Heat Capacities of Gases
When you heat a gas, you're adding energy — but where does that energy go? For a simple monatomic gas like helium, all added energy goes into making atoms move faster (translational kinetic energy). But for a diatomic molecule like oxygen or nitrogen, energy can also spin the molecule or make it vibrate. This extra 'storage space' for energy means diatomic gases require more heat to raise their temperature by the same amount — they have a higher heat capacity. Understanding this difference is one of the great triumphs of 19th-century physics. Heat capacity tells us how much energy a substance needs per degree of temperature rise. For gases, we distinguish two important versions: the heat capacity at constant volume (Cᵥ) and the heat capacity at constant pressure (Cₚ). These differ because, at constant pressure, an expanding gas does work on its surroundings, so extra energy must be supplied on top of what's needed to raise the temperature. The ratio γ = Cₚ/Cᵥ, called the adiabatic index, appears throughout thermodynamics in equations for sound speed, engine efficiency, and atmospheric physics. The key to predicting heat capacities from first principles is the Equipartition Theorem: at temperature T, each independent quadratic degree of freedom in the energy expression carries an average energy of ½k_BT, where k_B is Boltzmann's constant. Count the degrees of freedom for each type of molecule, multiply by ½k_BT, and you immediately get the internal energy per molecule — and therefore the heat capacity. This elegant connection between molecular structure and bulk thermodynamic properties is a cornerstone of statistical mechanics.
- What is a Degree of Freedom?
- Monatomic Gases (e.g., He, Ar, Ne)
- Diatomic Gases (e.g., N₂, O₂, H₂) at Moderate Temperatures
- Polyatomic Gases (e.g., CO₂, H₂O, CH₄)
- The Mayer Relation: Cₚ − Cᵥ = R
What is a Degree of Freedom?
A degree of freedom is any independent way a molecule can store energy. Translational motion in the x, y, and z directions gives 3 translational degrees of freedom for every molecule, regardless of its complexity. Think of it as three separate 'slots' in which kinetic energy can sit: ½mv_x², ½mv_y², and ½mv_z². The equipartition theorem guarantees that, on average, each slot holds exactly ½k_BT of energy at temperature T.
Monatomic Gases (e.g., He, Ar, Ne)
A monatomic gas is just single atoms — no bonds, no rotation that carries meaningful energy, no vibration. So f = 3 (pure translation). Total internal energy per mole: U = 3/2 · Nₐk_BT = 3/2 · RT. Heat capacity at constant volume: Cᵥ = 3/2 R ≈ 12.5 J mol⁻¹ K⁻¹. Heat capacity at constant pressure: Cₚ = Cᵥ + R = 5/2 R ≈ 20.8 J mol⁻¹ K⁻¹. The adiabatic index γ = 5/3 ≈ 1.67. Experiments on helium and argon confirm these values with remarkable precision.
Diatomic Gases (e.g., N₂, O₂, H₂) at Moderate Temperatures
A diatomic molecule is like a dumbbell: two atoms joined by a bond. In addition to 3 translational degrees of freedom, the molecule can rotate about two axes perpendicular to the bond axis (rotation about the bond axis itself is negligible because the moment of inertia is essentially zero for point-like atoms). This adds f = 2 rotational degrees of freedom, giving f = 5 total. U = 5/2 RT, so Cᵥ = 5/2 R ≈ 20.8 J mol⁻¹ K⁻¹, Cₚ = 7/2 R ≈ 29.1 J mol⁻¹ K⁻¹, and γ = 7/5 = 1.40. At very high temperatures, the bond can also vibrate (2 extra degrees of freedom: kinetic + potential), pushing Cᵥ toward 7/2 R. At very low temperatures, rotation can 'freeze out' and Cᵥ drops back toward 3/2 R. This temperature-dependence was a great puzzle until quantum mechanics explained it.
