Compressibility: Difference between revisions

Revision as of 18:05, 13 February 2008

The bulk modulus B gives the change in volume of a solid substance as the pressure on it is changed,

B = - V \frac{\partial p}{\partial V}

The compressibility K or $κ$ , is given by

κ = \frac{1}{B}

The isothermal compressibility, $κ_{T}$ is given by

κ_{T} = - \frac{1}{V} {\frac{\partial V}{\partial p} |}_{T} = \frac{1}{ρ} {\frac{\partial ρ}{\partial p} |}_{T}

(Note: in Hansen and McDonald the isothermal compressibility is written as $χ_{T}$ ). where $ρ$ is the particle number density given by

ρ = \frac{N}{V}

where $N$ is the total number of particles in the system, i.e.

N = \int_{V} ρ (r, t) d r

The adiabatic compressibility, $κ_{S}$ is given by

κ_{S} = - \frac{1}{V} {\frac{\partial V}{\partial p} |}_{S}

where $S$ is the entropy.

@@ Line 6: / Line 6: @@
 :<math>\kappa =\frac{1}{B}</math>
+==Isothermal compressibility==
 The  '''isothermal compressibility''',  <math>\kappa_T</math> is given by
@@ Line 19: / Line 19: @@
 :<math>N = \int_V \rho({\mathbf r},t)~{\rm d}{\mathbf r}</math>
+==Adiabatic compressibility==
+The  '''adiabatic compressibility''',  <math>\kappa_S</math> is given by
+:<math>\kappa_S =-\frac{1}{V} \left.\frac{\partial V}{\partial p}\right\vert_{S}</math>
+where <math>S</math> is the [[entropy]].
 ==See also==
 The [[compressibility equation]] in [[statistical mechanics]].
 [[category:classical thermodynamics]]