A sample of a monatomic ideal gas consisting of \(1.00 \times 10^{23}\) atoms in a volume of \(4.00 \times 10^{-3} \mathrm{~m}^{3}\) is initially at a temperature of \(290 \mathrm{~K}\). The gas is cooled until it reaches a pressure of \(5.00 \times 10^{4} \mathrm{~N} / \mathrm{m}^{2}\).

(a) What is the change in entropy for the gas and for each atom?

(b) What is the ratio of the final rms speed of each atom to its initial rms speed?

(c) How does the change in the entropy of the gas depend on this ratio \(v_{\text {rms }, \mathrm{f}} / v_{\mathrm{rms}, \mathrm{l}}\) ?