Equipartition and the Heat Capacities of Gases
Each way a molecule can store energy holds an average of half kT, so monatomic, diatomic and polyatomic gases have different heat capacities. Freezing out of modes at low temperature is the first hint that classical physics is incomplete.
What a learner can do afterwards
- Counts the degrees of freedom of a monatomic and a diatomic molecule and predicts the heat capacity
- Explains the difference between heat capacity at constant volume and at constant pressure
- Describes why measured heat capacities fall below the classical prediction at low temperature
1 · Read
Equipartition says each independent way a molecule stores energy holds an average of one half kT per molecule, or one half RT per mole. Count those ways, called degrees of freedom, and the heat capacity at constant volume is that count times R over 2, which is why some gases need more heat to warm up.
A lone atom drifts in three directions, so a monatomic gas has Cv of 3R over 2. A two atom molecule at room temperature also rotates about two axes, adding two more ways, so a diatomic gas has Cv of 5R over 2. Nitrogen and oxygen follow this rule.
Heat a gas in a rigid box and every joule stays inside as internal energy. Heat it while it pushes a piston and part of that energy leaves as expansion work. That is why Cp beats Cv by exactly R per mole.
Cool a gas far down and some rotations and shivers freeze out, so fewer ways share the energy. Measured heat capacities then fall below the classical count, which was an early hint that classical physics is incomplete.
Count the active ways to store energy, halve R times that count for Cv, and add R for Cp.
The rewritten readings were written for this dataset.
More to read
Sharing energy between molecular motions
Heat capacity measures the heat required for a one-degree temperature rise under specified conditions. Classical equipartition assigns an average energy of kT/2 per molecule to each independent quadratic energy term at equilibrium. Summing these contributions gives internal energy. Its temperature derivative at fixed volume gives Cv.
For an ideal monatomic gas, motion along three perpendicular directions contributes 3RT/2 per mole, giving Cv = 3R/2. A diatomic molecule also has two rotational degrees of freedom. When these are active but vibrations are not, Cv = 5R/2. This approximation describes many diatomic gases near room temperature.
Heating with and without expansion
At fixed volume, a gas does no expansion work. If no other work occurs, added heat increases its internal energy. At fixed pressure, heating also makes the gas expand and do work on its surroundings. An ideal gas therefore has Cp − Cv = R, where both heat capacities are molar values.
Around room temperature, nitrogen has three active translational modes and two rotational ones. These give Cv = 5R/2 and Cp = 7R/2. As cooling suppresses rotational excitation, the ideal-gas value of Cv approaches 3R/2. This does not describe nitrogen all the way to absolute zero: condensation can occur, and the gas model then fails.
Heat-capacity definitions and quantum limits
Heat capacity depends on how the measurement is made. Dividing by mass gives specific heat; dividing by amount in moles gives molar heat capacity. Holding pressure fixed differs from holding volume fixed. A gas that expands needs extra heat for each degree because it also transfers energy through work.
Classical equipartition assigns full energy contributions to vibrations regardless of temperature. Measurements disagree: diatomic gases have lower heat capacities when some modes cannot be excited. Vibrational energies come in discrete levels. At low temperatures, thermal energy cannot readily reach higher levels, so vibrations contribute little to heat capacity. Such discrepancies helped reveal the need for quantum physics.
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.