Compressibility: Difference between revisions
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The '''compressibility''', <math>Z</math>, is given by | |||
:<math>Z= \frac{pV}{Nk_BT}</math> | :<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, | 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> | ||
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:<math>\kappa =\frac{1}{B}</math> | :<math>\kappa =\frac{1}{B}</math> | ||
The ''isothermal compressibility'', <math>\kappa_T</math> is given by | The '''isothermal compressibility''', <math>\kappa_T</math> is given by | ||
:<math>\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}</math> | :<math>\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}</math> | ||
Revision as of 14:29, 22 May 2007
The 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 Z} , 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 Z= \frac{pV}{Nk_BT}}
The bulk modulus 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} 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 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 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}}
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 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 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} 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(r,t)~{\rm d}r}
See also
The compressibility equation in statistical mechanics.
Compressibility of an Ideal Gas
From the ideal gas law we see that
- 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 Z= \frac{pV}{Nk_BT}=1}