Solution 35
Let the length of the container be $L$ and the area of cross-section be $A$. The relaxed length of the spring is $L$. If the piston is at a distance $x$ from the left wall, the length of the spring is $L-x$. The extension/compression of the spring is $\Delta x = L – (L-x) = x$.
Force exerted by the spring $F = kx$. The pressure on the gas is:
$$ P = \frac{F}{A} = \frac{kx}{A} $$Since the volume of the gas is $V = Ax$, we have $x = V/A$. Substituting this into the pressure equation:
$$ P = \frac{k}{A} \left(\frac{V}{A}\right) \implies P = \left(\frac{k}{A^2}\right) V $$This implies $P \propto V$. Comparing this to the polytropic process equation $PV^x = \text{constant}$ (or $P V^{-1} = \text{constant}$), we find the polytropic index is $x = -1$.
The molar heat capacity for a polytropic process is given by $C = C_v + \frac{R}{1-x}$. For a monoatomic gas $C_v = \frac{3}{2}R$.
$$ C = \frac{3R}{2} + \frac{R}{1 – (-1)} = \frac{3R}{2} + \frac{R}{2} = 2R $$The heat capacity of the system (for $n$ moles) is: