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{{LSU_HBook_header}}
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==Ideal Gas Relations==
=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 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.
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.
 
 
==Fundamental Properties of an Ideal Gas==


===Property #1===  
===Property #1===  


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


<div align="center">
<div align="center">
Line 14: Line 17:


{{User:Tohline/Math/EQ_EOSideal00}}
{{User:Tohline/Math/EQ_EOSideal00}}
[<b>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter VII.3, Eq. (18)<br />
[http://adsabs.harvard.edu/abs/1968psen.book.....C D. D. Clayton (1968)], Eq. (2-7)<br />
[<b>[[User:Tohline/Appendix/References#CH87|<font color="red">H87</font>]]</b>], &sect;1.1, p. 5
</div>
</div>


Line 23: Line 30:


<div align="center">
<div align="center">
<math>
<math>~\epsilon = \epsilon(T) \, .</math>
\epsilon = \epsilon(T)
 
</math>.
[<b>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, Eq. (1)
</div>
 
==Specific Heats==
 
Drawing from Chapter II, &sect;1 of [<b>[[User:Tohline/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 [[User:Tohline/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 {{ User:Tohline/Math/VAR_Temperature01 }} &#8212; a consequence of the kinetic theory of gasses; see, for example, Chapter X of [<b>[[User:Tohline/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>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, Eq. (10)<br />
[<b>[[User:Tohline/Appendix/References#LL75|<font color="red">LL75</font>]]</b>], Chapter IX, &sect;80, Eq. (80.10)<br />
[<b>[[User:Tohline/Appendix/References#H87|<font color="red">H87</font>]]</b>], &sect;1.2, p. 9<br />
[<b>[[User:Tohline/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 [[User:Tohline/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>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, &sect;1, Eq. (9)<br />
[http://adsabs.harvard.edu/abs/1968psen.book.....C D. D. Clayton (1968)], Eq. (2-108)<br />
[<b>[[User:Tohline/Appendix/References#LL75|<font color="red">LL75</font>]]</b>], Chapter IX, &sect;80, immediately following Eq. (80.11)<br />
[<b>[[User:Tohline/Appendix/References#H87|<font color="red">H87</font>]]</b>], &sect;1.2, p. 9<br>
[<b>[[User:Tohline/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 {{User:Tohline/Math/VAR_Density01}} rather than in terms of its number density {{User:Tohline/Math/VAR_NumberDensity01}}.  Following [http://adsabs.harvard.edu/abs/1968psen.book.....C D. D. Clayton (1968)] &#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 {{ User:Tohline/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 {{ User:Tohline/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>
</div>


<font color="#007700">where {{ User:Tohline/Math/C_AvogadroConstant }} = 1/{{ User:Tohline/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">
<div align="center">
<span id="ConservingMass:Lagrangian"><font color="#770000">'''Conservative Form'''</font></span><br />
<span id="IdealGas:FormA"><font color="#770000">'''Form A'''</font></span><br />
of the Continuity Equation,
of the Ideal Gas Equation of State,
 
{{User:Tohline/Math/EQ_EOSideal0A}}
 
[<b>[[User:Tohline/Appendix/References#LL75|<font color="red">LL75</font>]]</b>], Chapter IX, &sect;80, Eq. (80.8)<br />
[<b>[[User:Tohline/Appendix/References#KW94|<font color="red">KW94</font>]]</b>], &sect;2.2, Eq. (2.7) and &sect;13, Eq. (13.1)
</div>
 
where {{User:Tohline/Math/C_GasConstant}} &equiv; {{ User:Tohline/Math/C_BoltzmannConstant }}{{ User:Tohline/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 [[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}}.
 
<!--
<div align="center">
<table border=1 cellpadding=8 width="80%">
<tr><td>
<font color="red">
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>
-->
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>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, immediately following Eq. (9)<br />
[<b>[[User:Tohline/Appendix/References#LL75|<font color="red">LL75</font>]]</b>], Chapter IX, &sect;80, immediately following Eq. (80.9)<br />
[<b>[[User:Tohline/Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;3.4, immediately following Eq. (72)<br />
[<b>[[User:Tohline/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="ConservingMass:Lagrangian"><font color="#770000">'''Form B'''</font></span><br />
of the Ideal Gas Equation of State


{{User:Tohline/Math/EQ_EOSideal02}}
{{User:Tohline/Math/EQ_EOSideal02}}
[<b>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, Eq. (5)<br />
[<b>[[User:Tohline/Appendix/References#HK94|<font color="red">HK94</font>]]</b>], &sect;1.3.1, Eq. (1.22)<br />
[<b>[[User:Tohline/Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], &sect;6.1.1, Eq. (6.4)
</div>
</div>


 
=Related Wikipedia Discussions=
==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_law Ideal Gas Law]
* [http://en.wikipedia.org/wiki/Ideal_gas Ideal Gas]
* [http://en.wikipedia.org/wiki/Ideal_gas Ideal Gas]
* [http://en.wikipedia.org/wiki/Equation_of_state Equation of State]






{{LSU_HBook_footer}}
{{LSU_HBook_footer}}

Latest revision as of 22:40, 9 March 2019

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 unit area) <math>~P</math> given by the following

