Discussion
Consider any substance whose mass specific heat capacity at constant pressure and mass specific heat capacity at constant volume are \(c_{P}\) and \(c_{V}\), respectively. If the gas constant is R, which ONE of the following relations is CORRECT for the substance at all conditions?
Answer: C
The relation true for every substance is \(c_{P} \ge c_{V}\).
The general thermodynamic relation is:
\[ c_{P}-c_{V} = \frac{Tv\beta^{2}}{\kappa_{T}} \]
Temperature, specific volume and the isothermal compressibility \(\kappa_{T}\) are always positive, and \(\beta^{2}\) cannot be negative, so the difference is never negative. It is zero when the expansivity \(\beta\) is zero, for example liquid water at 4 °C, or for an idealised incompressible substance.
So \(c_{P} \gt c_{V}\) fails in those cases, \(c_{P} - c_{V} = R\) holds only for an ideal gas, and \(c_{P} = c_{V}\) holds only in the special cases just named.
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