Seeking to explain thermodynamics based on moving and interacting atoms

Chapter 9 – Heat capacity (C)

Temperature tells you the average translational kinetic energy of the atoms in a system (Chapter 8). Heat capacity tells you how much energy (Q) must be transferred to that system to change its temperature by a given amount:

Q = m c ΔT c = specific heat capacity, per unit mass

Q = n C ΔT C = molar heat capacity, per mole

(Per molecule or atom, the heat capacity is simply C divided by Avogadro’s number; for the gases below, this puts it in units of Boltzmann’s constant, kB.)

High heat capacity means a substance requires a lot of energy to raise its temperature by a given amount. Low heat capacity means the opposite. This seemingly simple property turns out to be one of the most physically revealing quantities in thermodynamics, because what determines heat capacity is the structure of the atoms and molecules involved — specifically, how many ways they can absorb and store energy.

One practical note before turning to the physics. Heat capacity is reported per unit mass, per mole, or per atom, and the choice matters. Mass is the easiest quantity to measure, which is why Joseph Black built his original research on a mass basis; the means to convert mass to moles didn’t yet exist in his time. But it is the molar — or equivalently, per-atom — heat capacity that opens the door to the atomic interpretation developed in this chapter. When comparing substances, the molar basis is what reveals the underlying physics. Last note: be aware of the units on any physical property!

Why heat capacity doesn’t depend on mass

Start with the simplest possible case: a monatomic ideal gas. When two such atoms collide elastically, kinetic energy is conserved. Each atom may speed up or slow down, but their total kinetic energy stays the same, and so does their total momentum.

Now consider two systems of monatomic ideal gas, each with the same number of atoms N, in thermal contact across a boundary — System 1 hotter, System 2 colder. As atoms collide across the boundary, whatever kinetic energy one atom loses, the other gains. Summed over the vast number of collisions at the boundary, the kinetic energy lost by System 1 equals the kinetic energy gained by System 2. This is what Q represents: not a substance flowing between the systems, but the size of this shared change in total kinetic energy.

Since each system’s energy is simply N times the average kinetic energy per atom:

Δ⟨½m₁v₁²⟩ = −Δ⟨½m₂v₂²⟩

The atoms in System 1 might be helium and those in System 2 krypton. It doesn’t matter. From Chapter 8, temperature measures average translational kinetic energy by the same rule for every atom, heavy or light — at equilibrium, heavy atoms carry the same average kinetic energy as light ones, simply by moving more slowly. So equal and opposite changes in average kinetic energy mean equal and opposite changes in temperature:

ΔT₁ = −ΔT₂

With the same N, the same Q, and the same size of ΔT for both systems, Q = N C ΔT gives C₁ = C₂.

The heat capacity of any monatomic ideal gas, per atom, is the same as that of any other, regardless of atomic mass. This is not a coincidence or an empirical curiosity. It follows directly from two facts: temperature measures translational kinetic energy, and elastic collisions conserve it.

The monatomic ideal gas: Cv = (3/2)R

For a monatomic ideal gas, internal energy is purely kinetic. The atoms don’t interact, so there is no potential energy, and a single atom has no rotation worth counting. All of its energy is translational.

The derivation of the ideal gas law shows that the average translational kinetic energy of an atom is ⟨½mv²⟩ = (3/2)kBT — ½kBT for each of the three directions of motion, x, y, and z. For N atoms:

U = (3/2)NkBT = (3/2)nRT

At constant volume, no work is done, so all added energy goes into U: δQ = dU = nCv dT. Since U is linear in T:

Cv = (3/2)R monatomic ideal gas

This is the simplest heat capacity in all of thermodynamics — three directions of motion, nothing more, nothing hidden. Every other heat capacity in this chapter is this same idea, extended.

The crystalline solid: Cv = 3R

Now consider a crystalline solid. The atoms are locked in a three-dimensional lattice, each oscillating about its equilibrium position. As an atom moves away from center, a restoring force pulls it back, and kinetic and potential energy trade back and forth continuously while their total stays constant.

