DPDn

dPdn is a notation used in thermodynamics and statistical mechanics to denote the partial derivative of pressure with respect to the particle number, usually written as (∂P/∂n) at fixed temperature and volume. Here n represents the number of particles or moles in the system, depending on the chosen convention, and P is the system pressure. The quantity expresses how the pressure responds to adding or removing particles while keeping T and V constant. In many texts n is the mole number, so the derivative is (∂P/∂n)_{T,V}.

Context and interpretation

dPdn arises whenever an equation of state expresses pressure as a function of particle number, volume, and temperature: P = P(n, V, T). The derivative provides a measure of the system’s pressure sensitivity to particle insertion, a concept that is important in processes such as gas loading, adsorption, and phase-equilibria analysis. It is determined from the equation of state or from experimental data.

Example: ideal gas

For an ideal gas, P = nRT/V. Holding T and V constant yields dP/dn = RT/V, which can also be written as P/n. This straightforward relationship contrasts with real fluids, where molecular interactions cause dP/dn to depend nonlinearly on n, V, and T.

Relation to other quantities

dPdn is related to, but distinct from, other thermodynamic derivatives such as the isothermal compressibility κT = -1/V (∂V/∂P)_T,N. In practice, dPdn can be inferred from measured or modeled equations of state and used to assess system stability and response to particle exchange.