Temperature and Ideal Gases: Complete A Level H2 Physics Cheatsheet
A Level Physics (H2), Chapter 12 · 9478 Topic 12 · Read time: ~10 minutes
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Key Formulas
The thermodynamic temperature scale
The thermodynamic (kelvin) scale has an absolute zero (0 K) and does not depend on the property of any particular substance, unlike a mercury or resistance thermometer.
T/K = T/°C + 273.15. A temperature change of 1 K equals a change of 1 °C.
Trap: Every gas law and kinetic theory equation needs temperature in kelvin. Leaving it in °C is the most common mark lost in this chapter.
The ideal gas equation
An ideal gas obeys pV = NkT (counting particles, k = Boltzmann constant) or pV = nRT (counting moles, R = molar gas constant).
One mole of any substance contains 6.02 × 10²³ particles. N = nN_A, and Nk = nR.
For a fixed amount of gas, pV/T is constant, which covers the empirical gas laws in one equation.
Kinetic theory of gases
Basic assumptions: a very large number of identical particles in random motion; particles have negligible volume compared with the container; no forces between particles except during collisions; collisions are perfectly elastic; collision time is negligible compared with the time between collisions.
Pressure: each particle that bounces off a wall changes momentum, so the wall exerts a force on it and, by Newton's third law, it exerts a force on the wall. Many such collisions per second, averaged, give a steady force per unit area: pressure.
Starting with one particle moving in one dimension, then using ⟨cₓ²⟩ = ⅓⟨c²⟩ for three dimensions, gives pV = ⅓Nm⟨c²⟩.
Combining with pV = NkT: ½m⟨c²⟩ = 3/2 kT. The mean translational kinetic energy depends only on the thermodynamic temperature.
Trick: Two different gases at the same temperature have the same mean kinetic energy per particle. The lighter gas therefore has the higher r.m.s. speed.
Worked Example
A cylinder of volume 0.020 m³ contains helium at 300 K and 1.5 × 10⁵ Pa. k = 1.38 × 10⁻²³ J K⁻¹, mass of a helium atom = 6.64 × 10⁻²⁷ kg.
(a) Calculate the number of helium atoms. [2]
N = pV / kT = (1.5 × 10⁵ × 0.020) ÷ (1.38 × 10⁻²³ × 300)
= 7.25 × 10²³
(b) Calculate the mean translational kinetic energy of an atom. [1]
3/2 kT = 1.5 × 1.38 × 10⁻²³ × 300 = 6.21 × 10⁻²¹ J
(c) Calculate the r.m.s. speed of the atoms. [2]
½m⟨c²⟩ = 6.21 × 10⁻²¹ → ⟨c²⟩ = 2 × 6.21 × 10⁻²¹ ÷ 6.64 × 10⁻²⁷ = 1.87 × 10⁶
c(r.m.s.) = √(1.87 × 10⁶) = 1.37 × 10³ m s⁻¹
Why this question is a good test of the topic: it moves from a macroscopic measurement (p, V, T) to individual particles, the link the kinetic theory exists to make.
Frequently Asked Questions
Why must temperature be in kelvin?
The gas equations depend on absolute temperature, measured from absolute zero. Celsius values give wrong answers.
What is the difference between pV = NkT and pV = nRT?
They are the same equation. NkT counts particles using the Boltzmann constant; nRT counts moles using the molar gas constant. Nk = nR.
What are the assumptions of the kinetic theory?
Many identical particles in random motion, negligible particle volume, no forces except during collisions, perfectly elastic collisions, and negligible collision time.
What does the mean kinetic energy of a gas particle depend on?
Only on the thermodynamic temperature: ½m⟨c²⟩ = 3/2 kT.
How do gas particles exert pressure?
They change momentum when they hit the walls, so they exert a force on the walls. The average force per unit area is the pressure.
Kinetic theory derivations still shaky?
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