water is converted into steam is 1676\text{ c.c.} Given J = 4\cdot 2 \times 10^7\text{ ergs/cal}, one atmosphere = 10^6\text{ dyne/cm}^2 and latent heat of vapourisation of steam = 540\text{ cal/gm}.
Write a brief note on Chandrasekhar Limit.
Calculate the efficiency of an engine having compression ratio 13.8 and expansion ratio 6 and working on diesel cycle. Given \gamma = 1.4.
Consider one gm of ice at a temperature T_1\text{ K}. Show that when this ice changes into steam at a temperature T_2\text{ K}, the total gain in entropy is \Delta S = \frac{L_i}{T_1} + C \log_e \left( \frac{T_2}{T_1} \right) + \frac{L_s}{T_2} where L_i is latent heat of ice, C is specific heat of water, L_s is latent heat of steam. T_1 = 273\text{ K}.
Write the expression for the Fermi-Dirac distribution. Plot the Fermi-Dirac distribution at T=0 and for T_1>T_2>0. Now from the plot propose two alternative definitions of the Fermi level.
Eight indistinguishable balls are to be arranged in six distinguishable boxes. Calculate the total number of ways in which the above can be done.
