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: Write the process in polytropic form. The relation \(P = kV\) means \(P V^{-1} = k = \text{constant}\). Comparing with the polytropic law \(P V^{n} = \text{constant}\), we get \(n = -1\).

Step 2: For any polytropic process, the molar heat capacity is
\[C = C_v + \frac{R}{1 - n}.\]
Step 3: Substitute \(n = -1\):
\[C = C_v + \frac{R}{1 - (-1)} = C_v + \frac{R}{2}.\]
So the molar heat capacity exceeds \(C_v\) by \(R/2\).
\[\boxed{C = C_v + \dfrac{R}{2}}\]