The average kinetic energy of hydrogen atoms in a certain stellar atmosphere, assumed to be in thermal equilibrium, is 1\cdot 2\text{ eV}. Calculate the ratio of the number of atoms in the second excited state (n = 3) to the number in the ground state.
Two solids A and B have Debye temperatures 200\text{ K} and 300\text{ K} respectively. At T = 20\text{ K}, compare their specific heats.
Calculate the pressure at which water will boil at 150\text{ }^\circ\text{C}, given that the change in specific volume when 1\text{ gram} of water is converted into steam is 1676\text{ cm}^3. Given, latent heat of vaporization for steam = 540\text{ cal per gram}, J = 4\cdot 2 \times 10^7\text{ ergs/cal} and one atmospheric pressure = 10^6\text{ dynes/cm}^2.
A gas has only two particles a and b. Show with the help of diagrams how these two particles can be arranged in three energy states 1, 2, 3 using (i) Maxwell–Boltzmann, (ii) Fermi–Dirac and (iii) Bose–Einstein statistics.
What are the limitations of the first law of thermodynamics? One mole of a gas, assumed to be perfect, at 0\text{ }^\circ\text{C} is heated at constant pressure till its volume is twice its initial value. Calculate the amount of heat absorbed. Given, C_v = 20\cdot 9\text{ J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1} and R = 8\cdot 3\text{ J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}.
Why does the specific heat of solids depend on material at low temperatures but become independent of material at high temperatures?
Consider that an ideal non-interacting Fermi gas with internal energy '\text{U}' at temperature \text{T} is kept in a cubical box of volume \text{V}. Find the pressure for the gas in terms of \text{U} and \text{V}.
A reversible heat engine operates with three reservoirs at 300\text{ K}, 400\text{ K} and 1200\text{ K}. It absorbs 1200\text{ kJ} energy as heat from the reservoir at 1200\text{ K} and delivers 400\text{ kJ} work. Determine the heat interactions with the other two reservoirs.
A ternary system consists of three components (A, B and C) in equilibrium with two phases. Determine the number of degrees of freedom using the Gibb's phase rule and discuss the effect of pressure and temperature variations on the phase equilibrium.
Discuss briefly the considerations which led Van der Waals to modify the gas equation. What are the critical constants of a gas ? Calculate the values of these constants in terms of the constants of the Van der Waals equation.
Explain why, at equilibrium, the chemical potential of a component must be the same in all coexisting phases. Derive the equilibrium condition for a binary liquid-vapour system in terms of chemical potential.
For silver, the specific heat at constant pressure in the range of 50\text{ K} to 100\text{ K} is given by : \text{C}_{\text{p}} = 0\cdot 076\text{ T} - 0\cdot 00026\text{ T}^2 - 0\cdot 15\text{ cal mol}^{-1}\text{deg}^{-1} If 2 moles of silver are heated from 50\text{ K} to 100\text{ K}, calculate the change in entropy.
Consider \text{N} non-interacting ideal spinless particles (Bose gas) are occupying a volume \text{V}. Find out the temperature '\text{T}' below which B-E condensation takes place.
Consider a mixture of N_A molecules of a monatomic gas A and N_B molecules of a monatomic gas B. For this mixture, obtain the Helmholtz free energy and pressure. (The particle partition function for a monatomic gas is q = \left(\frac{2\pi m k T}{h^2}\right)^{\frac{3}{2}} V).
One gram of water (1\text{ cm}^3) becomes 1671\text{ cm}^3 of steam when boiled at constant pressure of 1\text{ atm} (1\cdot 013 \times 10^5\text{ Pa}). The heat of vapourisation at this pressure is \text{L} = 2\cdot 256 \times 10^6\text{ J/kg}. Calculate :
(i) The work done by the water when it vapourizes, and
(ii) Increase in its internal energy.
Assuming the Maxwell's velocity distribution formula, find out the value of :
(A) Mean velocity (\bar{\text{v}}),
(B) The most probable velocity (\text{v}_{\text{mp}}), and
(C) Root mean square speed (\text{v}_{\text{rms}}) in terms of the Boltzmann constant (\text{k}_{\text{B}}) and show that : \text{v}_{\text{rms}} > \bar{\text{v}} > \text{v}_{\text{mp}}. If nitrogen molecules are kept at 27^{\circ}\text{C}, find out the value of \text{v}_{\text{rms}}, \bar{\text{v}} and \text{v}_{\text{mp}}. Given : Molecular mass of nitrogen \text{M} = 28 \times 10^{-3}\text{ kg/mol} and gas constant \text{R} = 8\cdot 314\text{ J.mol}^{-1}\text{K}^{-1}.
Calculate the Fermi energy in electron-volt for sodium assuming that it has one free electron per atom. The density of sodium = 0.97\text{ gm/cc} and the atomic weight of sodium is 23.
What do you understand by macrostates and microstates? Briefly explain.
Find out the boiling temperature of water at the top of Mount Everest. Given : Pressure at the top of Everest is 0.36\text{ atm}. The density of water vapour at 100^\circ\text{C} is 0.598\text{ kg/m}^3. The latent heat is 2.257 \times 10^3\text{ J/g}.
Prove that the work done by a perfect gas during a quasi-static adiabatic expansion is given by W = \frac{P_i V_i}{\gamma - 1} \left[ 1 - \left( \frac{P_f}{P_i} \right)^{\frac{\gamma - 1}{\gamma}} \right] where \gamma is the ratio of specific heats.
