Editing SR/IdealGas (section)

==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 {{ Template:Math/VAR_Density01 }} rather than in terms of its number density {{ Template:Math/VAR_NumberDensity01 }}.  Following [<b>[[Appendix/References#Clayton68|<font color="red"> Clayton68 </font>]]</b>] &#8212; see his p. 82 discussion of ''The Perfect Monatomic Nondegenerate Gas'' &#8212; we will "<font color="#007700">let the mean molecular weight of the perfect gas be designated by {{ Template:Math/MP_MeanMolecularWeight }}.  Then the density is</font>

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<math>\rho = n_g \bar\mu m_u \, ,</math>
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<font color="#007700">where {{ Template:Math/C_AtomicMassUnit }} is the mass of 1 amu</font>" ([https://en.wikipedia.org/wiki/Unified_atomic_mass_unit atomic mass unit]).  "<font color="#007700">The number of particles per unit volume can then be expressed in terms of the density and the mean molecular weight as</font>

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<math>n_g = \frac{\rho}{\bar\mu m_u} = \frac{\rho N_A}{\bar\mu} \, ,</math>
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<font color="#007700">where {{ Template:Math/C_AvogadroConstant }} = 1/{{ Template:Math/C_AtomicMassUnit }} is Avogadro's number &hellip;</font>"  Substitution into the [[#Fundamental_Properties_of_an_Ideal_Gas|above-defined ''Standard Form of the Ideal Gas Equation of State'']] gives, what we will refer to as,

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<span id="IdealGas:FormA"><font color="#770000">'''Form A'''</font></span><br />
of the Ideal Gas Equation of State,

{{ Template:Math/EQ_EOSideal0A }}

[<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>], Chapter IX, &sect;80, Eq. (80.8)<br />
[<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>], &sect;2.2, Eq. (2.7) and &sect;13, Eq. (13.1)
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where {{ Template:Math/C_GasConstant}} &equiv; {{ Template:Math/C_BoltzmannConstant }}{{ Template:Math/C_AvogadroConstant }} is generally referred to in the astrophysics literature as the gas constant.  The definition of the gas constant can be found in the [[Appendix/VariablesTemplates|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 [[Appendix/References#C67|[<b><font color="red">C67</font></b>]]] or &sect;13.1 (p. 102) of [[Appendix/References#KW94|[<b><font color="red">KW94</font></b>]]] for particularly clear explanations of how to calculate {{ Template:Math/MP_MeanMolecularWeight }}.

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Exercise:
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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 [[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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Employing a couple of the expressions from the above discussion of specific heats, the right-hand side of ''Form A of the Ideal Gas Equation of State'' can be rewritten as,
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  <td align="right">
<math>\frac{\Re}{\bar\mu} \rho T</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(c_P - c_V)\rho \biggl(\frac{\epsilon}{c_V}\biggr)
=
(\gamma_g - 1)\rho\epsilon \, ,
</math>
  </td>
</tr>
</table>
<span id="gamma_g">where we have &#8212; as have many before us &#8212; introduced a key physical parameter,</span>
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  <td align="right">
<math>\gamma_g</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{c_P}{c_V} \, ,</math>
  </td>
</tr>
</table>

[<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, immediately following Eq. (9)<br />
[<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>], Chapter IX, &sect;80, immediately following Eq. (80.9)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, immediately following Eq. (72)<br />
[<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;3.7.1, Eq. (3.86)
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to quantify the ratio of specific heats.  This leads to what we will refer to as,
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<span id="IdealGasFormB"><font color="#770000">'''Form B'''</font></span><br />
of the Ideal Gas Equation of State

{{ Template:Math/EQ_EOSideal02 }}

[<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, Eq. (5)<br />
[<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;1.3.1, Eq. (1.22)<br />
[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], &sect;6.1.1, Eq. (6.4)
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