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

From VistrailsWiki
Jump to navigation Jump to search
 
(26 intermediate revisions by the same user not shown)
Line 1: Line 1:
{{LSU_HBook_header}}
{{LSU_HBook_header}}


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 />
=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/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.




Line 8: Line 10:
===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 15: 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 24: 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>
</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==
==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}}.  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  
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>
 
<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">
Line 39: Line 227:


{{User:Tohline/Math/EQ_EOSideal0A}}
{{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>
</div>
where {{User:Tohline/Math/C_GasConstant}} is the gas constant and {{User:Tohline/Math/MP_MeanMolecularWeight}} 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}}.
-->
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">
<div align="center">
<span id="ConservingMass:Lagrangian"><font color="#770000">'''Conservative Form'''</font></span><br />
<table border="0" cellpadding="5" align="center">
of the Continuity Equation,
 
<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/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]

Latest revision as of 22:40, 9 March 2019

Whitworth's (1981) Isothermal Free-Energy Surface
|   Tiled Menu   |   Tables of Content   |  Banner Video   |  Tohline Home Page   |

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

© 2014 - 2021 by Joel E. Tohline
|   H_Book Home   |   YouTube   |
Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS |
Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation