Work Done in Thermodynamic Processes: Formulas Explained

Work in thermodynamics usually means boundary work when a gas expands and pushes a piston or is compressed by it. It equals the area under a pressure-volume graph and depends not only on start and end points but also on the path taken between them.

Calculating work correctly is essential for engines, compressors, and power plants because work is the useful output we pay fuel to obtain. The same gas between same two states can do different work in different processes.

Key Concepts

Pressure P unit pascal, volume V unit cubic metre, work unit joule. Expansion work done by system is positive in engineering convention used here. On a P-V diagram work is area under the curve; clockwise cycles produce net work output. Reversible work assumes quasi-static change through equilibrium states. Work is a path function, not a state function, unlike internal energy.

Formulas

General: W_by = integral P dV. Isobaric: W = P(V2 - V1). Isothermal ideal gas: W = nRT ln(V2/V1) = P1V1 ln(V2/V1). Adiabatic ideal gas: W = (P1V1 - P2V2)/(gamma - 1). Isochoric: W = 0. Polytropic: PV^n = constant, W = (P1V1 - P2V2)/(n - 1) for n not equal 1.

Worked Example

A gas at constant pressure 2.0 x 10^5 pascal expands from 0.030 to 0.050 cubic metre. Find work.

Step 1: delta V = 0.050 - 0.030 = 0.020 cubic metre. Step 2: W = P delta V = 2.0 x 10^5 x 0.020 = 4000 joule. That is 4.0 kilojoule done by gas. If compression reversed the volumes, work done by gas would be -4000 joule, meaning 4000 joule work is done on gas.

Common Misconceptions

• Work depends only on endpoints: Wrong. Area under P-V curve changes with path, so work changes.
• Free expansion does lots of work: Wrong. Expansion into vacuum pushes against zero pressure, so boundary work is zero.
• Area on T-V diagram gives work: Wrong. Boundary work area is on P-V diagram only.

Real-World Applications

Car engine power comes from net area of its P-V loop repeated many times per second. Air compressors consume work to squeeze gas for tools and refrigeration. Steam turbines extract work from expanding steam. Bicycle pumps do work on air, warming it during compression, which you can feel in the pump barrel.

Thermodynamic work is integral of P dV and equals the P-V area. Learn the special formulas for isobaric, isothermal, adiabatic, and isochoric paths and you can compute engine output and compressor input reliably. Practising sign convention and units carefully avoids most numerical errors in examinations and design calculations.