Heat Capacity (C)
Temperature tells you the average translational kinetic energy of the atoms in a system. Heat capacity tells you how much energy (Q) must be transferred to that system to raise that temperature by a targeted amount. Using specific heat capacity (C) based on mass:
Q = m C ΔT C = heat capacity based on mass
Q = N C ΔT C = heat capacity based on moles
High heat capacity means a substance requires a lot of energy (Q) 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 data. Heat capacity is reported in different units — per unit mass, per mole, or per atom — and the choice matters for physical interpretation. Mass is the easiest quantity to measure directly, which is why Joseph Black built his original research on a mass basis; the means to convert mass to moles via atomic weight 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-level interpretation developed in this chapter. When comparing substances, as we do below, the molar basis is what reveals the underlying physics. Last note: be aware of the units on any physical property!
Heat capacity at the atomic level and why heat capacity doesn’t depend on mass — a derivation
The physical picture begins with the simplest possible case: a monatomic ideal gas. When two such atoms collide elastically (they bounce off each other with no loss of energy), kinetic energy is conserved. While each atom may experience a change in kinetic energy, the sum of the two remains constant. Total momentum is conserved as well.
Consider two systems of monatomic ideal gases, each containing the same number of atoms N, in thermal contact across a boundary — System 1 hotter, System 2 colder. As atoms collide across the boundary, kinetic energy is conserved in each collision: whatever one atom loses, the other gains. Summed over the vast number of collisions occurring at the boundary, the total kinetic energy lost by System 1 equals the total kinetic energy gained by System 2. This is what Q represents — not a substance flowing between the systems, but the magnitude of this shared change in total kinetic energy.
Since each system is a monatomic ideal gas, its total energy is simply N times the average kinetic energy per atom:
System 1: N1⟨½m₁v₁²⟩
System 2: N2⟨½m₂v₂²⟩
With N the same for both systems, and the change in total kinetic energy equal in magnitude for both:
∆⟨½m₁v₁²⟩ = – ∆⟨½m₂v₂²⟩
Because temperature is directly proportional to average translational kinetic energy — established earlier in this chapter — this means:
∆T₁ = – ∆T₂
And since Q = NC∆T, with N, Q and ∆T equal for both systems:
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 is a direct mathematical consequence of two facts already established in this chapter: temperature measures kinetic energy and elastic collisions conserve that kinetic energy. Heavier atoms and lighter atoms at the same temperature share the same average kinetic energy — achieved by the heavier atoms moving more slowly. Heat capacity, built on the relationship between added energy and temperature change, inherits this mass-independence directly.
Heat capacity data based on the ideal gas law
For a monatomic ideal gas, internal energy U is purely kinetic — there is no potential energy term, since the atoms don’t interact. There is no rotational energy. All energy gained or lost is 100% translational in nature.
Recall from the derivation of the ideal gas law that the average translational kinetic energy of an atom is directly proportional to temperature: ⟨½mv²⟩ = (3/2)kT. (A separate derivation based on the equipartition of energy in which each degree-of-freedom contributes 1/2 kT of energy leads to the same result since there are three degrees of freedom in this case, i.e., independent motion in the x, y, and z directions.) For N atoms, U = N⟨½mv²⟩ = (3/2)NkT = (3/2)RT.
At constant volume, no PdV work is done, so all added energy goes directly into U: Q = dU = CvdT. Since U here is linear in T, Cv = (∂U/∂T)v and
Cv = (3/2)R monatomic ideal gas
This is the simplest possible heat capacity in all of thermodynamics — three directions of motion, nothing more, nothing hidden. Atomic weight is irrelevant because temperature reflects kinetic energy: at equilibrium, atoms of different mass share the same average kinetic energy and simply move at different speeds. Every other heat capacity in this chapter is this same idea, extended.
