An ideal gas is taken from an initial state i to a final state f...

Given that the ratio of pressure to absolute temperature is constant:

$\dfrac{P}{T} = \text{constant}$

From the ideal gas equation, we have:

$PV = nRT$

$\Rightarrow \dfrac{P}{T} = \dfrac{nR}{V}$

Since $\dfrac{P}{T}$ is constant, $\dfrac{nR}{V}$ must also be constant. For a given amount of ideal gas, $n$ and $R$ are constants, which implies that the volume $V$ of the gas remains constant throughout the process.

Since the volume is constant ($dV = 0$), the process is isochoric.

The work done by the gas is given by:

$W = \int P \, dV = 0$