Pressure

Definition and physical interpretation

In thermodynamics, pressure $p$ is an intensive variable that characterizes the system’s mechanical tendency to expand or contract against its boundary . For a uniform isotropic fluid in mechanical equilibrium , it coincides with the familiar force-per-area on a surface element, averaged over microscopic fluctuations.

Thermodynamically, pressure is defined as the variable conjugate to volume in the energy balance for a simple compressible system.

Thermodynamic definition via fundamental differentials

For a single-component simple compressible system with state described by $(S,V,N)$, the differential form of the fundamental relation (see internal energy and the first law ) is

Here $p$ is defined by the partial derivative

Equivalently, in the entropy representation $S=S(U,V,N)$,

These definitions make sense at equilibrium ; away from equilibrium, “pressure” may require additional structure (e.g., stress tensors) and may not be a single state variable.

Work and sign conventions

For a quasistatic compression/expansion, the mechanical work contribution is

under the standard pressure–volume work sign convention (consistent with the general work sign convention and the fact that work is path-dependent ). With this choice, expansion ($dV>0$) corresponds to work done by the system, so $\delta W<0$.

Relations to free energies and response functions

Pressure can be expressed as a derivative of thermodynamic potentials obtained by a Legendre transform :

• From the Helmholtz free energy $F(T,V,N)=U-TS$,

  $$p = -\left(\frac{\partial F}{\partial V}\right)_{T,N}.$$

• From the Gibbs free energy $G(T,p,N)=U-TS+pV$,

  $$V = \left(\frac{\partial G}{\partial p}\right)_{T,N}.$$

A key linear response quantity controlled by pressure is the isothermal compressibility

which must satisfy $\kappa_T\ge 0$ for thermodynamic stability .

Finally, pressure is tied to the equation of state $p=p(T,V,N)$ (or equivalent forms), which encodes material-specific information beyond the general structure of the laws.