Editing SR/IdealGas

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=Ideal Gas Equation of State=

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<font size="-1">[[H_BookTiledMenu#Context|<b>Ideal Gas</b>]]</font>
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Much of the following overview of ideal gas relations is drawn from Chapter II of Chandrasekhar's classic text on ''Stellar Structure'' [[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 [[Appendix/EquationTemplates#Equations_of_State|key equations appendix]] of this H_Book.

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==Fundamental Properties of an Ideal Gas==

===Property #1=== 

An ideal gas containing {{ Template:Math/VAR_NumberDensity01 }} free particles per unit volume will exert on its surroundings an isotropic pressure (''i.e.'', a force per unit area) {{ Template:Math/VAR_Pressure01 }} given by the following

<div align="center">
<span id="IdealGas:StandardForm"><font color="#770000">'''Standard Form'''</font></span><br />
of the Ideal Gas Equation of State,

{{ Template:Math/EQ_EOSideal00 }}

[<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter VII.3, Eq. (18)<br />
[<b>[[Appendix/References#Clayton68 |<font color="red">Clayton68</font>]]</b>], Eq. (2-7)<br />
[<b>[[Appendix/References#H87|<font color="red">H87</font>]]</b>], &sect;1.1, p. 5
</div>

if the gas is in thermal equilibrium at a temperature {{ Template:Math/VAR_Temperature01 }}.

===Property #2=== 

The internal energy per unit mass {{ Template:Math/VAR_SpecificInternalEnergy01 }} of an ideal gas is a function ''only'' of the gas temperature {{ Template:Math/VAR_Temperature01 }}, that is,

<div align="center">
<math>~\epsilon = \epsilon(T) \, .</math>

[<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, Eq. (1)
</div>

==Specific Heats==

Drawing from Chapter II, &sect;1 of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>]:&nbsp;  "<font color="#007700">Let <math>\alpha</math> be a function of the physical variables.  Then the specific heat, <math>c_\alpha</math>, at constant <math>\alpha</math> is defined by the expression,</font>"
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>c_\alpha</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\biggl( \frac{dQ}{dT} \biggr)_{\alpha ~=~ \mathrm{constant}}</math>
  </td>
</tr>
</table>
The specific heat at constant pressure <math>c_P</math> and the specific heat at constant (specific) volume <math>c_V</math> prove to be particularly interesting parameters because they identify experimentally measurable properties of a gas.  

From the [[PGE/FirstLawOfThermodynamics#FundamentalLaw|Fundamental Law of Thermodynamics]], namely,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>dQ</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
d\epsilon + PdV \, ,
</math>
  </td>
</tr>
</table>
it is clear that when the state of a gas undergoes a change at constant (specific) volume <math>(dV = 0)</math>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{dQ}{dT} \biggr)_{V ~=~ \mathrm{constant}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{d\epsilon}{dT}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ c_V</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{d\epsilon}{dT} \, .</math>
  </td>
</tr>
</table>
Assuming <math>c_V</math> is independent of {{ Template:Math/VAR_Temperature01 }} &#8212; a consequence of the kinetic theory of gasses; see, for example, Chapter X of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] &#8212; and knowing that the specific internal energy is only a function of the gas temperature &#8212; see ''[[#Property_.232|Property #2]]'' above &#8212; we deduce that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\epsilon</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>c_V T \, .</math>
  </td>
</tr>
</table>

[<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, Eq. (10)<br />
[<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>], Chapter IX, &sect;80, Eq. (80.10)<br />
[<b>[[Appendix/References#H87|<font color="red">H87</font>]]</b>], &sect;1.2, p. 9<br />
[<b>[[Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;3.7.1, immediately following Eq. (3.80)
</div>

Also, from ''Form A of the Ideal Gas Equation of State'' (see below) and the recognition that <math>\rho = 1/V</math>, we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>P_\mathrm{gas}V</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{\Re}{\bar\mu} \biggr) T</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ PdV + VdP</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl(\frac{\Re}{\bar\mu} \biggr) dT \, .</math>
  </td>
</tr>
</table>
As a result, the [[PGE/FirstLawOfThermodynamics#FundamentalLaw|Fundamental Law of Thermodynamics]] can be rewritten as,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>dQ</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>c_\mathrm{V} dT + \biggl(\frac{\Re}{\bar\mu} \biggr) dT  - VdP \, .</math>
  </td>
</tr>
</table>
This means that the specific heat at constant pressure is given by the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>c_P \equiv \biggl( \frac{dQ}{dT} \biggr)_{P ~=~ \mathrm{constant}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>c_V + \frac{\Re}{\bar\mu} \, .</math>
  </td>
</tr>
</table>
</div>

That is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>c_P - c_V </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\Re}{\bar\mu} \, .</math>
  </td>
</tr>
</table>

[<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, &sect;1, Eq. (9)<br />
[<b>[[Appendix/References#Clayton68 |<font color="red">Clayton68</font>]]</b>], Eq. (2-108)<br />
[<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>], Chapter IX, &sect;80, immediately following Eq. (80.11)<br />
[<b>[[Appendix/References#H87|<font color="red">H87</font>]]</b>], &sect;1.2, p. 9<br>
[<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>], &sect;4.1, immediately following Eq. (4.15) 
</div>

==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>

<div align="center">
<math>\rho = n_g \bar\mu m_u \, ,</math>
</div>

<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>

<div align="center">
<math>n_g = \frac{\rho}{\bar\mu m_u} = \frac{\rho N_A}{\bar\mu} \, ,</math>
</div>

<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,

<div align="center">
<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)
</div>

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:
</font>
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}}.  
</td></tr>
</table>
</div>
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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,
<table border="0" cellpadding="5" align="center">

<tr>
  <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>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <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)
</div>

to quantify the ratio of specific heats.  This leads to what we will refer to as,
<div align="center">
<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)
</div>

=Related Wikipedia Discussions=
* [http://en.wikipedia.org/wiki/Equation_of_state#Classical_ideal_gas_law Equation of State: Classical ideal gas law]
* [http://en.wikipedia.org/wiki/Ideal_gas_law Ideal Gas Law]
* [http://en.wikipedia.org/wiki/Ideal_gas Ideal Gas]


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