The energy stored in a gas
If every molecule carries an average \tfrac{3}{2}k_BT of translational kinetic energy, then the whole gas's internal energy is just that summed over all N molecules. For a monatomic ideal gas (helium, argon — single atoms with only translational motion) the result is beautifully simple.
The internal energy of a monatomic ideal gas depends only on its temperature and amount — not at all on its volume or pressure. Temperature alone sets the energy.
This immediately predicts how hard a gas is to heat, its molar heat capacity. Diatomic gases like N₂ and O₂ store extra energy in rotation — two more degrees of freedom — so they need more heat per degree, exactly as equipartition foretells.
With f degrees of freedom, C_V = (f/2)R: so C_V = (3/2)R for monatomic (γ = 5/3) and (5/2)R for diatomic at room temperature (γ = 7/5 = 1.4). These predicted values match measured gases remarkably well.
Worked example: helium versus nitrogen
Problem. A container holds a mixture of helium (M = 4.0 g/mol) and nitrogen (M = 28 g/mol) at the same temperature, 300 K. (a) Which molecules move faster, and by how much? (b) If both gases can leak through a tiny pinhole, which escapes faster?
- Key idea: at the same temperature both gases have the same average kinetic energy ½m⟨v²⟩ = (3/2)k_BT. Equal energy, unequal mass — so the lighter gas must move faster.
- Take the ratio of rms speeds: v_He / v_N₂ = √(M_N₂ / M_He) = √(28/4) = √7 ≈ 2.65. Helium molecules are 2.65 times faster.
- Put numbers on it: v_rms(He) = √(3·8.314·300 / 0.004) ≈ 1370 m/s, versus ≈ 517 m/s for N₂ from the previous guide. The ratio checks out: 1370/517 ≈ 2.65.
- For the pinhole (effusion), the escape rate is proportional to molecular speed, so it too scales as 1/√M — this is Graham's law. Helium effuses √7 ≈ 2.65 times faster than nitrogen.
- Real-world check: this is exactly why helium balloons deflate within a day (its small, fast atoms slip out through the rubber), and how uranium isotopes were separated by repeated gaseous diffusion — tiny mass differences, amplified by 1/√M, over thousands of stages.
When gases stop being ideal
The ideal gas law is astonishingly good — but we promised honesty about its limits. Squeeze a gas hard or cool it toward condensing, and two neglected facts reassert themselves: molecules do occupy space, and they do attract one another (the weak forces that ultimately let gases become liquids). Johannes van der Waals patched the ideal law to include both, earning a Nobel Prize for it.
The van der Waals equation. The constant b subtracts the volume the molecules themselves take up; the term a n²/V² adds back the pressure lost to intermolecular attraction. Set a = b = 0 and the ideal gas law returns.
The bridge to thermodynamics and beyond
Return to the box one last time, now armed with meaning. Every slider is a piece of the theory you have built: temperature is the molecules' average kinetic energy, pressure is their collective drumming on the walls, and PV = nRT ties the two together. What looked like an animation in the first guide is now a working laboratory.
Where does this lead? Track the flow of energy in and out and you reach the first law of thermodynamics — kinetic theory is where its 'internal energy' comes from. Ask why heat flows from hot to cold but never back, and the fast-and-slow speed distribution becomes the seed of entropy and the second law of thermodynamics. Push the counting of molecular states further and you arrive at statistical mechanics, the deep engine beneath all of thermal physics. You began by asking what air is; you end holding the thread that ties the microscopic world of molecules to the macroscopic laws of heat.