On average, that energy splits equally between kinetic and potential. So when energy is added to a crystalline solid, only half goes into kinetic energy — and thus into temperature — while the other half goes into potential energy. It takes twice as much energy to achieve the same temperature rise as in a monatomic gas:

Cv = 3R crystalline solid

This is the Dulong–Petit law, identified by Pierre Dulong and Alexis Petit in 1819. Once again, atomic mass plays no role. What matters is structure.

Why energy shares equally: equipartition

The gas and the solid share a single principle, the equipartition of energy. At equilibrium, every independent term in a molecule’s energy that depends on the square of a velocity or a displacement holds on average the same energy, ½kBT.

Why equal shares? Collisions constantly pass energy among these terms, and the same conservation laws govern every collision, whatever kind of motion is involved. Not every collision passes energy into every mode equally readily, but that affects only how quickly equilibrium is reached, not where it ends up. And where it ends up is the most probable distribution (Chapters 5 and 6). Any distribution that concentrates energy in one mode at the expense of the others is overwhelmingly less probable than the one that shares it evenly.

Each such term is called a degree of freedom. A monatomic gas has three, one for each direction of translation, giving (3/2)kBT per atom. A diatomic molecule adds two for rotation. Its vibration adds two more — one for the kinetic energy of the vibrating atoms and one for the potential energy of the stretched bond — for a total of seven when all are active.

This is why polyatomic gases have higher molar heat capacities than monatomic ones. Energy added by collisions spreads across more modes. Rotation and vibration absorb energy, but only translation registers as temperature (Chapter 8). Less of the added energy reaches translation, so more must be added to move the temperature dial the same amount.

Degrees of freedom and the ratio γ = Cp/Cv

For an ideal gas with f active degrees of freedom, U = (f/2)nRT, and:

Cv = (f/2)R

Cp = Cv + R (ideal gas; see [1] for derivation)

γ = Cp/Cv = (f + 2)/f

For an ideal gas, then, knowing the degrees of freedom predicts the heat capacity.

A monatomic gas (f = 3) gives γ = 5/3 ≈ 1.67. A diatomic gas with translation and rotation but no vibration (f = 5) gives γ = 7/5 = 1.40. A diatomic gas with vibration fully active as well (f = 7) gives γ = 9/7 ≈ 1.29.

Maxwell, air, and a famous near-miss

This framework played a pivotal role in the history of the kinetic theory of gases. James Clerk Maxwell calculated γ for air assuming three rotational degrees of freedom and predicted γ = 1.33. The measured value was 1.408. Maxwell considered the discrepancy serious enough to threaten the kinetic theory of gases altogether. [2]

The resolution came from Boltzmann, who pointed out that a diatomic molecule has only two rotational degrees of freedom, not three. Rotation about the axis joining the two atoms involves a negligible moment of inertia and contributes nothing measurable. With f = 5, γ = 7/5 = 1.40, matching experiment closely. Even Maxwell, working at the frontier of this theory, momentarily got the atomic picture wrong. The correction came not from new data but from a more careful physical accounting of how a diatomic molecule can actually move.

Boltzmann’s argument was classical, and it worked. But the deeper reason it works — and the reason air’s vibration contributes nothing either — would have to wait for quantum mechanics.

When modes freeze out: the quantum boundary

Energy in rotation and vibration can only be gained in specific increments, or quanta. When the smallest available increment is large compared with kBT, collisions rarely carry enough energy to deliver it, and the mode stays unexcited. It is “frozen out” and contributes nothing to heat capacity.

This is why vibration is absent from the heat capacity of air at room temperature, and why γ for air sits at 1.40 rather than 1.29. As temperature rises, vibrational modes progressively unfreeze, the heat capacity rises, and γ drops accordingly. Rotation about a diatomic molecule’s own axis is the extreme case: its moment of inertia is so small that the required increment is never reached under ordinary conditions.

