Internal energy of an ideal gas: monatomic, diatomic and triatomic gases

Three identical vessels with helium, oxygen and water vapour at the same temperature. The molecules fly and spin, and their colour shows their energy. Give all three gases the same amount of heat and see which one warms up most, then choose a single gas and look at the “ways of moving” that make up the energy of its molecule.

molecule colour shows its energy: blue — below average, red — above → 1–3translational degrees of freedom ↻ 4–6rotational degrees of freedom

Molecular speeds in the scene are proportional to the real ones (1000 m/s → 1 unit per second), rotation is slowed down for clarity. Choose a single gas in the list and an enlarged molecule with numbered degrees of freedom appears next to the vessel.

What U is made of

Internal energy versus temperature: U(T)

Gas

Q amount of heat 1000 Дж

T temperature (set for all) 300 K

ν amount of substance 1.0 моль

Slow down the molecules

Check yourself

The vessels are closed by fixed pistons (V = const), so all the heat goes into internal energy. The heaters are identical: in comparison mode each gas receives the same Q. Vibrations of atoms inside molecules are ignored — they are not excited at ordinary temperatures.

The internal energy of an ideal gas is the total kinetic energy of its molecules: U = (i/2)·ν·R·T. Each degree of freedom of a molecule carries the same energy ½kT, and a monatomic molecule has 3 degrees of freedom, a diatomic one 5 and a triatomic one 6. So at the same temperature U of a triatomic gas is twice that of a monatomic one, and the same amount of heat warms the monatomic gas the most.

KEY FORMULAS

Internal energy of an ideal gas

$$U = \frac{i}{2}\,\nu R T = \frac{i}{2}\,\frac{m}{M}\,R T$$

The internal energy of an ideal gas is proportional to the number of degrees of freedom i of the molecule, the amount of substance ν and the absolute temperature T

▶ Detailed Explanation

Ideal-gas molecules neither attract nor repel at a distance, so all the internal energy is kinetic. U depends only on temperature: in an isothermal process the internal energy of an ideal gas does not change, however the pressure and volume change.

Example: 1 mol of helium at 300 K: U = 1.5·1·8.31·300 ≈ 3.7 kJ. The same amount of oxygen at the same temperature: U = 2.5·8.31·300 ≈ 6.2 kJ, water vapour: U = 3·8.31·300 ≈ 7.5 kJ. On the U(T) graph these are three straight lines with slopes (i/2)·ν·R.

Number of degrees of freedom of a molecule

$$i = i_{\text{пост}} + i_{\text{вращ}}: \qquad 3 = 3 + 0, \quad 5 = 3 + 2, \quad 6 = 3 + 3$$

A monatomic molecule has i = 3, a diatomic (and a linear one such as CO₂) i = 5, a non-linear triatomic or polyatomic one i = 6

▶ Detailed Explanation

Every molecule can move translationally along three axes — three degrees of freedom. Rotational degrees of freedom are added when the molecule has a noticeable moment of inertia about an axis. A monatomic molecule has a negligible moment of inertia, so rotation does not count. A diatomic one has two axes perpendicular to the bond, while about the bond itself the moment of inertia is almost zero — so there are two rotational degrees. A non-linear molecule has three axes of rotation.

In the simulator choose a single gas and look at the enlarged molecule: green arrows 1–3 are translation, orange rings are rotation, and for a diatomic molecule the grey ring around the bond axis is crossed out — it does not count.

Equipartition of energy over degrees of freedom

$$\langle \varepsilon \rangle = \frac{i}{2}\,k T, \qquad \langle \varepsilon_{\text{пост}} \rangle = \frac{3}{2}\,k T = \frac{m_0 \langle v^2 \rangle}{2}$$

Each degree of freedom carries on average the energy ½kT; the mean translational energy is the same for the molecules of all gases at the same temperature

▶ Detailed Explanation

k = 1.38·10⁻²³ J/K is Boltzmann’s constant, R = k·N_A. At 300 K each degree of freedom carries ½kT ≈ 2.07·10⁻²¹ J, and a helium molecule carries on average 6.2·10⁻²¹ J, oxygen 10.4·10⁻²¹ J, water 12.4·10⁻²¹ J.

