User:Tohline/SR/IdealGas

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>

[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 (specific) 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>~dQ</math>

<math>~=</math>

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

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

<math>~\epsilon</math>

<math>~=</math>

<math>~c_V T \, .</math>

[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 <math>~\rho = 1/V</math>, we can write,

<math>~P_\mathrm{gas}V</math>	<math>~=</math>	<math>~\biggl(\frac{\Re}{\bar\mu} \biggr) T</math>
<math>~\Rightarrow ~~~ PdV + VdP</math>	<math>~=</math>	<math>~\biggl(\frac{\Re}{\bar\mu} \biggr) dT \, .</math>

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

<math>~dQ</math>

<math>~=</math>

<math>~c_\mathrm{V} dT + \biggl(\frac{\Re}{\bar\mu} \biggr) dT - VdP \, .</math>

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

<math>~c_P \equiv \biggl( \frac{dQ}{dT} \biggr)_{P ~=~ \mathrm{constant}}</math>

<math>~=</math>

<math>~c_V + \frac{\Re}{\bar\mu} \, .</math>

That is,

<math>~c_P - c_V </math>

<math>~=</math>

<math>~\frac{\Re}{\bar\mu} \, .</math>

[C67], Chapter II, §1, Eq. (9)
[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 <math>~\rho</math> rather than in terms of its number density <math>~n_g</math>. 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 <math>~\bar{\mu}</math>. Then the density is

<math>~\rho = n_g \bar\mu m_u \, ,</math>

where <math>~m_u</math> 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

<math>~n_g = \frac{\rho}{\bar\mu m_u} = \frac{\rho N_A}{\bar\mu} \, ,</math>

where <math>~N_A</math> = 1/<math>~m_u</math> is Avogadro's number …"

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>

[LL75], Chapter IX, §80, Eq. (80.8)
[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>.

Drawing 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,

<math>~\frac{\Re}{\bar\mu} \rho T</math>

<math>~=</math>

<math>~ (c_P - c_V)\rho \biggl(\frac{\epsilon}{c_V}\biggr) = (\gamma_g - 1)\rho\epsilon \, , </math>

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

<math>~\gamma_g</math>

<math>~\equiv</math>

<math>~\frac{c_P}{c_V} \, ,</math>

[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

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

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

Related Wikipedia Discussions

• Equation of State: Classical ideal gas law
• Ideal Gas Law
• Ideal Gas

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