Counting molecular degrees of freedom
Count translational, rotational, and vibrational modes correctly for monoatomic and diatomic gases using degree-of-freedom logic.
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ContinueOne molecular vibration uses one independent coordinate but contains both kinetic and potential energy. That distinction explains why counting coordinates and predicting heat capacity are different tasks. First count the motions allowed by molecular geometry, then decide which energy contributions are thermally active.
At a glance
- There are two related but different “counting tasks”:
- Mechanical degrees of freedom: how many independent coordinates describe the molecule’s motion.
- Quadratic energy terms (equipartition count): how many ∝ x² or ∝ p² terms appear in the energy when a mode is thermally active.
- For a molecule with a atoms, there are 3a mechanical coordinates.
- Vibrational normal modes:
- Linear molecule with a ≥ 2: N_vib = 3a-5
- Non-linear molecule: N_vib = 3a-6
- For equipartition/heat-capacity work, an active vibrational mode contributes two quadratic terms (kinetic + potential), so it contributes kBT per molecule.
Separate coordinates from energy terms
Step 1: count mechanical degrees of freedom (geometry).
- Start from 3a.
- Subtract translations (always 3).
- Subtract rotations: 2 for linear molecules, 3 for non-linear molecules.
- What is left is the number of vibrational normal modes N_vib.
Step 2: convert to an equipartition “quadratic-term count” (energy bookkeeping).
- Translation contributes 3 quadratic kinetic-energy terms.
- Rotation contributes 2 (linear) or 3 (non-linear) quadratic kinetic-energy terms.
- Each vibrational normal mode contributes 2 quadratic terms (one kinetic + one potential) when the vibration is thermally active.
So if all modes are active, the equipartition count is
Here a counts atoms in one molecule, Nₛ counts molecules in the sample, and C_(V,m) is molar heat capacity. A monatomic gas is a separate point-particle case; the linear formula below assumes a ≥ 2, and the non-linear formula applies to a molecule with a genuine non-linear geometry. For a fixed active count in an ideal gas, the predictions follow from UY1: theorem of equipartition of energy:
- 3a counts mechanical coordinates, while f counts quadratic energy terms. The factor two for a vibration belongs to its kinetic and potential terms.
- A vibrational coordinate still exists when its heat-capacity contribution is small. See mode freezing.
- A linear molecule has two independent thermal rotations in the rigid-rotor model. Rotation about its own axis does not add a third.
Apply the count to atoms and molecules
Monatomic ideal gas
In this monatomic ideal-gas model, include only centre-of-mass motion; internal electronic excitations are neglected:
- Translational d.o.f. = 3
- Rotational d.o.f. = 0 (no extended structure for meaningful rotational thermal energy in this model)
- Vibrational modes = 0 (a single atom cannot vibrate internally)
So per atom, effective modes come only from translation.
Linear diatomic molecule
Mechanical count:
Energy/equipartition count (if vibration is active):
At ordinary temperatures, the vibrational mode is often frozen out, so a common UY1 working count is f = 3 + 2 = 5.
Use geometry and temperature separately
Diatomic heat-capacity regimes
Worked example 1
Counting active energy terms
Problem
Show full solution
Count quadratic terms
Method
Add three translational and two rotational terms; add two more when vibration is active.Reason
A diatomic molecule has one vibrational mode, but that mode has both kinetic and potential energy.Working
Answer
f_frozen = 3 + 2 = 5; f_active = 3 + 2 + 2 = 7.Convert the counts to molar heat capacities
Working
Answer
- Vibrations frozen: f = 5 ⇒ CV = (5/2)R.
- Vibrations active: f = 7 ⇒ CV = (7/2)R.
Both heat capacities have units J mol⁻¹K⁻¹.
