Gaseous systems composed of molecules with internal motion

1. Big Idea: Molecules Have Internal Motion

Up to this point in the book, gases were treated as if molecules were just point particles that only move through space (translational motion).

Real molecules, however, can also:

rotate
vibrate
have electrons in different energy levels
have nuclei with spin

All of that is called internal motion.

These internal motions give extra ways for a molecule to store energy, so they affect things like:

entropy
internal energy
specific heat
chemical potential

The goal of this section is to compute how these internal motions change the thermodynamic properties of a gas.

2. Partition Function with Internal Motion

For a dilute ideal gas (no interactions between molecules, and quantum effects not too strong), you can write the single-molecule partition function as

\[ Q_1(V,T) = \underbrace{\frac{V}{\lambda^3}}_{\text{translation}} \times \underbrace{j(T)}_{\text{internal}} \]

\(\lambda\) is the thermal de Broglie wavelength.
\(V/\lambda^3\) is the usual translational partition function.
\(j(T)\) is the internal partition function that contains all the internal states (electronic, vibrational, rotational, nuclear).

For \(N\) non-interacting, distinguishable molecules in an ideal gas you have

\[ Q_N(V,T) = \frac{1}{N!}[Q_1(V,T)]^N \]

From standard statistical mechanics, once you know the partition function, you can get all the thermodynamic quantities. Applied only to the internal part, you get:

Helmholtz free energy from internal motion:

\[ A_{\text{int}} = -NkT \ln j \]

Chemical potential contribution:

\[ \mu_{\text{int}} = -kT \ln j \]

Entropy contribution:

\[ S_{\text{int}} = Nk\left(\ln j + T \frac{\partial \ln j}{\partial T}\right) \]

Internal-energy contribution:

\[ U_{\text{int}} = NkT^2 \frac{\partial \ln j}{\partial T} \]

Constant-volume specific heat from internal motion:

\[ (C_V)_{\text{int}} = Nk \frac{\partial}{\partial T}\left[T^2 \frac{\partial \ln j}{\partial T}\right] \]

So the whole problem reduces to: find \(j(T)\) for the type of molecule you're interested in.

3. Breaking \(j(T)\) into Pieces

A molecule's internal state is determined by several "subsystems":

electronic state (how the electrons are arranged)
state of the nuclei (their spin orientations)
vibrational state (how the atoms vibrate relative to each other)
rotational state (overall spinning of the molecule)

In many situations these are approximately independent, so the total internal partition function factorizes:

\[ j(T) = j_{\text{elec}}(T) \, j_{\text{nuc}}(T) \, j_{\text{vib}}(T) \, j_{\text{rot}}(T) \]

Then the logarithm splits into a sum, and each part contributes additively to energy, entropy, and specific heat.

The chapter then goes through different kinds of molecules in rising complexity:

monatomic gases
diatomic molecules
polyatomic molecules

4. Monatomic Gases

4.1 What Internal Motion Do They Have?

A monatomic gas (He, Ne, Ar, …) is just one atom per molecule.

No rotations of a "molecule" (it's just a sphere ⇒ no rotational degrees of freedom).
No vibrations between atoms (only one atom ⇒ no vibrational modes).
What's left is electronic states (and nuclear spin).

So the internal partition function is essentially electronic + nuclear.

4.2 Simple Case: One Nondegenerate Ground State

Some atoms have:

orbital angular momentum \(L = 0\)
spin \(S = 0\)

So the ground electronic state is a single nondegenerate level ⇒ \(g_e = 1\).

The nucleus can have spin \(S_n\), which gives a degeneracy

\[ g_n = 2S_n + 1 \]

Since the atom has no vibrational/rotational states, the internal partition function is just

\[ j(T) = g_e g_n = 2S_n + 1 \]

This does not depend on temperature, so:

internal energy from these internal motions is zero,
internal specific heat from these internal motions is zero,
but entropy and chemical potential do get constant contributions (because of the degeneracy factor).

4.3 Fine Structure and Electronic Excitation

Real atoms often have:

spin ≠ 0
orbital angular momentum ≠ 0

Then the ground state splits into fine-structure levels with different total angular momentum \(J\), each with degeneracy \(2J+1\) and slightly different energy \(\epsilon_J\).

Partition function:

\[ j_{\text{elec}}(T) = \sum_J (2J+1) \, e^{-\epsilon_J/kT} \]

Two limiting cases:

High T compared to the splittings (\(kT \gg \epsilon_J\)): the exponentials are ~1, so

\[ j_{\text{elec}}(T) \approx \sum_J (2J+1) = (2L+1)(2S+1) \]

and internal specific heat from this tends to 0 (no T dependence).