Now consider a crystalline solid. The atoms are locked in a three-dimensional lattice, each one oscillating around its equilibrium position. As an atom moves away from its center point, a restoring force pulls it back. Kinetic and potential energy exchange continuously while total energy is conserved. According to the equipartition theorem, the total energy divides equally between kinetic and potential contributions across all available degrees of freedom. This means that when energy is added to a crystalline solid, only half goes into kinetic energy — and thus into temperature rise — 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. The heat capacity per mole at constant volume for a crystalline solid is:
Cv = 3R crystalline solid
This is the Dulong-Petit Law, identified by Pierre Dulong and Alexis Petit in 1819. Atomic weight is again irrelevant. What matters is structure not mass.
The physical basis for the equipartition of energy
Atoms don’t discriminate when they collide. Energy is conserved at every collision, and the outcome — whether it goes into faster translational motion, more rotational spinning, or stronger vibration — depends only on the geometry of the collision itself, not on which mode receives it. The atom doesn’t know or care. All modes of motion are equally accessible, and over vast numbers of collisions, energy distributes itself evenly among them. This is the physical basis of the equipartition theorem: each independent mode of motion — each degree of freedom — receives the same average energy, ½kT, where k is Boltzmann’s constant and T is temperature.
For a monatomic gas, three translational degrees of freedom are available — motion in x, y, and z — giving a total energy per atom of (3/2)kT, purely kinetic. For a diatomic molecule, rotation adds two more degrees of freedom, and vibration — which stores energy in both kinetic and potential form — adds two more still, for a total of seven. Each mode contributes ½kT. This is why polyatomic molecules have higher molar heat capacities than monatomic ones: when energy enters the system through collisions, it distributes itself across more modes. Less of it goes into translational kinetic energy — and thus into temperature — for the same total energy input. More energy is required to move the temperature dial. Heat capacity rises accordingly.
The underlying reason equipartition works is that the same energy and momentum conservation laws govern every collision regardless of what kind of atom is involved or what molecule it belongs to. The laws don’t change at the moment of impact. The result is that, given enough collisions, the average energy in each degree of freedom must equalize. Nature enforces this through sheer statistical weight — any distribution that concentrates energy in one mode at the expense of others is overwhelmingly less probable than the uniform one.
Degrees of Freedom and the Ratio γ = Cp/Cv
The pattern established above — more ways to store energy means higher heat capacity — can be made precise through the concept of degrees of freedom. A degree of freedom is simply one independent way a molecule can hold energy: 3 directions for translational motion (x, y, z) along with varying DOF’s for rotational and vibrational modes. The equipartition theorem assigns ½kT of energy to each degree of freedom at temperature T.
For a monatomic gas, there are three degrees of freedom — translation in the x, y, and z directions. Total internal energy is U = N(3/2)kT, giving Cv = 1.5R as established above.
This generalizes. For a system with f degrees of freedom, U = N(f/2)kT, and:
Cv = (f/2)R
Cp = Cv + R (see [1] below for derivation)
γ = Cp/Cv = (f + 2) / f
This single relation predicts heat capacity for any system once its degrees of freedom are known.
A monatomic gas (f = 3) gives γ = 5/3 ≈ 1.67. A diatomic gas with translation and rotation but no vibration (f = 5 — three translational plus two rotational, since rotation about the bond axis itself carries negligible energy) gives γ = 7/5 = 1.40. A diatomic gas with translation, rotation, and vibration (f = 7 — vibration contributes two degrees of freedom, kinetic and potential) 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, predicting γ = 1.33. The experimental value was 1.408 — a discrepancy Maxwell considered serious enough that he wrote it might be grounds to abandon 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 connecting the two atoms involves 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.
This same framework explains why vibrational degrees of freedom are often absent from room-temperature heat capacity data. As discussed above, vibrational energy levels are quantized, and at ordinary temperatures there is often insufficient thermal energy to populate the first excited vibrational state. The vibrational degrees of freedom are “frozen out,” and γ reflects only the translational and rotational contributions. As temperature rises, vibrational modes progressively unfreeze and γ shifts accordingly.
