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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>
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,
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<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>.
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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,
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<math>~\frac{\Re}{\bar\mu}</math> |
<math>~=</math> |
<math>~c_P - c_V \, ;</math> |
[H87], §1.2, p. 9
and the specific internal energy User:Tohline/Math/VAR SpecificInternalEnergy
Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
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Still need to explain:
Form B
of the Ideal Gas Equation of State,
<math>~P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math>
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