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Work (thermodynamics)

Energy transfer by organized macroscopic forces in thermodynamic processes, commonly expressed as mechanical, electrical or pressure-volume work; distinct from heat and path-dependent.

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: May 30, 2026

simple.wikipedia.org · Public domain

Overview

In thermodynamics, work is the transfer of energy that occurs when a force acts through a displacement or when a generalized force acts through a generalized displacement. Unlike heat, which is energy transferred because of a temperature difference, work is associated with macroscopic, coordinated motion or fields. The SI unit is the joule (J), and work can be expressed mechanically as W = \u222b F·dx in integral form.

Forms and calculation

Common kinds of thermodynamic work include:

Pressure–volume (PV) work: work due to volume change against an external pressure, often written W = \u222b P_ext dV. For a constant external pressure, W = P_ext \u0394V. For a reversible quasi-static expansion W_rev = \u222b P_int dV.
Shaft or mechanical work: useful work delivered by rotating devices such as turbines and compressors.
Electrical work: energy transferred by moving charge through a potential difference.
Other forms: surface-tension work, magnetic or electric field work, and flow/work associated with moving fluid across control-volume boundaries.

Key properties

Work is a path-dependent quantity (a process quantity), so it is not a state function; its differential is often written as δW to emphasize this. The amount of work exchanged depends on how the process is carried out, not solely on initial and final states. Reversible processes yield the maximum useful (non-dissipative) work for given end states, while irreversible processes produce less useful work.

Sign conventions and examples

Sign conventions vary by field: engineers commonly take work done by the system as positive, while many chemistry texts take work done on the system as positive. Examples:

Isobaric expansion: W = P\u0394V.
Quasi-static isothermal expansion of an ideal gas: W = nRT ln(V2/V1) (reversible case).

Relation to the first law and importance

The first law of thermodynamics balances changes in internal energy with heat and work: \u0394U = Q - W (or alternative sign conventions). Work represents an organized transfer of energy and is central to engines, refrigeration, and power conversion. Distinguishing work from heat helps classify energy flows and calculate efficiencies in cycles such as Carnot, Otto, and Rankine.

Historical and practical notes

The formalization of thermodynamic work grew from 19th-century studies of heat engines and mechanics. In practice, measuring work often involves measuring forces, torques, pressures, electrical potentials, or displacements. Because work depends on the process path and interaction details, careful specification of the system boundary and the external conditions is essential when computing it.

Smooth process

The work supplied frictionlessly and quasistatically is in the cylinder shown with cross-section $\left(\Rightarrow {\mathrm {d}}s={\frac {{\mathrm {d}}V}{A}}\right)$

because $F=p\cdot A$ (freedom from friction):

$\Rightarrow W_{\mathrm {1,2} }=\int \limits _{V_{1}}^{V_{2}}\delta W=-\int \limits _{V_{1}}^{V_{2}}p\cdot \mathrm {d} V$

with

$\delta W=-pdV$ the inexact differential of the volume work
: Pressure
$\mathrm {d} V$ : volume change.

In the p-V diagram, this change of state runs from point 1 to point 2, i.e. in the case of the compression shown in the negative volume direction without the minus sign in the formula, the compression work would have a negative sign. $\left({\mathrm {d}}V<0\right);$

The integral value, which corresponds to the area under the state curve, can be calculated if the function p = f(V) is known (see below).

Frictional process

In the real case, if a frictional force acts between the piston and the cylinder, the frictional work must be $W_{R}$ applied in addition to the volume change work during compression. This increases the internal energy of the system and thus the pressure compared to the frictionless process (if it is not dissipated to the outside as heat by cooling):

${p_{2}}'>p_{2}$

In the p-V diagram, the change of state now runs from point 1 to point 2'. This means that the work of change in volume, which corresponds to the area under the curve, also becomes greater, without the friction work itself being included in it:

$\Rightarrow W_{{1,2'}}>W_{{1,2}}$

The work to be done from the outside is therefore the sum of the now greater volume change work and the friction work:

$W_{{{\mathrm {ext}}}}=W_{{1,2'}}+W_{R}$

Calculation example

Assume the isothermal expansion of an ideal gas $\left(T={\text{konst.}}\right).$

Then, by substituting the thermal equation of state of ideal gases:

$p(V)=n\cdot R\cdot T\cdot {\frac {1}{V}}$

with

n is the amount of substance
R is the general gas constant
T the absolute temperature

solve the integral for the volume work:

${\begin{aligned}\Rightarrow W_{{\mathrm {1,2}}}&=-&n\cdot R\cdot T\cdot \ln {\frac {V_{2}}{V_{1}}}\\&=&n\cdot R\cdot T\cdot \ln {\frac {V_{1}}{V_{2}}}\end{aligned}}$

Using this equation, we see that during the expansion of an ideal gas, the volume work is negative, i.e. energy is released; this follows from the logarithm, which is negative for numbers less than one and positive for numbers greater than one:

${\begin{aligned}V_{2}>V_{1}\\\Leftrightarrow {\frac {V_{2}}{V_{1}}}>1\\\Leftrightarrow \ln {\frac {V_{2}}{V_{1}}}>0\\\Rightarrow W_{{\mathrm {1,2}}}<0\end{aligned}}$

Instead of n-R, one can also insert m-Rs above:

$n\cdot R=m\cdot R_{{\mathrm {s}}}$

where

m is the mass of the substance and
_{Rs is} its specific gas constant.

Open system

If the compression is carried out in an open system with the external pressure $p_{0}$ , the actual work to be done is as follows

$W_{1,2}=p_{0}\cdot (V_{2}-V_{1})$

since the external pressure multiplied by the surface area also results in a force. If the external pressure is higher than the internal pressure of the volume to be compressed, energy is gained; if it is lower, work must be done.