Low T compared to splittings (\(kT \ll \epsilon_J\)): only the lowest level matters:

\[ j_{\text{elec}}(T) \approx (2J_0 + 1) \, e^{-\epsilon_0/kT} \]

which again gives almost no \(C_V\) except near the temperature where \(kT\) is similar to the splitting.

Conclusion: for most monatomic gases, electronic/internal contributions to specific heat are:

negligible at low T,
negligible at very high T,
small bump near some intermediate T (where fine-structure splitting is of order \(kT\)).

For everyday temperatures, you usually ignore them.

5. Diatomic Molecules

Now molecules have two atoms (H₂, N₂, O₂, CO, etc.), so they can:

rotate
vibrate
still have electronic and nuclear states

5.1 Electronic States in Diatomics

For typical temperatures up to a few thousand K:

the first excited electronic state is usually far above \(kT\) (comparable to dissociation energy),
so practically only the ground electronic state contributes.

Often that ground electronic state is nondegenerate (or effectively so), so electronic contributions to \(C_V\) are again negligible in the usual temperature range.

There are some exceptions (like NO, with closely spaced electronic levels), where you get a noticeable bump in the specific heat, and the text writes down a 2-level partition function:

\[ j_{\text{elec}}(T) = g_0 + g_1 e^{-\Delta/kT} \]

and shows how this gives a specific heat term that peaks around \(T \sim \Delta/k\).

5.2 Vibrations of a Diatomic Molecule

Think of the two atoms connected by a "spring":

quantum energy levels: \((n + \tfrac{1}{2})\hbar\omega\)
define a characteristic temperature \(\Theta_v = \hbar\omega/k\)

The vibrational partition function (for one vibrational mode) is the same as for a quantum harmonic oscillator:

\[ j_{\text{vib}}(T) = \frac{e^{-\Theta_v/2T}}{1 - e^{-\Theta_v/T}} \]

From this, the vibrational contribution to \(C_V\) turns out to be:

\[ (C_V)_{\text{vib}} = Nk\left(\frac{\Theta_v}{T}\right)^2 \frac{e^{\Theta_v/T}}{(e^{\Theta_v/T}-1)^2} \]

Key points:

For \(T \ll \Theta_v\): vibrations are "frozen" (only ground state populated) ⇒ contribution ≈ 0.
For \(T \gg \Theta_v\): classical equipartition recovered: each vibrational mode gives \(Nk\) to \(C_V\) (because each vibration has 2 quadratic terms: kinetic + potential ⇒ 2 × ½kT).

So as T rises, there is a smooth "turn on" of vibrational specific heat from 0 up to \(Nk\).

5.3 Rotations of a Diatomic Molecule (Heteronuclear Case First)

A diatomic molecule can rotate about two perpendicular axes (perpendicular to the line connecting the atoms), so it has 2 rotational degrees of freedom.

Quantum theory gives rotational energy levels:

\[ \epsilon_{\text{rot}}(l) = \frac{l(l+1)\hbar^2}{2I}, \quad l = 0,1,2,\ldots \]

Each level has degeneracy \(2l+1\).

Define a rotational temperature:

\[ \Theta_r = \frac{\hbar^2}{2Ik} \]

Then the rotational partition function is

\[ j_{\text{rot}}(T) = \sum_{l=0}^{\infty} (2l+1) \, e^{-l(l+1)\Theta_r/T} \]

Again, two main regimes:

High T: \(T \gg \Theta_r\)

Levels become closely spaced, the sum can be approximated by an integral, and you get

\[ j_{\text{rot}}(T) \approx \frac{T}{\Theta_r} \]

Plugging into the formula for \(C_V\) gives

\[ (C_V)_{\text{rot}} \approx Nk \]

which matches equipartition: 2 rotational d.o.f. ⇒ each contributes ½k ⇒ total \(Nk\).

Low T: \(T \ll \Theta_r\)

Only the lowest level (\(l=0\)) is populated. Then

\[ (C_V)_{\text{rot}} \to 0 \]

at low temperature: rotations are frozen.

More detailed expansion (using Euler–Maclaurin) shows that at high but finite T, the rotational \(C_V\) is slightly above \(Nk\), with a maximum ≈ \(1.1Nk\) near \(T \approx 0.8\,\Theta_r\), then slowly approaches \(Nk\) from above.