Standard Form
of the Ideal Gas Equation of State,

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

[C67], Chapter VII.3, Eq. (18)
D. D. Clayton (1968), Eq. (2-7)
[H87], §1.1, p. 5

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)

Specific Heats

Drawing from Chapter II, §1 of [C67]:  "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,"

<math>~c_\alpha</math>

<math>~\equiv</math>

<math>~\biggl( \frac{dQ}{dT} \biggr)_{\alpha ~=~ \mathrm{constant}}</math>

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 Fundamental Law of Thermodynamics, namely,

<math>~dQ</math>

<math>~=</math>

<math>~ d\epsilon + PdV \, , </math>

it is clear that when the state of a gas undergoes a change at constant (specific) volume <math>~(dV = 0)</math>,

<math>~\biggl( \frac{dQ}{dT} \biggr)_{V ~=~ \mathrm{constant}}</math>

<math>~=</math>

<math>~\frac{d\epsilon}{dT}</math>

<math>~\Rightarrow ~~~ c_V</math>

<math>~=</math>

<math>~\frac{d\epsilon}{dT} \, .</math>

Assuming <math>~c_V</math> is independent of <math>~T</math> — a consequence of the kinetic theory of gasses; see, for example, Chapter X of [C67] — and knowing that the specific internal energy is only a function of the gas temperature — see Property #2 above — we deduce that,

<math>~\epsilon</math>

<math>~=</math>

<math>~c_V T \, .</math>

[C67], Chapter II, Eq. (10)
[LL75], Chapter IX, §80, Eq. (80.10)
[H87], §1.2, p. 9
[HK94], §3.7.1, immediately following Eq. (3.80)

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,

<math>~P_\mathrm{gas}V</math>

<math>~=</math>

<math>~\biggl(\frac{\Re}{\bar\mu} \biggr) T</math>

<math>~\Rightarrow ~~~ PdV + VdP</math>

<math>~=</math>

<math>~\biggl(\frac{\Re}{\bar\mu} \biggr) dT \, .</math>

As a result, the Fundamental Law of Thermodynamics can be rewritten as,

<math>~dQ</math>

<math>~=</math>

<math>~c_\mathrm{V} dT + \biggl(\frac{\Re}{\bar\mu} \biggr) dT - VdP \, .</math>

This means that the specific heat at constant pressure is given by the relation,

<math>~c_P \equiv \biggl( \frac{dQ}{dT} \biggr)_{P ~=~ \mathrm{constant}}</math>

<math>~=</math>

<math>~c_V + \frac{\Re}{\bar\mu} \, .</math>

That is,

<math>~c_P - c_V </math>

<math>~=</math>

<math>~\frac{\Re}{\bar\mu} \, .</math>

[C67], Chapter II, §1, Eq. (9)
D. D. Clayton (1968), Eq. (2-108)
[LL75], Chapter IX, §80, immediately following Eq. (80.11)
[H87], §1.2, p. 9
[KW94], §4.1, immediately following Eq. (4.15)

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>. Following D. D. Clayton (1968) — see his p. 82 discussion of The Perfect Monatomic Nondegenerate Gas — we will "let the mean molecular weight of the perfect gas be designated by <math>~\bar{\mu}</math>. Then the density is

<math>~\rho = n_g \bar\mu m_u \, ,</math>

where <math>~m_u</math> is the mass of 1 amu" (atomic mass unit). "The number of particles per unit volume can then be expressed in terms of the density and the mean molecular weight as

<math>~n_g = \frac{\rho}{\bar\mu m_u} = \frac{\rho N_A}{\bar\mu} \, ,</math>

where <math>~N_A</math> = 1/<math>~m_u</math> is Avogadro's number …" Substitution into the above-defined Standard Form of the Ideal Gas Equation of State gives, what we will refer 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>

[LL75], Chapter IX, §80, Eq. (80.8)
[KW94], §2.2, Eq. (2.7) and §13, Eq. (13.1)

where <math>~\Re</math><math>~k</math><math>~N_A</math> is generally referred to in the astrophysics literature as the gas constant. 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>.

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,

<math>~\frac{\Re}{\bar\mu} \rho T</math>

<math>~=</math>

<math>~ (c_P - c_V)\rho \biggl(\frac{\epsilon}{c_V}\biggr) = (\gamma_g - 1)\rho\epsilon \, , </math>

where we have — as have many before us — introduced a key physical parameter,

<math>~\gamma_g</math>

<math>~\equiv</math>

<math>~\frac{c_P}{c_V} \, ,</math>

[C67], Chapter II, immediately following Eq. (9)
[LL75], Chapter IX, §80, immediately following Eq. (80.9)
[T78], §3.4, immediately following Eq. (72)
[HK94], §3.7.1, Eq. (3.86)

to quantify the ratio of specific heats. This leads to what we will refer to as,

Form B
of the Ideal Gas Equation of State

<math>~P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math>

[C67], Chapter II, Eq. (5)
[HK94], §1.3.1, Eq. (1.22)
[BLRY07], §6.1.1, Eq. (6.4)

Related Wikipedia Discussions


Whitworth's (1981) Isothermal Free-Energy Surface

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