SR/IdealGas - Revision history

Jet53man: /* Consequential Ideal Gas Relations */

2021-10-08T16:35:51Z

Consequential Ideal Gas Relations

← Older revision		Revision as of 12:35, 8 October 2021
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	==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 {{ Template:Math/VAR_Density01 }} rather than in terms of its number density {{ Template: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 {{ Template:Math/MP_MeanMolecularWeight }}. Then the density is</font>		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>] — 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 {{ Template:Math/MP_MeanMolecularWeight }}. Then the density is</font>

	<div align="center">		<div align="center">

Jet53man: /* Specific Heats */

2021-10-08T16:33:42Z

Specific Heats

← Older revision		Revision as of 12:33, 8 October 2021
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	[<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>], Chapter II, §1, Eq. (9)<br />		[<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>], Chapter II, §1, Eq. (9)<br />
	[~~http:~~//~~adsabs.harvard.edu~~/~~abs/1968psen.book.....C D. D. Clayton (1968)~~], Eq. (2-108)<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, §80, immediately following Eq. (80.11)<br />		[<b>[[Appendix/References#LL75\|<font color="red">LL75</font>]]</b>], Chapter IX, §80, immediately following Eq. (80.11)<br />
	[<b>[[Appendix/References#H87\|<font color="red">H87</font>]]</b>], §1.2, p. 9<br>		[<b>[[Appendix/References#H87\|<font color="red">H87</font>]]</b>], §1.2, p. 9<br>

Jet53man: /* Property #1 */

2021-10-08T16:32:00Z

Property #1

← Older revision		Revision as of 12:32, 8 October 2021
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	[<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>], Chapter VII.3, Eq. (18)<br />		[<b>[[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>[[Appendix/References#Clayton68 \|<font color="red">Clayton68</font>]]</b>], Eq. (2-7)<br />
	[<b>[[Appendix/References#H87\|<font color="red">H87</font>]]</b>], §1.1, p. 5		[<b>[[Appendix/References#H87\|<font color="red">H87</font>]]</b>], §1.1, p. 5
	</div>		</div>

Jet53man: /* Ideal Gas Equation of State */

2021-07-28T03:38:13Z

Ideal Gas Equation of State

← Older revision		Revision as of 23:38, 27 July 2021
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	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
	=Ideal Gas Equation of State=		=Ideal Gas Equation of State=

			{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:white;" \|
			<font size="-1">[[H_BookTiledMenu#Context\|<b>Ideal Gas</b>]]</font>
			\|}

	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.		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.

			<br />
			<br />
			<br />
	==Fundamental Properties of an Ideal Gas==		==Fundamental Properties of an Ideal Gas==

Jet53man: /* Consequential Ideal Gas Relations */

2021-07-11T21:36:33Z

Consequential Ideal Gas Relations

← Older revision		Revision as of 17:36, 11 July 2021
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{\Re}{\bar\mu} \rho T</math>		<math>\frac{\Re}{\bar\mu} \rho T</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	(c_P - c_V)\rho \biggl(\frac{\epsilon}{c_V}\biggr)		(c_P - c_V)\rho \biggl(\frac{\epsilon}{c_V}\biggr)
	=		=
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\gamma_g</math>		<math>\gamma_g</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~\equiv</math>		<math>\equiv</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{c_P}{c_V} \, ,</math>		<math>\frac{c_P}{c_V} \, ,</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>

	[<b>[[~~User:Tohline/~~Appendix/References#C67\|<font color="red">C67</font>]]</b>], Chapter II, immediately following Eq. (9)<br />		[<b>[[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>[[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)<br />		[<b>[[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)		[<b>[[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,		to quantify the ratio of specific heats. This leads to what we will refer to as,
	<div align="center">		<div align="center">
	<span id="~~ConservingMass:Lagrangian~~"><font color="#770000">'''Form B'''</font></span><br />		<span id="IdealGasFormB"><font color="#770000">'''Form B'''</font></span><br />
	of the Ideal Gas Equation of State		of the Ideal Gas Equation of State

	{{~~User~~:~~Tohline/~~Math/EQ_EOSideal02}}		{{ Template:Math/EQ_EOSideal02 }}

	[<b>[[~~User:Tohline/~~Appendix/References#C67\|<font color="red">C67</font>]]</b>], Chapter II, Eq. (5)<br />		[<b>[[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>[[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)		[<b>[[Appendix/References#BLRY07\|<font color="red">BLRY07</font>]]</b>], §6.1.1, Eq. (6.4)
	</div>		</div>

Jet53man: /* Consequential Ideal Gas Relations */

2021-07-11T21:31:09Z

Consequential Ideal Gas Relations

← Older revision		Revision as of 17:31, 11 July 2021
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	==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}}. 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>		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 [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 {{ Template:Math/MP_MeanMolecularWeight }}. Then the density is</font>

	<div align="center">		<div align="center">
	<math>~\rho = n_g \bar\mu m_u \, ,</math>		<math>\rho = n_g \bar\mu m_u \, ,</math>
	</div>		</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>		<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">		<div align="center">
	<math>~n_g = \frac{\rho}{\bar\mu m_u} = \frac{\rho N_A}{\bar\mu} \, ,</math>		<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,		<font color="#007700">where {{ Template:Math/C_AvogadroConstant }} = 1/{{ Template: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">
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	of the Ideal Gas Equation of State,		of the Ideal Gas Equation of State,

	{{~~User~~:~~Tohline/~~Math/EQ_EOSideal0A}}		{{ Template:Math/EQ_EOSideal0A }}

	[<b>[[~~User:Tohline/~~Appendix/References#LL75\|<font color="red">LL75</font>]]</b>], Chapter IX, §80, Eq. (80.8)<br />		[<b>[[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)		[<b>[[Appendix/References#KW94\|<font color="red">KW94</font>]]</b>], §2.2, Eq. (2.7) and §13, Eq. (13.1)
	</div>		</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 §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}}.		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 §VII.3 (p. 254) of [[Appendix/References#C67\|[<b><font color="red">C67</font></b>]]] or §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 }}.

