Reversible work in thermodynamics

In classical mechanics, the amount of work done on an object, $dw$ , is the product of the external force, $F_{ext}$ , acting on the object and the distance moved in the direction of the force, $dx$ :

$dw=F_{ext}dx\;\;\;\;\;\;\;\;(1)$

Pressure-volume work, the most common form of work in chemical thermodynamics, uses the same concept and is defined as the transfer of energy between a system and its surroundings due to a force differential between the two. Another type of work usually encountered in chemical thermodynamics, which is discussed in the article on electrochemistry, is electrical work.

Consider a system containing an ideal gas in a frictionless, closed piston-cylinder with a weightless piston of area $A$ (see diagram above). Initially, the piston, which is part of the surroundings, is stationary and the gas molecules are evenly distributed in the system. This implies that the force exerted by the atmosphere on the piston balances the internal force on the piston.

When an additional external force is exerted on the piston, the gas is compressed. Substituting $p_{ext}=\frac{F_{ext}}{A}$ in eq1, the work done on the system is:

$dw=p_{ext}dV\;\;\;\;\;\;\;\;(2)$

However, there are two ways to move the piston to compress the gas. If the piston moves rapidly over a relatively great distance that results in a large change in volume, $\Delta V$ , there may be a momentary uneven distribution of molecules in the system, where the density of molecules adjacent to the piston wall is greater than that further away from the piston. In this case, we cannot equate the external pressure on the piston with the internal pressure. Instead, if the piston moves very slowly over an infinitesimal distance that results in an infinitesimal change in volume, $dV$ , such that the molecules in the system have enough time to equilibrate and redistribute themselves evenly throughout the infinitesimally smaller volume, the external pressure is equal to the internal pressure for the infinitesimal compression. As the internal pressure is the pressure of the gas $p$ :

$p_{ext}=p\;\;\;\;\;\;\;\;(3)$

Substituting eq3 in eq2,

$dw=pdV\;\;\;\;\;\;\;\;(4)$

The total work done for a series of infinitesimal compressions on the system from an initial volume $V_i$ to a final volume $V_f$ is:

$w=-\int_{V_i}^{V_f}pdV\;\;\;\;\;\;\;\;(5)$

As explained in the previous section, we call such a process of a series of infinitesimal changes that occur very slowly so that the system is always in equilibrium, a reversible process. According to IUPAC convention, the negative sign is added so that work done on the system by the surroundings is positive. In other words, if $dV<0$ , $w>0$ , we have reversible compression with work done by the surroundings on the system, and if $dV>0$ , $w<0$ , we have reversible expansion with work done by the system on the surroundings.