UPSC Physics Optional PYQs (CSE & IFoS)PYQs

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}.

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}.

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.

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.

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}.

The volume of a mole of liquid \text{He}^4 is 27 \times 10^{-6}\text{ m}^3 and the mass of a \text{He}^4 atom is 6.65 \times 10^{-27}\text{ kg}. Assuming that liquid \text{He}^4 is an ideal Bose gas, calculate (i) the concentration of boson in this volume. (ii) Bose temperature.

(i) Prove that for perfect gas the specific heat at constant pressure C_p is always greater than the specific heat at constant volume by a constant value R. (ii) Consider for hydrogen, the density at NTP is 0.0899\text{ gm/lit}, molecular weight 2.016\text{ gm} and specific heat at constant pressure 6.85\text{ cal/gm}. Calculate the specific heat of hydrogen gas at constant volume. Given : J = 4.18 \times 10^7\text{ erg}.

Deduce the thermodynamic relation : \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V Using the expression establish Clausius-Clapeyron latent heat equation \frac{dP}{dT} = \frac{L}{T(V_2 - V_1)}

Explain how the state of ionization of any particular element in a star changes with varying temperatures and pressures.

Distinguish between a perfect gas and a real gas. Derive van der Waals' equation of state and use it to obtain the expressions for the critical constants in terms of the constants of the van der Waals' equation.

Explain how we can produce refrigeration without using a compressor.

What is the concept of negative temperature in statistical mechanics? Explain in brief.

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.

Liquid {}^4\text{He} has normal boiling point at 4\cdot 2\text{ K}. It is observed that it boils at 1\cdot 2\text{ K} at a pressure of 1\text{ mm of Hg}. Calculate the average latent heat of vaporization of {}^4\text{He} in this temperature range. (\text{Given : Density of Hg} = 13\cdot 6 \times 10^3\text{ kg m}^{-3}).

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 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.

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.

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.

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}]