The translational energy is the same for all, but the speeds differ: v_rms = √(3RT/M). At 300 K helium molecules (M = 4 g/mol) fly at about 1370 m/s and oxygen ones (M = 32 g/mol) at about 480 m/s. In the animation helium is visibly “faster”, although the translational energy of the molecules is the same.

Change of internal energy and heating at V = const

$$\Delta U = \frac{i}{2}\,\nu R\,\Delta T, \qquad Q = \Delta U \;\Rightarrow\; \Delta T = \frac{2Q}{i\,\nu R}, \qquad C_V = \frac{i}{2} R$$

At constant volume the gas does no work, and all the heat received goes into internal energy; the larger i, the smaller the temperature rise

▶ Detailed Explanation

By the first law of thermodynamics Q = ΔU + A. The piston is fixed, the work A = 0, so Q = ΔU. Example: each gas (ν = 1 mol) receives Q = 1000 J. Helium warms by ΔT = 2·1000/(3·8.31) ≈ 80 K, oxygen by 48 K, water vapour by 40 K.

C_V = (i/2)·R is the molar heat capacity at constant volume: 12.5 J/(mol·K) for monatomic gases, 20.8 for diatomic and 24.9 for triatomic ones. The more “ways” a molecule has to store energy, the more heat it takes to raise the temperature by 1 K.

Internal energy in terms of pressure and volume

$$pV = \nu R T \;\Rightarrow\; U = \frac{i}{2}\,p V$$

Using the ideal gas law, the internal energy can be found from the pressure and volume of the gas

▶ Detailed Explanation

This form is handy in problems where p and V are given instead of the temperature. Example: a 10 L cylinder holds helium at 200 kPa. U = 1.5·2·10⁵·10⁻² = 3 kJ. If the same cylinder at the same pressure holds oxygen, U = 2.5·2·10³ = 5 kJ.

It also shows that in isobaric expansion (p = const) the internal energy grows in proportion to the volume, and in isochoric heating (V = const) in proportion to the pressure.

An ideal gas is a model in which molecules do not interact at a distance, so they have no potential energy of interaction. The internal energy of such a gas is simply the sum of the kinetic energies of all its molecules. It depends only on the temperature and the amount of substance: U = (i/2)·ν·R·T, where ν is the number of moles, R = 8.31 J/(mol·K) and i is the number of degrees of freedom of the molecule.

Degrees of freedom are the independent “ways” in which a molecule can move and store energy. A monatomic molecule (He, Ne, Ar) can only fly along the three axes x, y, z: i = 3. A diatomic one (H₂, N₂, O₂) also rotates about two axes perpendicular to the bond. Rotation about the bond axis itself carries almost no energy, so i = 3 + 2 = 5. A non-linear triatomic molecule such as H₂O rotates about all three axes: i = 3 + 3 = 6. Choose a single gas in the simulator — an enlarged molecule appears next to the vessel with arrows for translation and rings for rotation, and every degree of freedom is numbered.

The equipartition theorem says that each degree of freedom carries on average the energy ½kT. Hence the mean energy of a molecule is ⟨ε⟩ = (i/2)·k·T, and the mean translational energy is the same for all gases, (3/2)·k·T. Light helium molecules still fly faster than heavy oxygen molecules: v = √(3RT/M). You can see this both in the animation and in the measurements panel. The “What U is made of” bar splits the energy into blocks of ½νRT: green for translation, orange for rotation.

In comparison mode press “Give Q”. The vessels are closed by fixed pistons (V = const), so all the heat goes into internal energy: Q = ΔU = (i/2)·ν·R·ΔT. In helium the heat goes only into translational motion, and its temperature rises the most. In oxygen and water vapour part of the energy goes into rotation, and they warm up less: ΔT is inversely proportional to i. Beware of a trap: CO₂ is also triatomic, but its molecule is linear and has i = 5. Vibrations of atoms inside the molecule are not excited at ordinary temperatures and are ignored in the school model. The “Check yourself” box asks you to predict the outcome of an experiment and then test it on the model.

Internal energy of an ideal gas: monatomic, diatomic and triatomic gases

KEY FORMULAS

Internal energy of an ideal gas

Number of degrees of freedom of a molecule

Equipartition of energy over degrees of freedom

Change of internal energy and heating at V = const

Internal energy in terms of pressure and volume

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