Compressibility: Difference between revisions
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The ''' | The '''bulk modulus''' ''B'' 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> | |||
The '''compressibility''' ''K'' or <math>\kappa</math>, is given by | |||
The ''compressibility'' | |||
:<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]]. | ||
[[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,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = -V \frac{\partial p}{\partial V}}
The compressibility K or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa} , is given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa =\frac{1}{B}}
Isothermal compressibility[edit]
The isothermal compressibility, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa_T} is given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa_T =-\frac{1}{V} \left.\frac{\partial V}{\partial p}\right\vert_{T} = \frac{1}{\rho} \left.\frac{\partial \rho}{\partial p}\right\vert_{T}}
(Note: in Hansen and McDonald the isothermal compressibility is written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_T} ). where is the temperature, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} is the particle number density given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho = \frac{N}{V}}
where is the total number of particles in the system, i.e.
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = \int_V \rho({\mathbf r},t)~{\rm d}{\mathbf r}}
Adiabatic compressibility[edit]
The adiabatic compressibility, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa_S} is given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa_S =-\frac{1}{V} \left.\frac{\partial V}{\partial p}\right\vert_{S}}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is the entropy.