Problem — Area of a p-V cycle

Problem

A closed $pV$ cycle has an area of $1{,}5$ kJ and is traversed clockwise. What is the net work done by the gas in one cycle? And if it were traversed anticlockwise?

Solution

Work as oriented area. The net work done by the gas in a cycle is the integral of pressure over volume, i.e. the area enclosed in the $p$-$V$ plane:

$W = \oint p \, \dd V = \pm \, (\text{enclosed area})$

The sign depends on the direction of travel.

Clockwise (engine). Traversing the cycle clockwise, the expansion (upper branch, high pressure) occurs at pressures greater than the compression (lower branch): the gas does net positive work on the surroundings.

$\ev{W = +1{,}5 \ \text{kJ} \quad (\text{clockwise} \Rightarrow \text{engine})}$

Anticlockwise (reverse cycle). Reversing the direction, the compression occurs at high pressure and the expansion at low pressure: the gas receives net work from the surroundings. The magnitude is the same, only the sign changes:

$\ev{W = -1{,}5 \ \text{kJ} \quad (\text{anticlockwise} \Rightarrow \text{refrigerator/heat pump})}$

The area measures the magnitude of the work; it is the direction of travel that determines whether the engine produces work (engine) or consumes it (reverse cycle).

Collegamenti

Argomenti: Thermodynamics Concetti: Thermodynamic transformations · First law of thermodynamics Competenze: Reading graphs Metodi: P-V and ln P - ln V diagram Oggetti: Ideal gas