4.5 Appendix : Adiabatic Expansion of an Ideal Gas

4.5.1 Specific Heats and the Gas Constant

The molar specific heats of a gas at constant pressure and constant volume are defined by

$\begin{displaymath} Q = n C_p \Delta T \end{displaymath}$ (4.62)

and

$\begin{displaymath} Q = n C_V \Delta T \end{displaymath}$ (4.63)

where is the heat flowing into or out of moles of the gas resulting in a change in temperature $\Delta T$ .
Given that the work done by (or on) a gas is

$\begin{displaymath} W = \int_{V_i}^{V_f} P dV \end{displaymath}$ (4.64)

a gas can do no work, nor can it have work done on it, at constant volume.
With the first law of thermodynamics

$\begin{displaymath} \Delta E_{int} = Q - W \end{displaymath}$ (4.65)

and , we find

$\begin{displaymath} \Delta E_{int} = Q = n C_V \Delta T \end{displaymath}$ (4.66)

for a constant-volume process.
For an ideal gas, a central result of kinetic theory is

$\begin{displaymath} E_{int} = n C_V T \end{displaymath}$ (4.67)

where depends on the number of degrees of freedom (spatial dimensions, rotations, vibrations, e.g.) available to the gas. Together with Eq. 4.66, Eq. 4.67 leads us to the conclusion that a change in internal energy of an ideal gas in any process depends only on the change in temperature. Hence, for an ideal gas, Eq. 4.66 applies to any process.
For a constant-pressure process, the ideal gas law yields an expression for the work done by (on) a gas

$\begin{displaymath} W = P \Delta V = nR \Delta T \end{displaymath}$ (4.68)
For a constant-pressure process, Eq.'s 4.66 and 4.68 and the first law of thermodynamics yield

$\begin{displaymath} n C_V \Delta T = n C_p \Delta T - nR \Delta T \end{displaymath}$

or

$\begin{displaymath} R = C_p - C_V \end{displaymath}$ (4.69)

This relates a fundamental constant to constants and that are characteristics of the gas, so it not only true for constant-pressure processes. It is true in general.

4.5.2 Adiabatic Expansion

An adiabatic process is one in which no heat is exchanged with the environment ().
Adiabatic processes happen either very rapidly, so that there is no time for heat exchange, or in a well insulated system.
For an ideal gas, the first law of thermodynamics for an adiabatic process gives

$\begin{displaymath} n C_V dT = -PdV \end{displaymath}$ (4.70)
The ideal gas law gives

$\begin{displaymath} PdV + VdP = nR dT \end{displaymath}$ (4.71)

for any process.
Combining Eq. 4.70 and 4.71 to eliminate and using Eq. 4.69, we have

$\begin{displaymath} PdV + VdP = -\frac{C_p - C_V}{C_V} PdV \end{displaymath}$

which, divided by becomes

$\begin{displaymath} \frac{C_p}{C_V} \frac{dV}{V} + \frac{dP}{P} = 0 \end{displaymath}$ (4.72)
Integrating Eq. 4.72 we find

$\begin{displaymath} \frac{C_p}{C_V} \ln(V) + \ln(P) = \mathrm{constant} \end{displaymath}$

which is equivalent to

$\begin{displaymath} PV^\gamma = \mathrm{constant} \end{displaymath}$ (4.73)

where

$\begin{displaymath} \gamma = \frac{C_p}{C_V} \end{displaymath}$ (4.74)

4.5.3 About $\gamma$

The ratio of specific heats $\gamma$ of Eq. 4.74 can be related via kinetic theory to the number of degrees of freedom available to the molecules of the gas via

$\begin{displaymath} \gamma \approx \frac{d + 2}{d} \end{displaymath}$ (4.75)

See the chapter on the kinetic theory of gases in your favorite introductory physics text for the background.
A monatomic gas (noble gases He, Ne, Ar, e.g.) has three dimensions in which it can travel and hence only three degrees of freedom. Hence, we expect $\gamma \approx \frac{5}{3} = 1.67$ ( $\gamma_{\mathrm{He}}$ = 1.66, $\gamma_{\mathrm{Ne}}$ = 1.64, $\gamma_{\mathrm{Ar}}$ = 1.67).
A gas consisting of diatomic molecules (H, N, O, e.g.) can rotate. A ``dumbbell'' shape has one axis of symmetry, so it can only rotate about two axes. The bond between the atoms of a diatomic molecule behaves roughly like a (quantum mechanical) spring. Vibrations generally require more energy than rotations. In a temperature regime in which only rotations are excited in the gas, we expect $\gamma \approx \frac{7}{5} = 1.4$ ( $\gamma_{\mathrm{H}_2}$ = 1.41, $\gamma_{\mathrm{N}_2}$ = $\gamma_{\mathrm{O}_2}$ = 1.40). At higher temperatures, vibrations add a degree of freedom, giving $\gamma \approx \frac{8}{6} = 1.25$ .
Triatomic, and more complex molecules (CO, e.g.) have in general no axes of symmetry, and can rotate about three independent axes. Multiple vibrational modes are also expected, the number of unique modes depending on molecular structure. In a temperature regime in which only rotations are excited, we expect $\gamma \approx \frac{8}{6} = 1.25$ ( $\gamma_{\mathrm{CO}_2} = 1.30$ ).
All of these predictions are approximate, because real gases are only approximately ideal, and it is found empirically that $\gamma$ varies weakly with temperature.

Next: 5. Electromagnetic Waves Up: 4. Acoustic Waves Previous: 4.4 The Doppler Effect