Editing Compressibility (section)

==Thermodynamics==
{{Main|Compressibility factor}}

The [[isothermal]] compressibility is generally related to the [[isentropic]] (or [[adiabatic]]) compressibility by a few relations:<ref name=llcompress>{{cite book |last1=Landau |last2=Lifshitz |title=[[Course of Theoretical Physics]] Vol 5: Statistical Physics |date=1980 |publisher=Pergamon |pages=54–55 and 342}}</ref>

: <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 {{mvar|γ}} is the [[heat capacity ratio]], {{mvar|α}} is the volumetric [[coefficient of thermal expansion]], {{math|''ρ'' {{=}} ''N''/''V''}} 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:<ref name=llcompress/>
: <math>\beta_T = \frac{(\partial \rho / \partial \mu)_{V,T}}{\rho^2} = \frac{\langle(\Delta N)^2\rangle/V}{k_{\rm B} T \rho^2},</math>
where {{mvar|μ}} is the [[chemical potential]].

The term "compressibility" is also used in [[thermodynamics]] to describe deviations of the [[thermodynamic properties]] of a [[real gas]] from those expected from an [[ideal gas]].

The '''[[compressibility factor]]''' is defined as

: <math>Z=\frac{p V_m}{R T} </math>
where {{mvar|p}} is the [[pressure]] of the gas, {{mvar|T}} is its [[temperature]], and <math>V_m</math> is its [[molar volume]], all measured independently of one another.  In the case of an ideal gas, the compressibility factor {{mvar|Z}} is equal to unity, and the familiar [[ideal gas law]] is recovered:

: <math>p = \frac{RT}{V_m}</math>

{{mvar|Z}} can, in general, be either greater or less than unity for a real gas.

The deviation from ideal gas behavior tends to become particularly significant (or, equivalently, the compressibility factor strays far from unity) near the [[critical point (thermodynamics)|critical point]], or in the case of high pressure or low temperature.  In these cases, a generalized [[compressibility chart]] or an alternative [[equation of state]] better suited to the problem must be utilized to produce accurate results.