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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 free particles per unit volume will exert on its surroundings an isotropic pressure (i.e., a force per unit area) given by the following
Standard Form
of the Ideal Gas Equation of State,
[C67], Chapter VII.3, Eq. (18)
D. D. Clayton (1968), Eq. (27)
[H87], §1.1, p. 5
if the gas is in thermal equilibrium at a temperature .
Property #2
The internal energy per unit mass of an ideal gas is a function only of the gas temperature , that is,
[C67], Chapter II, Eq. (1)
Specific Heats
Drawing from Chapter II, §1 of [C67]: "Let be a function of the physical variables. Then the specific heat, , at constant is defined by the expression,"



The specific heat at constant pressure and the specific heat at constant (specific) volume prove to be particularly interesting parameters because they identify experimentally measurable properties of a gas.
From the Fundamental Law of Thermodynamics, namely,



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






Assuming is independent of — 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,



[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 , we can write,






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



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



That is,



[C67], Chapter II, §1, Eq. (9)
D. D. Clayton (1968), Eq. (2108)
[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 rather than in terms of its number density . Following D. D. Clayton (1968) — 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 . Then the density is
where 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
where = 1/ is Avogadro's number …" Substitution into the abovedefined 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,

[LL75], Chapter IX, §80, Eq. (80.8)
[KW94], §2.2, Eq. (2.7) and §13, Eq. (13.1)
where ≡ 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 .
Employing a couple of the expressions from the above discussion of specific heats, the righthand side of Form A of the Ideal Gas Equation of State can be rewritten as,



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



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