Linear triatomic molecule
Worked example 2
Triatomic mode count
Problem
Show full solution
Separate whole-molecule motion
Method
Subtract three translations and two rotations from 3a with a = 3.Reason
The remaining independent motions are internal vibrations, rather than motion of the molecule as a whole.Working
Answer
3a = 9, Nₜᵣₐₙₛ = 3, Nᵣₒₜ = 2, N_vib = 3a-5 = 4.Count the active energy terms
Working
Answer
There are four vibrational normal modes. Each contributes two terms, so
f = 3 + 2 + 2(4) = 13.Thus f = 13, distinct from the nine mechanical coordinates. If all vibrations are frozen but rotation is active, the reduced count is f = 5; which approximation applies depends on the molecular energy spacings and temperature.
Practice set
Check your understanding 1
In the point-particle model of a monatomic ideal gas, list the translational, rotational and internal vibrational mode counts.
Show hint
One atom can move in three directions but has no internal arrangement of atoms.
Show answer
Translation: 3; rotation: 0; internal vibration: 0. Only centre-of-mass motion contributes in this model.
Check your understanding 2
A linear molecule has two atoms. Find its total mechanical degrees of freedom and its number of vibrational normal modes.
Show hint
Start with 3a, then subtract whole-molecule translation and rotation.
Show answer
There are 3(2) = 6 mechanical degrees of freedom. Subtracting three translations and two rotations leaves 6-3-2 = 1 vibrational mode; its two energy terms do not make it two normal modes.
Check your understanding 3
A linear molecule has four atoms. How many vibrational normal modes does it have?
Show hint
Subtract the five whole-molecule motions from 3a.
Show answer
N_vib = 3(4)-5 = 7. These are normal modes, irrespective of whether they are thermally active.
Use a partly excited bent molecule
Guided practice 3
Separate geometry from partial activation
Problem
A bent triatomic ideal-gas molecule has three translational and three rotational motions active. Its vibrational normal modes have different spacings. In a stipulated plateau approximation, two vibrations are fully classical; all remaining vibrations contribute negligible heat capacity. Count its vibrational normal modes and active quadratic energy terms, then find its molar heat capacity at constant volume. Use R = 8.314 J mol⁻¹ K⁻¹. Explain why the frozen vibration is still part of the mechanical count.
Work through the model, then compare your reasoning
Hints
Hint 1
Check the model
Make two separate counts: motions that exist, then thermal energy terms included by the approximation.Show solution step by step
Count the allowed motions
Method
Use the non-linear molecular geometry.Reason
The number of coordinates does not depend on thermal excitation.Working
Answer
3a = 9 coordinates split into three translations, three rotations and 9-3-3 = 3 vibrations.
Count thermal contributions
Method
Include the stipulated two active vibrations.Reason
An inactive heat-capacity contribution is not a missing coordinate.Working
Answer
f = 3 + 3 + 2(2) = 10, giving C_(V,m) = 5R = 41.57 J mol⁻¹ K⁻¹. Full vibrational activation would give f = 12 and C_(V,m) = 6R; all vibrations frozen would give f = 6 and C_(V,m) = 3R. These plateau approximations do not make intermediate activation an abrupt switch.
Interpret a measured heat capacity
Check your understanding 4
A dilute bent triatomic gas has classical translation and rotation and measured molar CV = 4.50R at one temperature. A solution declares it has nine mechanical coordinates because 2CV/R = 9, then claims exactly one and a half vibrational modes are active. What can the geometry and this heat-capacity measurement actually establish?
Show answer
Triatomic geometry gives nine mechanical coordinates independently of this measurement; three are vibrational normal modes for a bent molecule. Classical translation and rotation contribute 3R to molar heat capacity, leaving 1.50R from vibrational thermal response within this model. The effective capacity count 2CV/R = 9 happens to match the coordinate count numerically, but the quantities have different meanings. Partial excitation can give non-integer multiples of a fully classical mode contribution. A single capacity value does not identify which modes are excited or their individual gaps.
What to keep
Geometry counts coordinates and normal modes; energy bookkeeping counts active quadratic terms. A vibration contributes two terms in its classical limit. Intermediate heat capacity describes a thermal response, not a fractional number of mechanical coordinates.
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Syllabus and review details
No official syllabus alignment is listed for this lesson.