1 litre of hydrogen at 127^{\circ}\mathrm{C} and 10^{6} dynes \mathrm{/cm}^{2} pressure expands isothermally until its volume is doubled and then expands adiabatically until its volume is redoubled. Calculate the resulting pressure. \gamma = 1.42
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.
Write down the salient features of the Einstein's theory of lattice heat capacity. Further write down the expression for specific heat in Einstein's theory and explain its high and low temperature limits.
The molecules of a gas obey Maxwell-Boltzmann distribution. Calculate the fraction of molecules of the gas within 1\% of the most probable speed at STP. Interpret your result.
Consider a system of N particles and a phase space consisting of only two states with energies 0 and \varepsilon(>0). Obtain the expressions for the partition function and the internal energy of the system, if it obeys M-B statistics.
The viscosity in a liquid arises due to friction between adjacent layers. What causes viscosity in a gas? Explain.
The molecules of a gas obeying Maxwell-Boltzmann distribution move with an average speed of 450\ \mathrm{m\ s^{-1}}. If the coefficient of viscosity of the gas \eta is 16.6\times10^{-6}\ \mathrm{N\,s\,m^{-2}}, density of the gas \rho is 1.25\ \mathrm{kg\ m^{-3}} and number density is 2.7\times10^{25}\ \mathrm{m^{-3}}, calculate the mean free path and diameter of the gas molecules.
m gram of water at temperature T_1 is isobarically and adiabatically mixed with an equal mass of water at temperature T_2. Show that the change in entropy is given by \Delta S=2mC_p\ln\left(\dfrac{T_{\mathrm{av}}}{T_{\mathrm{geo}}}\right), where T_{\mathrm{av}}=\dfrac{T_1+T_2}{2} and T_{\mathrm{geo}}=\sqrt{T_1T_2}.
Write down van der Waals' equation of state for n moles of a gas and calculate the temperature at which 5 moles of the gas at 5\ \mathrm{atm} pressure will occupy a volume of 20 litres. Given, R=8.31\times10^7\ \mathrm{erg\,mol^{-1}\,K^{-1}}, a=1.34\times10^{12}\ \mathrm{dyne\,cm^4\,mol^{-2}}, b=31.2\ \mathrm{cm^3\,mol^{-1}} and 1\ \mathrm{atm}=1.013\times10^6\ \mathrm{dyne\,cm^{-2}}.
What do you understand by the term 'phase transition'? Using Clausius-Clapeyron equation, show that for first-order phase transitions, vapour pressure decreases exponentially with temperature. You can assume that the vapour behaves like an ideal gas and latent heat remains constant with temperature.
For a Van der Waals gas, write down the equation of state. Determine the coefficient of critical expansion \beta.
A Van der Waals gas undergoes Joule-Kelvin expansion with a pressure drop of 50\ \mathrm{atm}. If its initial temperature is 300^{\circ}\mathrm{K}, determine its final temperature. (Given Van der Waals constant a=0.136\ \mathrm{Pa\,m^6\,mol^{-1}}, b=36.5\times10^{-6}\ \mathrm{m^3\,mol^{-1}}, C_p=30\ \mathrm{J\,K^{-1}\,mol^{-1}}, R=8.3\ \mathrm{J\,K^{-1}\,mol^{-1}}.)
The vapour pressure of an organic substance is 50\times10^3\ \mathrm{Pa} at 40^{\circ}\mathrm{C}. Its normal boiling point is 80^{\circ}\mathrm{C}. If the substance in vapour phase can be treated like an ideal gas, find the latent heat of vaporization of the substance.
Consider a system of free gas particles having f degrees of freedom. Use equipartition theorem to establish the relation f=\frac{2}{\left(\dfrac{C_p}{C_V}-1\right)}, where C_p and C_V are molar specific heats at constant pressure and constant volume respectively. Obtain the values of \dfrac{C_p}{C_V} for diatomic and triatomic gases.
One kg of water at 20^\circ\mathrm{C} is converted into ice at -10^\circ\mathrm{C} at constant pressure. Heat capacity of water is 4{,}200\ \mathrm{J\,kg^{-1}\,K^{-1}} and that of ice is 2{,}100\ \mathrm{J\,kg^{-1}\,K^{-1}}. Heat of fusion of ice at 0^\circ\mathrm{C} is 335\times10^3\ \mathrm{J\,kg^{-1}}. Calculate the total change in entropy of the system.
Explain the four thermodynamic relations of Maxwell. Using the same, obtain the Clausius-Clapeyron equation \frac{\mathrm{d}P}{\mathrm{d}T}=\frac{L}{T(V_2-V_1)}.
Show that both Fermi-Dirac and Bose-Einstein distribution functions at an energy E are given by: f(E) \simeq \exp\left[\frac{\mu-E}{k_{\mathrm B}T}\right], where f(E) is much smaller than unity, \mu and k_{\mathrm B}T are the chemical potential and thermal energy of the atom.
Define Enthalpy and show that it remains constant in a throttling process.
N particles obeying Classical Statistics are distributed among three states having energies \varepsilon_1 = 0, \varepsilon_2 = k_{\mathrm B}T and \varepsilon_3 = 2k_{\mathrm B}T, where k_{\mathrm B} is Boltzmann constant. If the total equilibrium energy of the system is 1000k_{\mathrm B}T, calculate the value of N.
A thermally insulated ideal gas is compressed quasi-statically from an initial state with volume V_0 and pressure P_0 to a final state of volume V_f and pressure P_f. Show that the work done on the gas in the process is given by W=\frac{C_V}{R}\left(P_fV_f-P_0V_0\right) where C_V and R having standard meanings.
