Difference between revisions of "User:Tohline/SR/IdealGas"
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[<b>[[User:Tohline/Appendix/References#CH87|<font color="red">H87</font>]]</b>], §1.1, p. 5 | |||
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<math>~\epsilon = \epsilon(T) \, .</math> | <math>~\epsilon = \epsilon(T) \, .</math> | ||
[<b>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, Eq. (1) | |||
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{{User:Tohline/Math/EQ_EOSideal0A}} | {{User:Tohline/Math/EQ_EOSideal0A}} | ||
[<b>[[User:Tohline/Appendix/References#KW94|<font color="red">KW94</font>]]</b>], §2.2, Eq. (2.7) and §13, Eq. (13.1) | |||
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where {{User:Tohline/Math/C_GasConstant}} is the gas constant and {{User:Tohline/Math/MP_MeanMolecularWeight}} <math>\equiv</math> {{User:Tohline/Math/VAR_Density01}}/({{User:Tohline/Math/C_AtomicMassUnit}}{{User:Tohline/Math/VAR_NumberDensity01}}) is the mean molecular weight of the gas. The definition of the gas constant can be found in the [[User:Tohline/Appendix/Variables_templates|Variables Appendix]] of this H_Book; its numerical value can be obtained by simply scrolling the computer mouse over its symbol in the text of this paragraph. See §VII.3 (p. 254) of [[User:Tohline/Appendix/References#C67|[<b><font color="red">C67</font></b>]]] or §13.1 (p. 102) of [[User:Tohline/Appendix/References#KW94|[<b><font color="red">KW94</font></b>]]] for particularly clear explanations of how to calculate {{User:Tohline/Math/MP_MeanMolecularWeight}}. | where {{User:Tohline/Math/C_GasConstant}} is the gas constant and {{User:Tohline/Math/MP_MeanMolecularWeight}} <math>\equiv</math> {{User:Tohline/Math/VAR_Density01}}/({{User:Tohline/Math/C_AtomicMassUnit}}{{User:Tohline/Math/VAR_NumberDensity01}}) is the mean molecular weight of the gas. The definition of the gas constant can be found in the [[User:Tohline/Appendix/Variables_templates|Variables Appendix]] of this H_Book; its numerical value can be obtained by simply scrolling the computer mouse over its symbol in the text of this paragraph. See §VII.3 (p. 254) of [[User:Tohline/Appendix/References#C67|[<b><font color="red">C67</font></b>]]] or §13.1 (p. 102) of [[User:Tohline/Appendix/References#KW94|[<b><font color="red">KW94</font></b>]]] for particularly clear explanations of how to calculate {{User:Tohline/Math/MP_MeanMolecularWeight}}. | ||
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Ideal Gas Equation of State
Much of the following overview of ideal gas relations is drawn from Chapter II of Chandrasekhar's classic text on Stellar Structure [C67], which was originally published in 1939. A guide to parallel print media discussions of this topic is provided alongside the ideal gas equation of state in the key equations appendix of this H_Book.
Fundamental Properties of an Ideal Gas
Property #1
An ideal gas containing <math>~n_g</math> free particles per unit volume will exert on its surroundings an isotropic pressure (i.e., a force per unity area) <math>~P</math> given by the following
if the gas is in thermal equilibrium at a temperature <math>~T</math>.
Property #2
The internal energy per unit mass <math>~\epsilon</math> of an ideal gas is a function only of the gas temperature <math>~T</math>, that is,
<math>~\epsilon = \epsilon(T) \, .</math>
[C67], Chapter II, Eq. (1)
Consequential Ideal Gas Relations
Throughout most of this H_Book, we will define the relative degree of compression of a gas in terms of its mass density <math>~\rho</math> rather than in terms of its number density <math>~n_g</math>. Hence, in place of the above "standard form" of the ideal gas equation of state, we more commonly will adopt the following expression, which will be referred 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> |
[KW94], §2.2, Eq. (2.7) and §13, Eq. (13.1)
where <math>~\Re</math> is the gas constant and <math>~\bar{\mu}</math> <math>\equiv</math> <math>~\rho</math>/(<math>~m_u</math><math>~n_g</math>) is the mean molecular weight of the gas. The definition of the gas constant can be found in the Variables Appendix of this H_Book; its numerical value can be obtained by simply scrolling the computer mouse over its symbol in the text of this paragraph. See §VII.3 (p. 254) of [C67] or §13.1 (p. 102) of [KW94] for particularly clear explanations of how to calculate <math>~\bar{\mu}</math>.
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Exercise: If <math>~\Re</math> is defined as the product of the Boltzmann constant <math>~k</math> and the Avogadro constant <math>~N_A</math>, as stated in the Variables Appendix of this H_Book, show that "Form A" and the "Standard Form" of the ideal gas equation of state provide equivalent expressions only if <math>~(\bar\mu)^{-1}</math> gives the number of free particles per atomic mass unit, <math>~m_u</math>. |
Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
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Still need to explain:
Form B
of the Ideal Gas Equation of State,
<math>~P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math>
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