Irreversible PV work falls under non-equilibrium thermodynamics, which is hard or sometimes impossible to calculate using simple equations. This is due to the difficulty of defining the properties of the system during the process. For example, during an irreversible expansion of a gas against a piston, the piston accelerates away from the system, resulting in regions of varying pressures in the system. To overcome this problem, broad assumptions are made to expresss irreversible PV work mathematically.

Consider the irreversible expansion of a system consisting of a gas in an isolated vertical cylinder. The frictionless piston, which is part of the surroundings, has mass and is held stationary with catches (see diagram above).

If the force exerted by the gas on the bottom surface of the piston is greater than the weight of the piston, the gas expands and pushes the piston up when the catches are removed. The piston moves over a distance , until the force exerted by the expanded gas on the bottom surface of the piston equals to the weight of the piston. This implies that the gas is expanding against a constant force , which is due to the weight of the piston.

The change in energy of the surroundings is

where is acceleration due to gravity and is the kinetic energy of the piston.

Since the initial and final are both zero, and eq8 becomes

Noting that energy in the universe is conserved, where , and that , eq9 becomes

Since the system is isolated, there is no transfer of heat. The change in energy of the system, according to the first law of thermodynamics, is therefore the change in work done on the system:

The integral form is:

If , the gas expands freely into the vacuum. In this case, and therefore . In general, irreversible PV work against constant pressure is estimated using eq11.

Expansion work by a system on its surroundings is always greater when the process is carried out reversibly than irreversibly. This can be seen by plotting eq6 from the previous article and eq11 on the same PV graph (see upper diagram above), where the area under AC (reversible) is greater than the area under BC (irreversible). for the irreversible process is made equal to the final pressure at when the piston stops, similar to our piston illustration above.

Conversely, compression work by the surroundings on the system is always greater when the process is carried out irreversibly (BA) than reversibly (CA, see lower diagram above). for the irreversible process is now made equal to the final pressure at when the piston stops.

It is important to remember that the above is a crude attempt to associate an irreversible process with an equation.

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