Difference between revisions of "User:Tohline/SR/IdealGas"
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[<b>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter VII.3, Eq. (18)<br /> | [<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>], §1.1, p. 5 | [<b>[[User:Tohline/Appendix/References#CH87|<font color="red">H87</font>]]</b>], §1.1, p. 5 | ||
</div> | </div> | ||
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</tr> | </tr> | ||
</table> | </table> | ||
The specific heat at constant pressure <math>~c_P</math> and the specific heat at constant volume <math>~c_V</math> prove to be particularly interesting parameters because they identify experimentally measurable properties of a gas. | 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, | From the [[User:Tohline/PGE/FirstLawOfThermodynamics#FundamentalLaw|Fundamental Law of Thermodynamics]], namely, | ||
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</tr> | </tr> | ||
</table> | </table> | ||
it is clear that when the state of a gas undergoes a change at constant volume <math>~(dV = 0)</math>, | 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"> | <table border="0" cellpadding="5" align="center"> | ||
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</table> | </table> | ||
Assuming <math>~c_V</math> is independent of {{ User:Tohline/Math/VAR_Temperature01 }} — 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>] — and knowing that the specific internal energy is only a function of the gas temperature — see ''[[#Property_.232|Property #2]]'' above — we deduce that, | Assuming <math>~c_V</math> is independent of {{ User:Tohline/Math/VAR_Temperature01 }} — 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>] — and knowing that the specific internal energy is only a function of the gas temperature — see ''[[#Property_.232|Property #2]]'' above — we deduce that, | ||
<div align="center"> | <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# | [<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, §80, Eq. (80.10)<br /> | |||
[<b>[[User:Tohline/Appendix/References#H87|<font color="red">H87</font>]]</b>], §1.2, p. 9<br /> | |||
[<b>[[User:Tohline/Appendix/References#HK94|<font color="red">HK94</font>]]</b>], §3.7.1, immediately following Eq. (3.80) | |||
</div> | </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></tr> | <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> | </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, | |||
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<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
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<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~\frac{ | <math>~c_P \equiv \biggl( \frac{dQ}{dT} \biggr)_{P ~=~ \mathrm{constant}}</math> | ||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
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</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~c_V + \frac{\Re}{\bar\mu} \, .</math> | ||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
</div> | </div> | ||
That is, | |||
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<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
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<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~ | <math>~c_P - c_V </math> | ||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
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</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~\frac{\Re}{\bar\mu} \, .</math> | ||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
[<b>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, Eq. ( | [<b>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, §1, Eq. (9)<br /> | ||
[<b>[[User:Tohline/Appendix/References#LL75|<font color="red">LL75</font>]]</b>], Chapter IX, §80, Eq. (80. | [http://adsabs.harvard.edu/abs/1968psen.book.....C D. D. Clayton (1968)], Eq. (2-108)<br /> | ||
[<b>[[User:Tohline/Appendix/References#H87|<font color="red">H87</font>]]</b>], §1.2, p. 9 | [<b>[[User:Tohline/Appendix/References#LL75|<font color="red">LL75</font>]]</b>], Chapter IX, §80, immediately following Eq. (80.11)<br /> | ||
[<b>[[User:Tohline/Appendix/References#H87|<font color="red">H87</font>]]</b>], §1.2, p. 9<br> | |||
[<b>[[User:Tohline/Appendix/References#KW94|<font color="red">KW94</font>]]</b>], §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)] — see his p. 82 discussion of ''The Perfect Monatomic Nondegenerate Gas'' — 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 …</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=" | <span id="IdealGas:FormA"><font color="#770000">'''Form A'''</font></span><br /> | ||
of the Ideal Gas Equation of State, | of the Ideal Gas Equation of State, | ||
{{User:Tohline/Math/ | {{User:Tohline/Math/EQ_EOSideal0A}} | ||
[<b>[[User:Tohline/Appendix/References#LL75|<font color="red">LL75</font>]]</b>], Chapter IX, §80, Eq. (80.8)<br /> | |||
[<b>[[User:Tohline/Appendix/References#KW94|<font color="red">KW94</font>]]</b>], §2.2, Eq. (2.7) and §13, Eq. (13.1) | |||
</div> | |||
where {{User:Tohline/Math/C_GasConstant}} ≡ {{ 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 §VII.3 (p. 254) of [[User:Tohline/Appendix/References#C67|[<b><font color="red">C67</font></b>]]] or §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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<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> | ||
--> | |||
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 — as have many before us — introduced a key physical parameter,</span> | |||
<div align="center"> | <div align="center"> | ||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
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</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~\frac{c_P}{c_V} \, | <math>~\frac{c_P}{c_V} \, ,</math> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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[<b>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>], Chapter II, immediately following Eq. (9)<br /> | [<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, §80, immediately following Eq. (80.9)<br /> | [<b>[[User:Tohline/Appendix/References#LL75|<font color="red">LL75</font>]]</b>], Chapter IX, §80, immediately following Eq. (80.9)<br /> | ||
[<b>[[User:Tohline/Appendix/References#T78|<font color="red">T78</font>]]</b>], §3.4, immediately following Eq. (72) | [<b>[[User:Tohline/Appendix/References#T78|<font color="red">T78</font>]]</b>], §3.4, immediately following Eq. (72)<br /> | ||
[<b>[[User:Tohline/Appendix/References#HK94|<font color="red">HK94</font>]]</b>], §3.7.1, Eq. (3.86) | |||
</div> | </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}} | |||
[<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>], §1.3.1, Eq. (1.22)<br /> | |||
[<b>[[User:Tohline/Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], §6.1.1, Eq. (6.4) | |||
</div> | |||
=Related Wikipedia Discussions= | =Related Wikipedia Discussions= |
Latest revision as of 22:40, 9 March 2019
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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,
<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)
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