[UPSC IFoS 2025] Thermodynamics (Q11, 15+5=20M)

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}.