Compressibility

The compressibility is a characteristic of a body, defining its relative variation of Volume under the effect of a Pression applied. It is a very large for the Gaz, low for the Liquide S and very low value for the solid S usual.

It is defined as

$\backslash \; chi\; =\; -\; \backslash \; frac\; \{1\}\; \{V\}\; \backslash \; frac\; \{D\; V\}\; \{D\; P\}$

where V is the volume of the body and P the pressure applied. It is thus expressed in Pa ^-1.

The variation of volume with the pressure being generally negative, this definition makes compressibility χ positive.

This definition is generally insufficient: under the effect of a compression, the bodies tend to warm up, and thus one defines a compressibility Isotherme for a body remaining in constant Température:

$\backslash \; chi\_T\; =\; -\; \backslash \; frac\; \{1\}\; \{V\}\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; V\}\; \{\backslash \; partial\; P\}\; \backslash \; right)\; \_T$,

the T in index of the derivative indicating that the derivative is taken at constant temperature.

It will be noticed that the coefficient of compressibility is the reverse of the module of compressibility of the medium, also called module of elasticity and generally noted K .

It is an always positive intensive variable, the volume of the body which can decrease only when the pressure is increased.

Case of a perfect gas

If it is considered that one is in a perfect gas, one can apply the equation

,

one fixes T and N ; statement is thus constant. By using the Napierian logarithm, one obtains:

;

that is to say

$\backslash \; frac\; \{dP\}\; \{P\}\; +\; \backslash \; frac\; \{FD\}\; \{V\}\; =\; 0$

then

$\backslash \; frac\; \{FD\}\; \{dP\}\; =\; -\; \backslash \; frac\; \{V\}\; \{P\}$

and finally

$\backslash \; chi\_T\; =\; -\; \backslash \; frac\; \{1\}\; \{V\}\; \backslash \; left\; (-\; \backslash \; frac\; \{V\}\; \{P\}\; \backslash \; right)$

thus

$\backslash \; chi\_T\; =\; \backslash \; frac\; \{1\}\; \{P\}$.

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