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The Ideal Gas

Source lecture(s): SC133 Lec 28

Intuition

Model a gas as a swarm of tiny billiard balls: pointlike, non-interacting except for elastic bounces. Absurdly simple — yet it predicts real gas behavior superbly at ordinary conditions, and it does something profound: it connects the world of thermometers and pressure gauges to the invisible world of molecular collisions. Pressure is drumfire — trillions of wall impacts per second — and temperature is the average violence of the drumming.

The equation of state

\[\boxed{\,PV = nRT = Nk_BT\,}\]

\(n\) moles (\(R = 8.314\,\text{J/(mol·K)}\)) or \(N\) molecules (\(k_B = 1.38\times10^{-23}\,\text{J/K}\)); \(T\) in kelvin, always. Reference: equation page. One mole at 0 °C, 1 atm: 22.4 L — the same for any ideal gas, which is the miracle: the law doesn't care what the molecules are.

The old empirical laws are corollaries: Boyle (\(PV\) const at fixed \(T\)), Charles (\(V \propto T\) at fixed \(P\)), Gay-Lussac (\(P \propto T\) at fixed \(V\)).

Kinetic theory: pressure from collisions

One molecule of mass \(m\), speed component \(v_x\), bouncing between walls of a box of side \(L\): each wall hit transfers momentum \(2mv_x\) (impulse) every \(2L/v_x\) seconds. Summing over \(N\) molecules and averaging:

\[PV = \tfrac13 N m \overline{v^2}\]

Comparing with \(PV = Nk_BT\) yields the golden dictionary entry:

\[\boxed{\,\overline{K}_\text{trans} = \tfrac12 m\overline{v^2} = \tfrac32 k_B T\,}\]

Temperature is average translational kinetic energy. Absolute zero = zero jiggle (classically). Every gas at the same \(T\) has the same average molecular KE — lighter molecules simply move faster (molecular speeds).

Internal energy and heat capacities

Equipartition: each quadratic degree of freedom carries \(\tfrac12 k_BT\) per molecule.

Gas \(f\) \(E_\text{int}\) \(C_V\) \(\gamma = C_P/C_V\)
Monatomic (He, Ar) 3 \(\tfrac32 nRT\) \(\tfrac32 R\) 5/3
Diatomic (N₂, O₂), ~300 K 5 \(\tfrac52 nRT\) \(\tfrac52 R\) 7/5

with \(C_P = C_V + R\) (constant pressure also pays for expansion work — the first law in action). The ratio \(\gamma\) governs adiabatic processes and the speed of sound.

Worked example: air in a dive tank

A 12 L tank holds air at 200 bar, 20 °C. How many moles — and what volume at the surface (1 bar)?

\(n = PV/RT = \frac{2\times10^7\times0.012}{8.314\times293} \approx 98.5\,\text{mol}\) (≈2.9 kg of air). At 1 bar, same \(T\): \(V = nRT/P \approx 2.4\,\text{m}^3\) — the tank "contains" 2400 surface-litres, which is how dive durations are computed.

When ideality fails

Near condensation (molecules attract) and at high density (molecules have size), use van der Waals corrections. And in a plasma, "ideal" gets redefined — long-range Coulomb forces change the game entirely.

Common mistakes

  • Celsius in the gas law. \(PV = nRT\) demands kelvin; 20 °C is 293 K, not 20.
  • "Heavier gases are hotter/faster." Same \(T\) ⇒ same average KE; heavier means slower, not more energetic.
  • Confusing \(C_V\) and \(C_P\) — constant-pressure heating costs more because the gas also does work.
  • Applying ideal-gas results to liquids — no free flight between collisions, no ideal-gas law.

Knowledge graph position

Prerequisites: Collisions & impulse, Temperature. Leads to: Molecular speeds, Second law, plasma physics.

Quiz

Q1 (computational). What is the average translational KE of any gas molecule at 300 K, and the rms speed of N₂ (\(m = 4.65\times10^{-26}\) kg)?

Answer

\(\overline K = \tfrac32 k_BT = 6.2\times10^{-21}\,\text{J}\); \(v_\text{rms} = \sqrt{3k_BT/m} = \sqrt{3(1.38\times10^{-23})(300)/4.65\times10^{-26}} \approx 517\,\text{m/s}\) — faster than sound, as it must be (sound is carried by these molecules).

Q2 (conceptual). A sealed rigid flask is heated from 300 K to 600 K. What happens to pressure, and to the average molecular speed?

Answer

\(P\) doubles (\(P \propto T\) at fixed \(V, N\)); \(v_\text{rms} \propto \sqrt T\) grows by \(\sqrt2 \approx 1.41\) — pressure rises both because impacts are harder and more frequent.

Q3 (multiple choice). Two flasks, same \(T\), \(V\), \(P\): one holds helium, one holds nitrogen. Which contains more molecules? (a) He (b) N₂ (c) equal

Answer

(c). Avogadro's insight, embedded in \(PV = Nk_BT\): the count is fixed by \(P, V, T\) regardless of species.