	<!--		<!--

Jet53man: /* Specific Heats */

2021-07-11T21:25:13Z

Specific Heats

← Older revision		Revision as of 17:25, 11 July 2021
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	</tr>		</tr>
	</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 {{ Template:Math/VAR_Temperature01 }} — a consequence of the kinetic theory of gasses; see, for example, Chapter X of [<b>[[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">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\epsilon</math>		<math>\epsilon</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~c_V T \, .</math>		<math>c_V T \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>

	[<b>[[~~User:Tohline/~~Appendix/References#C67\|<font color="red">C67</font>]]</b>], Chapter II, Eq. (10)<br />		[<b>[[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>[[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>[[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)		[<b>[[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,		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">		<table border="0" cellpadding="5" align="center">

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

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~dQ</math>		<math>dQ</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~c_\mathrm{V} dT + \biggl(\frac{\Re}{\bar\mu} \biggr) dT - VdP \, .</math>		<math>c_\mathrm{V} dT + \biggl(\frac{\Re}{\bar\mu} \biggr) dT - VdP \, .</math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~c_P \equiv \biggl( \frac{dQ}{dT} \biggr)_{P ~=~ \mathrm{constant}}</math>		<math>c_P \equiv \biggl( \frac{dQ}{dT} \biggr)_{P ~=~ \mathrm{constant}}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~c_V + \frac{\Re}{\bar\mu} \, .</math>		<math>c_V + \frac{\Re}{\bar\mu} \, .</math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~c_P - c_V </math>		<math>c_P - c_V </math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{\Re}{\bar\mu} \, .</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, §1, Eq. (9)<br />		[<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>], Chapter II, §1, Eq. (9)<br />
	[http://adsabs.harvard.edu/abs/1968psen.book.....C D. D. Clayton (1968)], Eq. (2-108)<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, §80, immediately following Eq. (80.11)<br />		[<b>[[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>[[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)		[<b>[[Appendix/References#KW94\|<font color="red">KW94</font>]]</b>], §4.1, immediately following Eq. (4.15)
	</div>		</div>

Jet53man: /* Specific Heats */

2021-07-11T21:21:42Z

Specific Heats

← Older revision		Revision as of 17:21, 11 July 2021
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	==Specific Heats==		==Specific Heats==

	Drawing from Chapter II, §1 of [<b>[[~~User:Tohline/~~Appendix/References#C67\|<font color="red">C67</font>]]</b>]:  "<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>"		Drawing from Chapter II, §1 of [<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>]:  "<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">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~c_\alpha</math>		<math>c_\alpha</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~\equiv</math>		<math>\equiv</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl( \frac{dQ}{dT} \biggr)_{\alpha ~=~ \mathrm{constant}}</math>		<math>\biggl( \frac{dQ}{dT} \biggr)_{\alpha ~=~ \mathrm{constant}}</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</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.		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 [[PGE/FirstLawOfThermodynamics#FundamentalLaw\|Fundamental Law of Thermodynamics]], namely,

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

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

Jet53man: /* Property #2 */

2021-07-11T21:19:39Z

Property #2

← Older revision		Revision as of 17:19, 11 July 2021
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	===Property #2===		===Property #2===

	The internal energy per unit mass {{~~User~~:~~Tohline/~~Math/VAR_SpecificInternalEnergy01}} of an ideal gas is a function ''only'' of the gas temperature {{~~User~~:~~Tohline/~~Math/VAR_Temperature01}}, that is,		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">		<div align="center">
	<math>~\epsilon = \epsilon(T) \, .</math>		<math>~\epsilon = \epsilon(T) \, .</math>

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

Jet53man: /* Property #1 */

2021-07-11T21:18:19Z

Property #1

← Older revision		Revision as of 17:18, 11 July 2021
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	[<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>], Chapter VII.3, Eq. (18)<br />		[<b>[[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 />		[http://adsabs.harvard.edu/abs/1968psen.book.....C D. D. Clayton (1968)], Eq. (2-7)<br />
	[<b>[[Appendix/References#~~CH87~~\|<font color="red">H87</font>]]</b>], §1.1, p. 5		[<b>[[Appendix/References#H87\|<font color="red">H87</font>]]</b>], §1.1, p. 5
	</div>		</div>

← Older revision		Revision as of 12:33, 8 October 2021
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	[<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>], Chapter II, §1, Eq. (9)<br />		[<b>[[Appendix/References#C67\|<font color="red">C67</font>]]</b>], Chapter II, §1, Eq. (9)<br />
	[~~http:~~//~~adsabs.harvard.edu~~/~~abs/1968psen.book.....C D. D. Clayton (1968)~~], Eq. (2-108)<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, §80, immediately following Eq. (80.11)<br />		[<b>[[Appendix/References#LL75\|<font color="red">LL75</font>]]</b>], Chapter IX, §80, immediately following Eq. (80.11)<br />
	[<b>[[Appendix/References#H87\|<font color="red">H87</font>]]</b>], §1.2, p. 9<br>		[<b>[[Appendix/References#H87\|<font color="red">H87</font>]]</b>], §1.2, p. 9<br>