The same effect appears in solids. At low temperatures, the lattice vibrations freeze out, the heat capacity falls below the Dulong–Petit value, and it approaches zero at absolute zero. Einstein explained this in 1907, and Debye refined it in 1912. This is where the classical atomic picture reaches its boundary and quantum mechanics takes over.

A numerical check: helium, krypton, and iron

The claim that heat capacity depends on structure rather than mass can be checked directly against data. Helium and krypton are both monatomic gases, differing in atomic mass by a factor of about 21 — helium at 4 g/mol, krypton at 84 g/mol. Their molar heat capacities at constant pressure are identical: 20.79 J/mol·K, which is (5/2)R, corresponding to Cv = (3/2)R. [3] Mass plays no role.

Now compare helium gas with solid iron. Iron’s molar heat capacity at room temperature is about 25 J/mol·K, or about 3.0R per atom [4] — almost exactly double helium’s Cv of 1.5R, just as Dulong–Petit predicts for a solid with six degrees of freedom per atom (three kinetic, three potential). For a solid, Cp and Cv are nearly equal, so the comparison holds either way. Although iron’s atoms are fourteen times heavier than helium’s, the heat capacity per atom is governed entirely by the number of ways each atom can hold energy — three for a free atom in a gas, six for an atom locked in a vibrating lattice.

These numbers are not coincidences. They are direct experimental confirmation that heat capacity is a window into atomic structure, not a property of mass.

Heat capacity at constant volume versus constant pressure

One further distinction matters throughout thermodynamics. Heat capacity can be measured at constant volume (Cv) or at constant pressure (Cp). At constant volume, all added energy goes into the internal energy of the system. At constant pressure, the system is also free to expand as it heats, pushing back its surroundings, and that expansion costs energy. So more energy is usually needed at constant pressure to achieve the same temperature rise.

For any substance, Cp is at least as large as Cv. The two are equal only when the substance doesn’t expand on heating, as with liquid water near 4 °C, where its density peaks. For an ideal gas, the difference is exactly:

Cp − Cv = R per mole, ideal gas

For liquids and solids, which barely expand, the difference is small.

The ideal-gas difference became the basis for Robert Mayer’s discovery of heat–work equivalence. He equated the extra heat required at constant pressure with the work done by the gas as it expands.

The distinction carries direct consequences throughout thermodynamics:

Heat capacity is not a secondary property. It is the quantitative link between atomic motion and much of thermodynamics.

Heat capacity and the number of ways to store energy

The pattern is clear. Heat capacity is governed by how many ways a system can store energy. A monatomic gas stores it only as translation — three degrees of freedom. A crystalline solid stores it as kinetic and potential energy in three directions — six. A polyatomic gas adds rotation and, at high enough temperatures, vibration. The more ways a system can absorb energy without registering it as temperature, the more energy it takes to move the temperature dial, and the higher its heat capacity.

References

[1] For an ideal gas, the first law gives δQ = dU + PdV. At constant volume, dV = 0, so all of the added heat increases the internal energy: δQ = Cv dT per mole. At constant pressure, the added heat must both increase the internal energy and supply the work of expansion: Cp dT = Cv dT + PdV. For one mole of ideal gas, PV = RT, so at constant pressure PdV = RdT. Substituting gives Cp dT = Cv dT + RdT, and dividing through by dT gives Cp = Cv + R. The heat capacity at constant pressure exceeds that at constant volume by exactly R because some of the added heat performs expansion work against the surrounding pressure.

[2] Maxwell’s calculation and the discrepancy with experimental γ for air are discussed in Brush, S.G., The Kind of Motion We Call Heat, North-Holland, 1976, and in Garber, Brush, and Everitt (eds.), Maxwell on Molecules and Gases, MIT Press, 1986.

[3] Standard molar heat capacity values for helium and krypton: NIST Chemistry WebBook, SRD 69.

[4] Heat capacity per atom for iron and the Dulong–Petit comparison: Atkins, P.W. and de Paula, J., Physical Chemistry, 10th ed., Oxford University Press, 2014, Chapter 2.

END