Halliday, Resnick & Walker — Entropy and the Second Law of Thermodynamics: Problem 51

As a sample of nitrogen gas (\(N_2\)) undergoes a temperature increase at constant volume, the distribution of molecular speeds increases. That is, the probability distribution function \(P(v)\) for the molecules spreads to higher speed values, as suggested in Fig. 19.6.1b. One way to report the spread in \(P(v)\) is to measure the difference \(\Delta v\) between the most probable speed \(v_P\) and the rms speed \(v_{rms}\). When \(P(v)\) spreads to higher speeds, \(\Delta v\) increases. Assume that the gas is ideal and the \(N_2\) molecules rotate but do not oscillate. For 1.5 mol, an initial temperature of 250 K, and a final temperature of 500 K, what are (a) the initial difference \(\Delta v_i\), (b) the final difference \(\Delta v_f\), and (c) the entropy change \(\Delta S\) for the gas?