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 <math>\kappa</math> (denoted in some fields) may be expressed as

:<math>\beta =-\frac{1}{V}\frac{\partial V}{\partial p}</math>,

where is volume and 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.

Definition

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:

:<math>\beta_T=-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_T,</math>

where the subscript indicates that the partial differential is to be taken at constant temperature.

Isentropic compressibility is defined:

:<math>\beta_S=-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_S,</math>

where is entropy. For a solid, the distinction between the two is usually negligible.

Since the density of a material is inversely proportional to its volume, it can be shown that in both cases

:<math>\beta=\frac{1}{\rho}\left(\frac{\partial \rho}{\partial p}\right).</math>

For instance, for an ideal gas,

:<math>pV=nRT,\, \rho=n/V </math>. Hence <math>\rho=p/RT </math>.

Consequently, the isothermal

compressibility of an ideal gas is

:<math>\beta=1/(\rho RT)= 1/P </math>.

The ideal gas (where the particles do not interact with each other) is an abstraction. The particles in real materials interact with each other. Then, the relation between the pressure, density and temperature is known as the equation of state denoted by some function <math>F</math>. The Van der Waals equation is an example of an equation of state for a realistic gas.

:<math>\rho=F(p,T)</math>.

Knowing the equation of state, the compressibility can be determined for any substance.

Relation to speed of sound

The speed of sound is defined in classical mechanics as:

:<math>c^2=\left(\frac{\partial p}{\partial\rho}\right)_S</math>

It follows, by replacing partial derivatives, that the isentropic compressibility can be expressed as:

:<math>\beta_S=\frac{1}{\rho c^2}</math>

Relation to bulk modulus

The inverse of the compressibility is called the bulk modulus, often denoted (sometimes or <math>\beta</math>).).

The compressibility equation relates the isothermal compressibility (and indirectly the pressure) to the structure of the liquid.

Thermodynamics

The isothermal compressibility is generally related to the isentropic (or adiabatic) compressibility by a few relations:

: <math>\frac{\beta_T}{\beta_S} = \frac{c_p}{c_v} = \gamma,</math>

: <math>\beta_S = \beta_T - \frac{\alpha^2 T}{\rho c_p}, </math>

: <math>\frac{1}{\beta_S} = \frac{1}{\beta_T} + \frac{\Lambda^2 T}{\rho c_v} ,</math>

where is the heat capacity ratio, is the volumetric coefficient of thermal expansion, is the particle density, and <math>\Lambda = (\partial P/\partial T)_{V}</math> is the thermal pressure coefficient.

In an extensive thermodynamic system, the application of statistical mechanics shows that the isothermal compressibility is also related to the relative size of fluctuations in particle density:

! Material !! <math>\beta_T</math> (m<sup>2</sup>/N or Pa<sup>−1</sup>)

| Plastic clay || –

| Stiff clay || –

| Medium-hard clay || –

| Loose sand || –

| Dense sand || –

| Dense, sandy gravel || –

| Ethyl alcohol ||

| Carbon disulfide || 4.6

| Rock, sound || <

| Glycerine

In transition regions, where this pressure dependent dissociation is incomplete, both beta (the volume/pressure differential ratio) and the differential, constant pressure heat capacity greatly increases. For moderate pressures, above 10,000 K the gas further dissociates into free electrons and ions. for the resulting plasma can similarly be computed for a mole of initial air, producing values between 2 and 4 for partially or singly ionized gas. Each dissociation absorbs a great deal of energy in a reversible process and this greatly reduces the thermodynamic temperature of hypersonic gas decelerated near the aerospace object. Ions or free radicals transported to the object surface by diffusion may release this extra (nonthermal) energy if the surface catalyzes the slower recombination process.

Negative compressibility

For ordinary materials, the bulk compressibility (sum of the linear compressibilities on the three axes) is positive, that is, an increase in pressure squeezes the material to a smaller volume. This condition is required for mechanical stability. However, under very specific conditions, materials can exhibit a compressibility that can be negative.