Question

An ideal gas is taken through a process in which the pressure and volume change according to the equation \(P = kV\). The molar heat capacity of the gas for the process is given by:

• A. \(C = C_v + \dfrac{R}{3}\)
• B. \(C = C_v + R\)
• C. \(C = C_v + \dfrac{R}{2}\)
• D. \(C = C_v + 2R\)

Hint:

Rewrite \(P = kV\) as \(PV^{-1} = \text{const}\) (polytropic \(n = -1\)) and use \(C = C_v + \dfrac{R}{1-n}\).

Answer 1

The Correct Option is C

Step 1: Use the first law directly. For one mole, $dQ = C\,dT$, $dU = C_v\,dT$, and $dW = P\,dV$, so $C = C_v + P\dfrac{dV}{dT}$.

Step 2: The gas obeys $P = kV$ and the ideal-gas law $PV = RT$. Substituting $P = kV$ gives $kV^2 = RT$, hence $V^2 = \dfrac{RT}{k}$ and $V = \sqrt{RT/k}$.

Step 3: Differentiate $kV^2 = RT$ with respect to $T$: $2kV\dfrac{dV}{dT} = R$, so $\dfrac{dV}{dT} = \dfrac{R}{2kV}$.

Step 4: Then $P\dfrac{dV}{dT} = (kV)\cdot\dfrac{R}{2kV} = \dfrac{R}{2}$. Therefore
\[C = C_v + \frac{R}{2}.\]
\[\boxed{C = C_v + \dfrac{R}{2}}\]