Animated Solution for Chemistry - Chemical Thermodynamics: In thermodynamics the P-V work done is given by $w=-\int dV{P}_{\text{ext}}$. For a system undergoing a particular process, the work done is, $w=-\int dV\left(\frac{RT}{V-b}-\frac{a}{V^2}\right)$. This equation is applicable to a

Let's start by recalling the fundamental definition of pressure-volume work in thermodynamics. When a gas expands or compresses against an external pressure, the work done by the system is given by the integral:

Here, $P_{\text{ext}}$ represents the external pressure acting on the piston, and $dV$ is the infinitesimal change in volume.

Now, let's look at the specific expression provided in the problem:

By comparing this with our general formula, it is immediately obvious that the external pressure $P_{\text{ext}}$ has been replaced by the term inside the parentheses:

Does this expression look familiar? It should! It is the pressure of a real gas derived from the Van der Waals equation of state for exactly 1 mole of gas:

Rearranging this for the pressure of the gas ($P_{\text{gas}}$), we get exactly the term we saw in the integral. This confirms that the system contains a gas that obeys the Van der Waals equation. Therefore, Option (A) is correct.

Here is the critical conceptual leap. We established that $P_{\text{ext}}$ was replaced by $P_{\text{gas}}$. Under what thermodynamic conditions are we allowed to say that the external pressure is equal to the internal gas pressure?

This equality holds true only when the process is carried out infinitely slowly, such that the system is always in mechanical equilibrium with its surroundings. This is the very definition of a reversible process.

Because the equation relies on this substitution, it is valid for any reversible process. It doesn't matter if the temperature is kept constant (isothermal) or if no heat is exchanged (adiabatic); as long as the process is reversible, the equation holds. Thus, Options (B) and (C) are also correct.

Conversely, in an irreversible process, the external pressure is typically constant or differs significantly from the internal pressure, making this substitution invalid. Therefore, Option (D) is incorrect.