Internal pressure

Internal pressure is the rate at which a system’s internal energy changes with volume at constant temperature: \(\pi_T=(\partial U/\partial V)_T\). It is a thermodynamic property, not the mechanical pressure exerted on a container.

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Thermodynamic identity: An equation relating thermodynamic derivatives through the laws of thermodynamics. It gives the route from the energy definition to a measurable equation-of-state expression.
Internal energy: The total microscopic kinetic and potential energy of a system. Internal pressure measures how this energy changes when volume changes at fixed temperature.
Van der Waals equation: An equation of state that models gases using finite molecular size and attractive interactions. Its attraction term produces a nonzero internal pressure in the fluid model.
Ideal gas: A model gas whose particles have negligible volume and no intermolecular forces. Its internal energy depends only on temperature, so its internal pressure is zero.
Thermodynamic equation of state: A relation among thermodynamic state variables and response properties. It expresses internal pressure using temperature and the change of mechanical pressure with temperature.
Thermodynamic pressure: The intensive state variable associated with reversible volume work in thermodynamics. It is distinct from internal pressure, though its temperature dependence helps determine it.
Joule expansion: The free expansion of a gas into a vacuum in an insulated container. The temperature change in this experiment is linked to internal pressure through the Joule coefficient.
Mechanical pressure: Force per unit area exerted by a fluid or other material on a surface. It describes force on boundaries, unlike internal pressure’s energy response to volume.
Maxwell relations: Identities among thermodynamic derivatives derived from equality of mixed second derivatives of thermodynamic potentials. They connect internal pressure’s energy derivative to derivatives of pressure and temperature.
Partial derivative: A derivative describing change in one variable while specified other variables are held fixed. The constant-temperature constraint is essential to interpreting the defining derivative.