A Numerical Check: Helium, Krypton, and Iron
The claim that heat capacity per atom depends on structure rather than mass can be checked directly against data. Helium and krypton are both monatomic noble gases, differing in atomic mass by a factor of 21 — helium at 4 g/mol, krypton at 84 g/mol. Their measured molar heat capacities at constant pressure are 20.78 J/mol·K and 20.95 J/mol·K respectively [3] — essentially identical, as the framework above predicts. Both are monatomic gases with f = 3; atomic mass plays no role in Cv per mole.
Now compare helium gas to solid iron. Helium, monatomic gas, has Cv = 1.5R per atom. Iron, a crystalline solid, has Cv ≈ 3.0R per atom [4] — almost exactly double, consistent with the Dulong-Petit value of f = 6 for a crystalline solid (three kinetic plus three potential degrees of freedom from lattice vibration). Despite iron’s atomic mass being fourteen times that of helium, the heat capacity per atom is governed entirely by the number of degrees of freedom available — three for a free monatomic 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 — how many ways an atom or molecule can hold energy — rather than a property of mass.
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 energy only as translational kinetic energy — three directions of motion, three degrees of freedom. A crystalline solid stores energy as both kinetic and potential energy in three directions — six degrees of freedom total. The more ways a system can absorb energy without registering it as a temperature rise, the higher its heat capacity.
This explains why polyatomic gas molecules have higher molar heat capacities than monatomic gases. As established in the temperature section above, rotation and vibration do not contribute to temperature, which is defined by translational kinetic energy alone. But they do absorb energy. Energy added to a polyatomic gas can go into rotation or vibration in addition to translational speed; thus, molar heat capacities are high for such gases. It takes more heat to move the temperature needle.
At very low temperatures, quantum mechanics introduces an important correction. Energy transitions between vibrational and rotational states are quantized — only specific energy increments are permitted. Below a certain temperature there is insufficient thermal energy available to excite these transitions. The modes become frozen out and cease contributing to heat capacity. Heat capacity drops below the Dulong-Petit value and approaches zero at absolute zero. Einstein showed in 1907, with further refinement by Debye in 1912, how quantum effects explain the observed departures from Dulong-Petit at low temperatures. This is where the classical atomic picture reaches its boundary and quantum mechanics takes over.
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 changing the internal energy of the system — kinetic and potential contributions as described above. At constant pressure, the system is also free to expand as it heats, doing PdV work against the surroundings. That expansion costs energy. Thus, more energy must be added at constant pressure to achieve the same temperature rise as at constant volume and so for any substance, Cp > Cv. For an ideal gas the difference is exactly:
Cp − Cv = R per mole, ideal gas
This difference between Cp and Cv for an ideal gas became the basis for Robert Mayer’s discovery of heat-work equivalence. He equated the difference between Cp and Cv (heat) to the work done by the expansion (work).
This distinction carries direct consequences throughout thermodynamics. The constant-pressure calorimeter measures ΔH using Cp. The bomb calorimeter measures ΔU using Cv. Absolute entropy is calculated by integrating Cp/T from absolute zero — Cp rather than Cv because entropy is most naturally measured at constant pressure, the condition under which most chemical and physical processes occur. And the efficiency of adiabatic compression, the speed of sound in a gas, and the behavior of a turbine all depend on the ratio Cp/Cv. Heat capacity itself is not a secondary property. It is the quantitative link between atomic motion and a wide range of thermodynamic properties.
References
[1] For an ideal gas, the first law of thermodynamics gives Q = dU + PdV. At constant volume, dV = 0, so all of the added heat increases the internal energy and dU = CvdT. At constant pressure, the added heat must both increase the internal energy and provide the work required for expansion, so dQ = CpdT = CvdT+PdV. Using the ideal gas law for one mole, PV = RT, differentiation at constant pressure yields PdV=RdT. Substituting this into the first-law expression gives CpdT=CvdT+RdT, and dividing through by dT produces:
Cp = Cv + R
This result shows that the heat capacity at constant pressure exceeds the heat capacity at constant volume by exactly R because, in addition to raising the gas’s internal energy, some of the added heat must perform 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.
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