At high temperature, a diatomic molecule has 3 translational degrees (x, y, z motion), 2 rotational degrees (two perpendicular axes), and 2 vibrational quadratic terms (1 kinetic + 1 potential, since both ½mv² and ½kx² are quadratic). That's 7 total quadratic terms, each contributing (1/2)kT, giving U = (7/2)kT per molecule and C_v = (7/2)R. The common mistake is counting 6 — vibrational modes always contribute two quadratic terms (kinetic and potential), not one.
Question 2 Multiple Choice
Nitrogen gas (N₂) at room temperature has a measured C_v of approximately (5/2)R, not (7/2)R. What explains this?
AN₂ molecules lack vibrational modes entirely due to their bond structure
BRotational modes are also frozen out at room temperature for N₂
CVibrational modes are 'frozen out' because the quantum energy level spacing ħω >> kT at room temperature, so those modes cannot absorb thermal energy
DThe equipartition theorem does not apply to diatomic molecules
Quantum mechanics discretizes the energy levels of each mode. Vibrational modes in N₂ have large energy spacing ħω. At room temperature, kT is much smaller than this spacing, so molecules cannot climb to the first vibrational excited state — the mode stays frozen in its ground state and contributes nothing to heat capacity. Rotational modes in N₂ have much smaller energy spacing and are fully active at room temperature, giving the observed (5/2)R.
Question 3 True / False
A harmonic oscillator has both kinetic energy ½mv² and potential energy ½kx². By the equipartition theorem, the total average thermal energy of this oscillator is (1/2)kT.
TTrue
FFalse
Answer: False
Each quadratic term independently contributes (1/2)kT. The kinetic energy ½mv² is one quadratic term contributing (1/2)kT, and the potential energy ½kx² is a second quadratic term also contributing (1/2)kT. The total average energy is therefore kT, not (1/2)kT. This is why vibrational modes contribute twice as much to heat capacity as translational or rotational modes — they have two quadratic terms, not one.
Question 4 True / False
The equipartition theorem gives reliable predictions for heat capacities of most real gases at any temperature.
TTrue
FFalse
Answer: False
The equipartition theorem is a classical result assuming all modes can absorb energy continuously. Quantum mechanics restricts modes to discrete energy levels — if kT is much smaller than the level spacing ħω, the mode is frozen and contributes nothing. This quantum freezing makes classical equipartition an overestimate at low temperatures or for modes with large energy spacing (like vibrations in light diatomic molecules). The theorem gives good predictions only when kT >> ħω for the relevant mode.
Question 5 Short Answer
Why does a vibrational degree of freedom contribute kT (not (1/2)kT) to average molecular energy, while a translational degree of freedom contributes only (1/2)kT?
Think about your answer, then reveal below.
Model answer: The equipartition theorem assigns (1/2)kT to each quadratic term in the energy expression. Translational motion in one direction has a single quadratic term — ½mv² — so it contributes (1/2)kT. A vibrational mode has two quadratic terms: kinetic energy ½mv² and potential energy ½kx². Each independently gets (1/2)kT, summing to kT for the complete vibrational mode.
This is why counting 'degrees of freedom' requires care — a vibrational mode counts as two contributions to energy, not one. Mistakenly treating vibration as one degree of freedom predicts C_v = (6/2)R for a fully activated diatomic, when the correct answer is (7/2)R.