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 unity area) <math>~P</math> given by the following

Standard Form
of the Ideal Gas Equation of State,

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>.

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,

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

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 1/<math>~\bar{\mu}</math> gives the number of free particles per atomic mass unit, <math>m_\mathrm{u}</math>, where the the ratio of the atomic mass unit to the proton mass is, <math>m_\mathrm{u}</math>/<math>~m_p</math> = 0.992777.

Conservative Form
of the Continuity Equation,

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

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