Difference between revisions of "User:Tohline/SR/IdealGas"

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(→‎Consequential Ideal Gas Relations: Insert corrected label for "Form B" of the ideal gas equation of state)
(→‎Ideal Gas Equation of State: Clean up text and references in this chapter)
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=Ideal Gas Equation of State=
=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'' [{{User:Tohline/Math/REF_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 [http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates key equations appendix] of this H_Book.<br />
Much of the following overview of ideal gas relations is drawn from Chapter II of Chandrasekhar's classic text on ''Stellar Structure'' [[User:Tohline/Appendix/References#C67|[<b><font color="red">C67</font></b>]]], 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 [[User:Tohline/Appendix/Equation_templates#Equations_of_State|key equations appendix]] of this H_Book.




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<math>
<math>~\epsilon = \epsilon(T) \, .</math>
\epsilon = \epsilon(T)
</math>.
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{{User:Tohline/Math/EQ_EOSideal0A}}
{{User:Tohline/Math/EQ_EOSideal0A}}
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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 [http://www.vistrails.org/index.php/User:Tohline/PGE 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 &sect;VII.3 (p. 254) of {{User:Tohline/Math/REF_C67}} or &sect;13.1 (p. 102) of {{User:Tohline/Math/REF_KW94}} 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 &sect;VII.3 (p. 254) of [[User:Tohline/Appendix/References#C67|[<b><font color="red">C67</font></b>]]] or &sect;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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Exercise:
Exercise:
</font>
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If {{User:Tohline/Math/C_GasConstant}} is defined as the product of the Boltzmann constant {{User:Tohline/Math/C_BoltzmannConstant}} and the Avogadro constant {{User:Tohline/Math/C_AvogadroConstant}}, as stated in the [http://www.vistrails.org/index.php/User:Tohline/PGE 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 1/{{User:Tohline/Math/MP_MeanMolecularWeight}} gives the number of free particles per atomic mass unit, {{User:Tohline/Math/C_AtomicMassUnit}}.   
If {{User:Tohline/Math/C_GasConstant}} is defined as the product of the Boltzmann constant {{User:Tohline/Math/C_BoltzmannConstant}} and the Avogadro constant {{User:Tohline/Math/C_AvogadroConstant}}, as stated in the [[User:Tohline/Appendix/Variables_templates|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, {{User:Tohline/Math/C_AtomicMassUnit}}.   
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{{LSU_WorkInProgress}}
Still need to explain:


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Revision as of 23:31, 12 July 2015

Whitworth's (1981) Isothermal Free-Energy Surface
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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

Standard Form
of the Ideal Gas Equation of State,

<math>~P = n_g k T</math>

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>


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,

LSU Key.png

<math>~P_\mathrm{gas} = \frac{\Re}{\bar{\mu}} \rho T</math>

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>.


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>.



Work-in-progress.png

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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Whitworth's (1981) Isothermal Free-Energy Surface

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