SR/IdealGas

Ideal Gas Equation of State

Ideal Gas

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 $n_g$ free particles per unit volume will exert on its surroundings an isotropic pressure (i.e., a force per unit area) $P$ given by the following

Standard Form
of the Ideal Gas Equation of State,

$P=n_{g}kT$

[C67], Chapter VII.3, Eq. (18)
[Clayton68], Eq. (2-7)
[H87], §1.1, p. 5

if the gas is in thermal equilibrium at a temperature $T$.

Property #2

The internal energy per unit mass $ϵ$ of an ideal gas is a function only of the gas temperature $T$, that is,

$ϵ=ϵ(T).$

[C67], Chapter II, Eq. (1)

Specific Heats

Drawing from Chapter II, §1 of [C67]:  "Let $\alpha$ be a function of the physical variables. Then the specific heat, $c_{\alpha }$, at constant $\alpha$ is defined by the expression,"

$c_{\alpha }$

$\equiv$

$(\frac{dQ}{dT}{)}_{\alpha =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}}$

The specific heat at constant pressure $c_P$ and the specific heat at constant (specific) volume $c_V$ prove to be particularly interesting parameters because they identify experimentally measurable properties of a gas.

From the Fundamental Law of Thermodynamics, namely,

$dQ$

$=$

$dϵ+PdV,$

it is clear that when the state of a gas undergoes a change at constant (specific) volume $(dV=0)$,

$(\frac{dQ}{dT}{)}_{V=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}}$	$=$	$\frac{dϵ}{dT}$
$\Rightarrow c_V$	$=$	$\frac{dϵ}{dT}.$

Assuming $c_V$ is independent of $T$ — 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,

$ϵ$

$=$

$c_{V}T.$

[C67], Chapter II, Eq. (10)
[LL75], Chapter IX, §80, Eq. (80.10)
[H87], §1.2, p. 9
[HK94], §3.7.1, immediately following Eq. (3.80)

Also, from Form A of the Ideal Gas Equation of State (see below) and the recognition that $\rho =1/V$, we can write,

$P_{\mathrm{g}\mathrm{a}\mathrm{s}}V$	$=$	$\left(\frac{\mathrm{\Re }}{\overline{\mu }}\right)T$
$\Rightarrow PdV+VdP$	$=$	$\left(\frac{\mathrm{\Re }}{\overline{\mu }}\right)dT.$

As a result, the Fundamental Law of Thermodynamics can be rewritten as,

$dQ$

$=$

$c_{\mathrm{V}}dT+\left(\frac{\mathrm{\Re }}{\overline{\mu }}\right)dT-VdP.$

This means that the specific heat at constant pressure is given by the relation,

$c_P\equiv (\frac{dQ}{dT}{)}_{P=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}}$

$=$

$c_V+\frac{\mathrm{\Re }}{\overline{\mu }}.$

That is,

$c_P-c_V$

$=$

$\frac{\mathrm{\Re }}{\overline{\mu }}.$

[C67], Chapter II, §1, Eq. (9)
[Clayton68], Eq. (2-108)
[LL75], Chapter IX, §80, immediately following Eq. (80.11)
[H87], §1.2, p. 9
[KW94], §4.1, immediately following Eq. (4.15)

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 $\rho$ rather than in terms of its number density $n_g$. Following [ Clayton68 ] — see his p. 82 discussion of The Perfect Monatomic Nondegenerate Gas — we will "let the mean molecular weight of the perfect gas be designated by $\overline{\mu }$. Then the density is

$\rho =n_g\overline{\mu }{m}_u,$

where $m_u$ is the mass of 1 amu" (atomic mass unit). "The number of particles per unit volume can then be expressed in terms of the density and the mean molecular weight as

$n_g=\frac{\rho }{\overline{\mu }{m}_u}=\frac{\rho N_A}{\overline{\mu }},$

where $N_A$ = 1/$m_u$ is Avogadro's number …" Substitution into the above-defined Standard Form of the Ideal Gas Equation of State gives, what we will refer to as,

Form A
of the Ideal Gas Equation of State,

$P_{\mathrm{g}\mathrm{a}\mathrm{s}}=\frac{\mathrm{\Re }}{\overline{\mu }}\rho T$

[LL75], Chapter IX, §80, Eq. (80.8)
[KW94], §2.2, Eq. (2.7) and §13, Eq. (13.1)

where $\mathrm{\Re }$ ≡ $k$$N_A$ is generally referred to in the astrophysics literature as the gas constant. 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 $\overline{\mu }$.

Employing a couple of the expressions from the above discussion of specific heats, the right-hand side of Form A of the Ideal Gas Equation of State can be rewritten as,

$\frac{\mathrm{\Re }}{\overline{\mu }}\rho T$

$=$

$(c_P-c_V)\rho \left(\frac{ϵ}{c_V}\right)=({\gamma }_g-1)\rho ϵ,$

where we have — as have many before us — introduced a key physical parameter,

${\gamma }_g$

$\equiv$

$\frac{c_P}{c_V},$

[C67], Chapter II, immediately following Eq. (9)
[LL75], Chapter IX, §80, immediately following Eq. (80.9)
[T78], §3.4, immediately following Eq. (72)
[HK94], §3.7.1, Eq. (3.86)

to quantify the ratio of specific heats. This leads to what we will refer to as,

Form B
of the Ideal Gas Equation of State

$P=({\gamma }_{\mathrm{g}}-1)ϵ\rho$

[C67], Chapter II, Eq. (5)
[HK94], §1.3.1, Eq. (1.22)
[BLRY07], §6.1.1, Eq. (6.4)

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