Difference between revisions of "User:Tohline/SR/EOS"
(Draw on the Wikipedia discussion of the Ideal_gas_law) |
(extend discussion to "Form A") |
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<math>P = \biggl( \frac{N}{V} \biggr) kT</math> | <math>P = \biggl( \frac{N}{V} \biggr) kT</math> | ||
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where {{User:Tohline/Math/VAR_Pressure01}}, {{User:Tohline/Math/VAR_Temperature01}} and {{User:Tohline/Math/C_BoltzmannConstant}} are as we have defined them in our accompanying [http://www.vistrails.org/index.php/User:Tohline/Appendix/Variables_templates variables appendix], and the ratio <math>(N/V)</math> specifies the number density of free particles that make up the gas. The number density of free particles also can be written as a ratio of the mass density, {{User:Tohline/Math/VAR_Density01}}, to the average mass per particle, <math>\bar{m}</math>, that is, the ideal gas equation of state may be written in the form, | where {{User:Tohline/Math/VAR_Pressure01}}, {{User:Tohline/Math/VAR_Temperature01}} and {{User:Tohline/Math/C_BoltzmannConstant}} are as we have defined them in our accompanying [http://www.vistrails.org/index.php/User:Tohline/Appendix/Variables_templates variables appendix], and the ratio <math>(N/V)</math> specifies the number density of free particles that make up the gas. The number density of free particles also can be written as a ratio of the mass density, {{User:Tohline/Math/VAR_Density01}}, to the average mass per free particle, <math>\bar{m}</math>, that is, the ideal gas equation of state may be written in the form, | ||
<div align="center"> | <div align="center"> | ||
<math>P = \biggl( \frac{\rho}{\bar{m}} \biggr) kT</math> | <math>P = \biggl( \frac{\rho}{\bar{m}} \biggr) kT</math> . | ||
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It is customary in astrophysical discussions to write the average mass per free particle as {{User:Tohline/Math/MP_MeanMolecularWeight}}<math>m_u</math>, that is, as a product of the atomic mass unit, <math>m_u</math> = 1/{{User:Tohline/Math/C_AvogadroConstant}}, and a dimensionless coefficient of order unity referred to as the the mean molecular weight, {{User:Tohline/Math/MP_MeanMolecularWeight}}. This leads to what we will refer to as, | |||
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<font color="red">Form A</font><br /> | |||
of the ideal gas equation of state<br /> | |||
{{User:Tohline/Math/EQ_EOSideal0A}} . | |||
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Revision as of 06:48, 24 January 2010
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Equations of State
Ideal Gas
In Wikipedia, the ideal gas equation of state is referred to as the ideal gas law. The Wikipedia discussion points out that, as derived from first principles in statistical mechanics, this "gas law" can most naturally be written in the form,
<math>P = \biggl( \frac{N}{V} \biggr) kT</math>
where <math>~P</math>, <math>~T</math> and <math>~k</math> are as we have defined them in our accompanying variables appendix, and the ratio <math>(N/V)</math> specifies the number density of free particles that make up the gas. The number density of free particles also can be written as a ratio of the mass density, <math>~\rho</math>, to the average mass per free particle, <math>\bar{m}</math>, that is, the ideal gas equation of state may be written in the form,
<math>P = \biggl( \frac{\rho}{\bar{m}} \biggr) kT</math> .
It is customary in astrophysical discussions to write the average mass per free particle as <math>~\bar{\mu}</math><math>m_u</math>, that is, as a product of the atomic mass unit, <math>m_u</math> = 1/<math>~N_A</math>, and a dimensionless coefficient of order unity referred to as the the mean molecular weight, <math>~\bar{\mu}</math>. This leads to what we will refer to as,
Form A
of the ideal gas equation of state
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<math>~P_\mathrm{gas} = \frac{\Re}{\bar{\mu}} \rho T</math> |
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