PV Diagrams and Thermodynamic Work Reference Cheat Sheet

Key Facts

• Thermodynamic work done by a gas is $W = \int_{V_i}^{V_f} P\,dV$, which equals the signed area under a curve on a PV diagram.
• For an isobaric process with constant pressure, the work is $W = P\Delta V = P(V_f - V_i)$.
• For an isochoric process with constant volume, the work is $W = 0$ because $\Delta V = 0$.
• For an ideal gas undergoing an isothermal process, the work is $W = nRT\ln\left(\frac{V_f}{V_i}\right)$.
• For an adiabatic ideal gas process, $PV^{\gamma} = \text{constant}$ and $W = \frac{P_iV_i - P_fV_f}{\gamma - 1}$.
• Expansion gives $\Delta V > 0$ and usually $W > 0$, meaning the gas does work on the surroundings.
• Compression gives $\Delta V < 0$ and usually $W < 0$, meaning work is done on the gas.
• For a complete cycle on a PV diagram, the net work is the enclosed area, with clockwise cycles giving $W_{\text{net}} > 0$ and counterclockwise cycles giving $W_{\text{net}} < 0$.

Vocabulary

PV Diagram

A graph of pressure $P$ versus volume $V$ that represents the thermodynamic state and process of a gas.

Thermodynamic Work

Energy transferred when a gas changes volume under pressure, calculated by $W = \int P\,dV$.

Isobaric Process

A thermodynamic process that occurs at constant pressure, so work is $W = P\Delta V$.

Isochoric Process

A thermodynamic process that occurs at constant volume, so the gas does no work and $W = 0$.

Isothermal Process

A thermodynamic process that occurs at constant temperature, with ideal gas work $W = nRT\ln\left(\frac{V_f}{V_i}\right)$.

Adiabatic Process

A thermodynamic process with no heat transfer, so $Q = 0$ and for an ideal gas $PV^{\gamma} = \text{constant}$.