5.4 Putting Translation + Rotation + Vibration Together

For a diatomic heteronuclear gas, depending on temperature range:

Very low T (both rotation and vibration frozen):

Only translation contributes:

\[ C_V = \frac{3}{2}Nk, \quad C_P = \frac{5}{2}Nk, \quad \gamma = \frac{C_P}{C_V} = \frac{5}{3} \]

Intermediate T where rotation is active but vibration is still frozen (\(\Theta_r \ll T \ll \Theta_v\)):

Translation (\(3/2\, Nk\)) + Rotation (\(Nk\)) gives:

\[ C_V = \frac{5}{2}Nk, \quad C_P = \frac{7}{2}Nk, \quad \gamma = \frac{7}{5} \]

High T where both rotation and vibration are active (\(T \gg \Theta_r, \Theta_v\)):

Vibration adds \(Nk\), so:

\[ C_V = \frac{7}{2}Nk, \quad C_P = \frac{9}{2}Nk, \quad \gamma = \frac{9}{7} \]

Experiments on real gases show curves that move between these plateaus as T increases.

6. Homonuclear Diatomics and Nuclear Spin Statistics (Ortho–Para)

For a homonuclear diatomic molecule (like H₂, D₂, etc.) the two nuclei are identical. Quantum mechanically:

the total wavefunction must be either symmetric or antisymmetric under exchange of the two nuclei, depending on whether they are bosons or fermions,
the total wavefunction = (nuclear spin part) × (rotational part) × (others),
symmetry of one part constrains the symmetry of the other.

Consequence: even and odd rotational levels are not equivalent; some are combined with symmetric nuclear states and others with antisymmetric ones.

This leads to:

different partition functions for "even-\(l\)" and "odd-\(l\)" rotational states:

\[ r_{\text{even}} = \sum_{l=0,2,4,\ldots} (2l+1) e^{-l(l+1)\Theta_r/T} \]

\[ r_{\text{odd}} = \sum_{l=1,3,5,\ldots} (2l+1) e^{-l(l+1)\Theta_r/T} \]

different combinations depending on whether the nuclei are fermions (H₂) or bosons (D₂).

This gives the famous ortho and para forms of hydrogen:

"ortho-hydrogen" = molecules with nuclear spins parallel (one symmetry type),
"para-hydrogen" = spins antiparallel (other symmetry type).

The ratio of ortho to para states is controlled by the partition function structure. At very low T, one type dominates (para for H₂, ortho for D₂), and this affects the specific heat.

The text discusses:

how at very low T the ratio of odd to even levels behaves as \(\sim \exp(-2\Theta_r/T)\),
why, in practice, you often do not reach this equilibrium ratio (conversion between ortho and para can be very slow, taking about a year at room temperature unless a catalyst is used),
as a result, samples of hydrogen can be in nonequilibrium mixtures, and their observed \(C_V\) is a weighted average of two curves (even- and odd-level contributions).

This explains discrepancies between naive equilibrium theory and experiments in old data.

7. Polyatomic Molecules

Now things get more complicated, but the logic is the same.

For a nonlinear polyatomic molecule with \(n\) atoms:

total degrees of freedom: \(3n\)
3 translational
3 rotational
the rest (\(3n-6\)) are vibrational

For a linear polyatomic molecule:

rotations: 2
vibrations: \(3n-5\)

7.1 Rotations

Because the moments of inertia are large, the energy spacing between rotational levels is small compared to \(kT\) at most temperatures, so we can usually treat rotations classically.

The classical rotational partition function involves the three principal moments of inertia (\(I_1, I_2, I_3\)), and gives, after calculation:

\[ (C_V)_{\text{rot}} = \frac{3}{2}Nk \]

So a nonlinear polyatomic molecule contributes \(3/2\, Nk\) to \(C_V\) from rotations (3 rotational degrees of freedom).

7.2 Vibrations

Each normal vibrational mode behaves like a harmonic oscillator with its own frequency \(\omega_i\) and characteristic temperature \(\Theta_i = \hbar\omega_i/k\).

The total vibrational partition function is a product over modes; the total vibrational \(C_V\) is a sum of terms of the same form as the diatomic vibrational expression:

\[ (C_V)_{\text{vib}} = Nk\sum_i \left(\frac{\Theta_i}{T}\right)^2 \frac{e^{\Theta_i/T}}{(e^{\Theta_i/T}-1)^2} \]

Important features:

Each mode "turns on" around \(T \sim \Theta_i\).
Different modes may have very different \(\Theta_i\), often around \(10^3\) K.
As T increases, modes become active one by one ⇒ \(C_V\) can show a series of plateaus and rises over a wide T range.

Example: CO₂ has four normal modes with specific \(\Theta_i\) quoted in a footnote; in practice the gas decomposes before all modes are fully classical.