Compressibility

In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a fluid or solid as a response to a pressure (or mean stress) change. In its simple form, the compressibility (denoted β in some fields) may be expressed as where V is volume and p is pressure. The choice to define compressibility as the negative of the fraction makes compressibility positive in the (usual) case that an increase in pressure induces a reduction in volume. The reciprocal of compressibility at fixed temperature is called the isothermal bulk modulus. The specification above is incomplete, because for any object or system the magnitude of the compressibility depends strongly on whether the process is isentropic or isothermal. Accordingly, isothermal compressibility is defined: where the subscript T indicates that the partial differential is to be taken at constant temperature. Isentropic compressibility is defined: where S is entropy. For a solid, the distinction between the two is usually negligible. Since the density ρ of a material is inversely proportional to its volume, it can be shown that in both cases The speed of sound is defined in classical mechanics as: It follows, by replacing partial derivatives, that the isentropic compressibility can be expressed as: The inverse of the compressibility is called the bulk modulus, often denoted K (sometimes B or ).). The compressibility equation relates the isothermal compressibility (and indirectly the pressure) to the structure of the liquid. Compressibility factor The isothermal compressibility is generally related to the isentropic (or adiabatic) compressibility by a few relations: where γ is the heat capacity ratio, α is the volumetric coefficient of thermal expansion, ρ = N/V is the particle density, and is the thermal pressure coefficient.