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

From VistrailsWiki
Jump to navigation Jump to search
Line 77: Line 77:
</table>
</table>


[<b>[[User:Tohline/Appendix/References#CH87|<font color="red">H87</font>]]</b>], &sect;1.2, p. 9
[<b>[[User:Tohline/Appendix/References#H87|<font color="red">H87</font>]]</b>], &sect;1.2, p. 9
</div>
</div>


and the specific internal energy {{ User:Tohline/Math/VAR_SpecificInternalEnergy }}
and the specific internal energy of the gas, {{ User:Tohline/Math/VAR_SpecificInternalEnergy01 }}, is related to the gas termperature through the expression,
<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
</div>





Revision as of 19:12, 25 October 2018

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)
[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)

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>. 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

Form A
of the Ideal Gas Equation of State,

LSU Key.png

<math>~P_\mathrm{gas} = \frac{\Re}{\bar{\mu}} \rho T</math>

[KW94], §2.2, Eq. (2.7) and §13, Eq. (13.1)

where <math>~\Re</math> is the gas constant and <math>~\bar{\mu}</math> <math>\equiv</math> <math>~\rho</math>/(<math>~m_u</math><math>~n_g</math>) is the mean molecular weight of the gas. 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>.

Exercise: If <math>~\Re</math> is defined as the product of the Boltzmann constant <math>~k</math> and the Avogadro constant <math>~N_A</math>, as stated in the 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, <math>~m_u</math>.

From an accompanying discussion, we know that for any ideal gas the universal gas constant <math>~\Re</math> is related to the specific heat at constant pressure of the gas, <math>~c_P</math>, and to the specific heat at constant volume of the gas, <math>~c_V</math>, through the expression,

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

<math>~=</math>

<math>~c_P - c_V \, ;</math>

[H87], §1.2, p. 9

and the specific internal energy of the gas, <math>~\epsilon</math>, is related to the gas termperature through the expression,

<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



Work-in-progress.png

Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
|   Go Home   |


Still need to explain:

Form B
of the Ideal Gas Equation of State,

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

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