Van der Waal's gas equation for an adiabatic process and its Carnot engine efficiency

TL;DR

This paper derives the Van der Waal's equation for adiabatic processes and calculates the Carnot engine efficiency for such a gas, highlighting differences from ideal gases and educational insights for undergraduates.

Contribution

It introduces a specific form of Van der Waal's equation for adiabatic processes and analyzes Carnot efficiency with this real gas model, filling a gap in undergraduate literature.

Findings

Derived Van der Waal's adiabatic equation with a new exponent

Calculated Carnot efficiency for Van der Waal's gas

Showed efficiency is independent of the working substance

Abstract

There has been many studies on gases which obeys Van der Waal's equation of state. However there is no specific and direct studies of Van der Waal's gas which undergoes adiabatic processes are available in the undergraduate text books and also in literature. In an adiabatic process there is no heat energy exchange between the system and its surroundings. In this article, we find that the Van der Waal's equation for the adiabatic process as $\left(P+\frac{n^2a}{V^2}\right) \left(V-nb\right)^{\Gamma}=\mbox{constant}$, where $P$ is the pressure, $V$ is the volume, $n$ is the number of moles of the Van der Waal's gas, $a$ and $b$ are Van der Waal's constant and $\Gamma$ is a factor which relates the specific heat at constant pressure and at constant volume. We use this relation explicitly and obtained the efficiency of a Carnot engine whose working substance obeys Van der Waal's equation of state. Our simplest approach may provide clear idea to the undergraduate students that $\Gamma$ is different from $\gamma$ of the ideal gas for an adiabatic process. We also shown that the efficiency of the Carnot engine is independent of the working substance.