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,

[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)

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 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>~ d\epsilon + PdV \, , </math>

it is clear that when the state of a gas undergoes a change at constant 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,

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>

[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,

[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>

[C67], Chapter II, Eq. (10)
[LL75], Chapter IX, §80, Eq. (80.10)
[H87], §1.2, p. 9

Combining these two relationships with Form A of the Ideal Gas Equation of State, we derive what will be referred 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)

where the ratio of specific heats,

<math>~\gamma_g</math>

<math>~\equiv</math>

[C67], Chapter II, immediately following Eq. (9)
[LL75], Chapter IX, §80, immediately following Eq. (80.9)
[T78], §3.4, immediately following Eq. (72)

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