Discussion
The temperature of an ideal monoatomic gas is changed from \(T_{1}\) to \(T_{2}\), keeping (i) pressure constant, and (ii) keeping its volume constant. Then the ratio of the change in entropy \((\Delta S)p/(\Delta S)V\) is :
Answer: C
The ratio is 5/3.
Heating at constant pressure or constant volume between the same temperatures gives
\[ \begin{aligned} (\Delta S)_{p} &= nC_{p}\ln\frac{T_{2}}{T_{1}} \\ (\Delta S)_{V} &= nC_{V}\ln\frac{T_{2}}{T_{1}} \\ \frac{(\Delta S)_{p}}{(\Delta S)_{V}} &= \frac{C_{p}}{C_{V}} = \gamma \end{aligned} \]
For a monatomic ideal gas \(C_{V} = \frac{3}{2}R\) and \(C_{p} = \frac{5}{2}R\), so the ratio is 5/3.
2/3 is \(R/C_{V}\), not a ratio of heat capacities (the inverse ratio \(C_{V}/C_{p}\) would be 3/5); 1.0 would mean the two heat capacities are equal; and 1/2 has no basis.
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