Problem — Rectangular cycle in the p-V plane

Problem

For the thermodynamic cycle in the figure (ideal gas, $A\to B\to C\to D\to A$), identify the phases in which the gas absorbs and releases heat. Estimate the work per cycle as the enclosed area.

Solution

The four phases. The cycle is a rectangle with alternating isobaric sides (constant pressure, horizontal segments) and isochoric sides (constant volume, vertical segments). For an ideal gas $pV = nRT$, so the temperature is highest at the top-right vertex and lowest at the bottom-left one.

• $A \to B$ (isobaric, low $p$, $V$ increases): expansion, $T$ increases $\Rightarrow$ heat absorbed ($Q>0$).
• $B \to C$ (isochoric, large $V$, $p$ increases): $T$ increases $\Rightarrow$ heat absorbed ($Q>0$).
• $C \to D$ (isobaric, high $p$, $V$ decreases): compression, $T$ decreases $\Rightarrow$ heat released ($Q<0$).
• $D \to A$ (isochoric, small $V$, $p$ decreases): $T$ decreases $\Rightarrow$ heat released ($Q<0$).

Work as area. The net work done by the gas in a cycle is the area enclosed in the $p$-$V$ plane, with sign given by the direction of travel. The rectangle has base $\Delta V$ and height $\Delta p$:

$\abs{W_{\text{net}}} = \Delta p \cdot \Delta V$

Sign. The path $A\to B\to C\to D\to A$ is anticlockwise (right at the bottom, up on the right, left at the top, down on the left). An anticlockwise cycle in the $p$-$V$ plane gives net negative work (work done ON the gas): the gas gives back more work in the high-pressure compression than it receives in the low-pressure expansion. This is a reverse cycle (refrigerator/heat pump), not an engine.

$W_{\text{net}} = \oint p \, \dd V < 0, \qquad \abs{W_{\text{net}}} = \text{area of the rectangle}$

$\ev{\text{Absorbs in } A\!\to\!B,\, B\!\to\!C;\ \text{releases in } C\!\to\!D,\, D\!\to\!A;\ \abs{W} = \Delta p\,\Delta V}$

Collegamenti

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