Math 2.0

Classical thermodynamics, heat transfer Definition

Compressibility

Builds on Isothermal processIsentropic processPartial derivativeIdeal gasHeat capacity ratio

Definition

Let a simple system with a fixed amount of matter have volume V>0V > 0, pressure pp, temperature TT and entropy SS. Assume that VV is a differentiable function of the state. The isothermal compressibility is κT=−1V(∂V∂p)T,\kappa_T = -\frac{1}{V} \left( \frac{\partial V}{\partial p} \right)_T, the relative change of volume per unit pressure along an isothermal process. The isentropic compressibility is κS=−1V(∂V∂p)S,\kappa_S = -\frac{1}{V} \left( \frac{\partial V}{\partial p} \right)_S, the same quantity along an isentropic process. Both derivatives are partial derivatives.

Remarks

The bulk modulus is K=1/κK = 1/\kappa. The two compressibilities are related by κS/κT=CV/Cp=1/γ\kappa_S / \kappa_T = C_V / C_p = 1/\gamma, where γ\gamma is the heat capacity ratio.

Example: for an ideal gas, V=nRT/pV = nRT/p, so (∂V/∂p)T=−nRT/p2=−V/p(\partial V / \partial p)_T = -nRT/p^2 = -V/p and κT=1/p\kappa_T = 1/p. Along a reversible adiabat pVγpV^\gamma is constant, so (∂V/∂p)S=−V/(γp)(\partial V / \partial p)_S = -V/(\gamma p) and κS=1/(γp)\kappa_S = 1/(\gamma p).

Non-example: in the two-phase region of a pure fluid at fixed temperature, the volume changes while the pressure stays at the vapour pressure. There (∂p/∂V)T=0(\partial p / \partial V)_T = 0, so κT\kappa_T is not finite.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHeat capacity ratio, concept, no evidence levelHeat capacity ratioconceptIdeal gas, concept, no evidence levelIdeal gasconceptIsentropic process, concept, no evidence levelIsentropic processconceptIsothermal process, concept, no evidence levelIsothermal processconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial derivative, definition, no evidence levelPartial derivativedefinitionCompressibility, definition, no evidence levelCompressibilitydefinition
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  1. Euclidean space object
  2. Function concept
  3. Heat capacity ratio concept
  4. Ideal gas concept
  5. Isentropic process concept
  6. Isothermal process concept
  7. Function of several variables concept
  8. Limit of a function of several variables concept
  9. Partial derivative definition
  10. Compressibility definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Compressibility
About this page
TagXG8W326PPJ
URIm2://XG8W326PPJ
Kinddefinition
WritingW0 stub
MSC80A10, 76N15
DefinesCompressibility
UsesIsothermal process, Isentropic process, Partial derivative, Ideal gas, Heat capacity ratio
Changed3 November 2026

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