Discussion
The specific heat capacity at constant volume Cv of a mole of an ideal gas is related to the gas constant R and the ratio of the specific heats Υ as
Answer: A
The relation is \(C_{v} = \frac{R}{\Upsilon - 1}\), where the paper writes \(\Upsilon\) for the ratio of specific heats usually called \(\gamma\).
For one mole of an ideal gas, \(C_{p} - C_{v} = R\) and \(\frac{C_{p}}{C_{v}} = \gamma\). Substituting \(C_{p} = \gamma C_{v}\):
\[ \begin{aligned} \gamma C_{v} - C_{v} &= R \\ C_{v} &= \frac{R}{\gamma - 1} \end{aligned} \]
Then \(C_{p} = \frac{\gamma R}{\gamma - 1}\). \(\frac{R}{1 - \gamma}\) and \(\frac{R\gamma}{1 - \gamma}\) are negative because \(\gamma \gt 1\), so they cannot be heat capacities. \(\frac{R}{1 + \gamma}\) has the wrong sign in the denominator.
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