Editing Compressibility (section)

==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 process|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 {{mvar|T}} 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 {{mvar|S}} is entropy. For a solid, the distinction between the two is usually negligible.

Since the [[density]] {{mvar|ρ}} 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 derivative]]s, 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 {{mvar|K}} (sometimes {{mvar|B}} or <math>\beta</math>).).
The [[compressibility equation]] relates the isothermal compressibility (and indirectly the pressure) to the structure of the liquid.