Polyatomic Gases (e.g., CO₂, H₂O, CH₄)
Polyatomic molecules have 3 translational degrees of freedom plus up to 3 rotational degrees of freedom (a non-linear molecule like H₂O or CH₄ can rotate about all three axes). This gives f ≥ 6, so Cᵥ ≥ 3R ≈ 24.9 J mol⁻¹ K⁻¹ and γ < 4/3. Additional vibrational modes activate at higher temperatures, so polyatomic gases show a marked rise in Cᵥ with temperature. This is why CO₂ is such an effective greenhouse gas: its many vibrational modes can absorb infrared radiation corresponding to those quantised vibrational energies.
The Mayer Relation: Cₚ − Cᵥ = R
At constant volume, all added heat goes into raising internal energy: dU = nCᵥdT. At constant pressure, the gas expands and does work W = nRΔT (from the ideal gas law PV = nRT). First law: dU = dQ − dW, so dQ = dU + dW = nCᵥdT + nRdT = n(Cᵥ + R)dT. Therefore Cₚ = Cᵥ + R. This Mayer relation is universal for ideal gases — it doesn't depend on the molecular structure. R = 8.314 J mol⁻¹ K⁻¹ is the extra energy needed per mole per kelvin to do the work of expansion against constant external pressure.
The Key Equation
- C_V — Molar heat capacity at constant volume (J mol⁻¹ K⁻¹): Energy needed to raise 1 mole by 1 K with volume fixed.
- C_P — Molar heat capacity at constant pressure (J mol⁻¹ K⁻¹): Energy needed to raise 1 mole by 1 K at constant pressure; includes work of expansion.
- f — Degrees of freedom (dimensionless): Number of independent quadratic energy modes active at the given temperature (3 for monatomic, 5 for diatomic at room temperature, ≥6 for polyatomic).
- R — Universal gas constant (J mol⁻¹ K⁻¹): R = Nₐ k_B = 8.314 J mol⁻¹ K⁻¹.
- \gamma — Adiabatic index (heat capacity ratio) (dimensionless): Ratio Cₚ/Cᵥ; determines how steeply pressure rises during adiabatic compression. γ = 5/3 (monatomic), 7/5 (diatomic), ~4/3 (polyatomic).
Worked Example
Problem. A cylinder contains 2.00 mol of nitrogen gas (N₂) at 300 K. (a) How much heat is needed to raise the temperature by 50 K at constant volume? (b) How much heat is needed for the same temperature rise at constant pressure? Treat N₂ as an ideal diatomic gas at room temperature (f = 5).
Answer. Q_V ≈ 2080 J; Q_P ≈ 2910 J; extra 831 J goes into expansion work.
Origins · 1 minJulius Robert von Mayer and how this was discovered
Mayer was one of the first scientists to articulate the law of conservation of energy. In 1842 he recognised that heat and mechanical work are equivalent forms of energy. He derived the relation Cₚ − Cᵥ = R for ideal gases from purely thermodynamic reasoning — without knowing anything about molecular structure — simply by noting that expanding gas against constant pressure requires extra energy. This 'Mayer relation' is fundamental to thermal physics. Mayer was a physician, not a professional physicist. He got his initial clue about energy conservation from observing that the venous blood of sailors near the tropics was unusually bright red — suggesting the body needed to burn less oxygen to stay warm in a hot climate. This medical observation launched one of physics' most important principles.
Related Concepts
- The Equipartition Theorem — Heat capacities of gases are derived directly from equipartition: each degree of freedom contributes ½k_BT to average energy, so knowing f gives Cᵥ immediately.
- First Law of Thermodynamics — The distinction between Cᵥ and Cₚ — and the Mayer relation — follows directly from the first law applied to constant-volume vs. constant-pressure processes.
- Adiabatic Processes — The adiabatic index γ = Cₚ/Cᵥ determines how pressure and volume change in an adiabatic process (PV^γ = constant), which governs sound propagation and engine cycles.
- Kinetic Theory of Gases — Kinetic theory provides the microscopic foundation: internal energy is the sum of molecular kinetic (and vibrational potential) energies, linking molecular motion directly to heat capacity.