List of top Physics Questions on Thermodynamic and Statistical Physics asked in GATE PH

Two identical particles with a fixed total energy \(E = 2\hbar\omega\) are in thermal equilibrium in a one-dimensional harmonic oscillator potential \(\frac{1}{2}m\omega^2x^2\). Let the entropy of the particles be denoted by \(S_F\) if they are fermions with spin \(\frac{1}{2}\) (\(S_z = \pm\frac{\hbar}{2}\)) and by \(S_B\) if they are bosons with spin 0. Then, which of the following options is correct? (\(k_B\) is the Boltzmann constant)

A gas of \(N\) classical particles that can occupy energy levels \(\epsilon_1\) and \(\epsilon_2 = \epsilon_1+\Delta\) is in equilibrium with a reservoir at temperature \(T\). From the schematics shown below, choose the correct dependence of the internal energy \(U\) on \(T\).

The specific heat \(C_p(T)\) of one mole of a material as a function of temperature \(T\) is given as \(C_p(T) = AT + BT^3\), where \(A = 0.695\) mJ.mol\(^{-1}\).K\(^{-2}\) and \(B = 0.045\) mJ.mol\(^{-1}\).K\(^{-4}\). When \(T\) is changed from 1 K to 10 K at constant pressure, the change in entropy \(\Delta S\) in mJ.mol\(^{-1}\).K\(^{-1}\) (rounded off to one decimal place) is

Schematic variation of the specific heat \(C_p\) of an ideal gas of diatomic molecules with temperature \(T\) is shown in the figure below. For rotational energy \(E_R\) and vibrational energy \(E_v\) of the molecule, which of the following options is/are correct? Here \(k_B\) is the Boltzmann constant.

The formula for energy \(E\) of a photon gas at temperature \(T\) in a two-dimensional box at equilibrium with \(g_{2d}(\nu)\) denoting the density of states of photons is given below, where symbols \(\nu\), \(h\) and \(k_B\) have their standard meaning. The specific heat (\(C_V\)) of this photon gas obeys
\[ E = \int_0^{\infty} d\nu \, g_{2d}(\nu) \, \frac{h\nu}{\exp\left(\dfrac{h\nu}{k_B T}\right) - 1} \]

A gas of non-interacting \(^4\text{He}\) atoms (of mass \(m\)) is in a three-dimensional trap whose energy levels can be approximated by those of a harmonic oscillator potential \(V(x,y,z) = \dfrac{1}{2} m \omega^2 (x^2+y^2+z^2)\). The chemical potential of the gas at \(T=0\) K is

Consider the Otto cycle for an ideal gas engine consisting of two quasistatic adiabatic and two quasistatic isochoric processes. The correct temperature-entropy (T-S) phase diagram for the cycle is: