Compressibility: Difference between revisions

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The '''compressibility''', <math>Z</math>, is given by
The '''bulk modulus''' ''B'' gives the change in volume of a solid substance as the [[pressure]] on it is changed,
:<math>Z= \frac{pV}{Nk_BT}</math>
 
The '''bulk modulus''' <math>B</math> gives the change in volume of a solid substance as the pressure on it is changed,


:<math>B = -V \frac{\partial p}{\partial V}</math>
:<math>B = -V \frac{\partial p}{\partial V}</math>


The ''compressibility'' <math>K</math> or <math>\kappa</math>, is given by
The '''compressibility''' ''K'' or <math>\kappa</math>, is given by


:<math>\kappa =\frac{1}{B}</math>
:<math>\kappa =\frac{1}{B}</math>
 
==Isothermal compressibility==
The  '''isothermal compressibility''',  <math>\kappa_T</math> is given by
The  '''isothermal compressibility''',  <math>\kappa_T</math> is given by


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(Note: in Hansen and McDonald the isothermal compressibility is written as <math>\chi_T</math>).
(Note: in Hansen and McDonald the isothermal compressibility is written as <math>\chi_T</math>).
where <math>\rho</math> is the ''particle number density'' given by
where <math>T</math> is the [[temperature]], <math>\rho</math> is the ''particle number density'' given by


:<math>\rho  = \frac{N}{V}</math>
:<math>\rho  = \frac{N}{V}</math>
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where <math>N</math> is the total number of particles in the system, i.e.
where <math>N</math> is the total number of particles in the system, i.e.


:<math>N = \int_V \rho(r,t)~{\rm d}r</math>
:<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==
==See also==
The [[compressibility equation]] in [[statistical mechanics]].
The [[compressibility equation]] in [[statistical mechanics]].
==Compressibility of an Ideal Gas==
From the [[Equation of State: Ideal Gas | ideal gas law]]  we see that


:<math>Z= \frac{pV}{Nk_BT}=1</math>
[[category:classical thermodynamics]]
[[category:classical thermodynamics]]

Latest revision as of 17:07, 13 February 2008

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

The compressibility K or , is given by

Isothermal compressibility[edit]

The isothermal compressibility, is given by

(Note: in Hansen and McDonald the isothermal compressibility is written as ). where is the temperature, is the particle number density given by

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

Adiabatic compressibility[edit]

The adiabatic compressibility, is given by

where is the entropy.

See also[edit]

The compressibility equation in statistical mechanics.