3.3 Work in thermodynamic processes

Work in Thermodynamic Processes

Definition of thermodynamic work

Thermodynamic work is energy transferred between a system and its surroundings when a force acts through a displacement. The most common example is a gas pushing a piston inside a cylinder: as the gas expands, it does work on the piston.

Work occurs whenever a system changes an external parameter like volume or position. Two main types show up in thermodynamics courses:

• Boundary work (PdV work) is work done by a system as its volume changes against an external pressure. This is the type you'll encounter most often. It's expressed as:

$W = \int P \, dV$

where $P$ is the external pressure and $dV$ is the infinitesimal change in volume. When the gas expands ($dV > 0$), the system does positive work on the surroundings. When it's compressed ($dV < 0$), work is done on the system.

• Shaft work is work transmitted through a rotating shaft, like in a turbine or pump. It's expressed as:

$W = \int \tau \, d\theta$

where $\tau$ is the torque and $d\theta$ is the angular displacement.

Calculation of work in specific processes

The formula you use to calculate work depends entirely on what's held constant during the process. Here are the four key process types:

Isothermal process (constant temperature)

For an ideal gas at constant $T$, pressure and volume are inversely related (Boyle's Law), so the work integral evaluates to:

$W = nRT \ln \frac{V_2}{V_1}$

where $n$ is the number of moles, $R$ is the universal gas constant, $T$ is the temperature, and $V_1$ and $V_2$ are the initial and final volumes. If the gas expands ($V_2 > V_1$), the natural log is positive and the system does positive work.

Isobaric process (constant pressure)

Since pressure doesn't change, it comes out of the integral, giving:

$W = P(V_2 - V_1)$

This is the simplest work calculation. On a P-V diagram, it's just the area of a rectangle.

Isochoric process (constant volume)

No work is done because $dV = 0$. The volume never changes, so there's no displacement for a force to act through. Any energy transfer in an isochoric process happens entirely as heat.

Adiabatic process (no heat transfer)

With $Q = 0$, all energy exchange occurs as work. For an ideal gas undergoing a reversible adiabatic process:

$W = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1}$

where $\gamma = C_p / C_v$ is the ratio of specific heats. Notice that the work depends on both the initial and final states of pressure and volume.

Work analysis on P-V diagrams

P-V diagrams are one of the most useful tools in thermodynamics because the work done in any process equals the area under the curve on a P-V diagram.

• For a single process (A to B), calculate the area under the path between those two states. Different paths between the same two states yield different amounts of work, which is why work is a path function, not a state function.
• For a complete thermodynamic cycle, the net work equals the area enclosed by the cycle on the P-V diagram:

$W_{\text{net}} = \oint P \, dV$

Sign convention for cycles: A clockwise cycle on a P-V diagram represents a heat engine that does positive net work on the surroundings. A counterclockwise cycle represents a refrigerator or heat pump where net work is done on the system.

Reversible vs. irreversible work

Reversible work is the maximum useful work you can extract from a process between two equilibrium states. It requires the process to proceed infinitely slowly (quasi-statically), so the system stays in equilibrium with its surroundings at every step. This is an idealization; no real process is truly reversible.

Irreversible work is the actual work obtained in a real process. It's always less than the reversible work for a work-producing device (and always more than the reversible work for a work-consuming device like a compressor).

The gap between reversible and irreversible work is called lost work, and it comes from irreversibilities such as:

• Friction between moving parts
• Heat transfer across a finite temperature difference
• Unrestrained (free) expansion of a gas
• Mixing of different substances

Thermodynamic efficiency captures this gap as a ratio:

$\eta = \frac{W_{\text{actual}}}{W_{\text{reversible}}}$

For any real process, $\eta < 1$ (less than 100%). This isn't a design flaw you can engineer away; it's a fundamental consequence of the second law of thermodynamics.