Ch 2 · 15 Ideal Gases · A Level Physics 9702
Topic 2 of 3 in Thermal Physics — A Level Physics 9702 — 67 lessons.
Introduction
A gas is billions of tiny molecules flying about and bouncing off the walls. From that simple picture, Newton's laws give the gas pressure, and comparing it with experiment reveals what temperature really is: a measure of the average kinetic energy of the molecules. (Syllabus topic 15: sections 15.1–15.3.)
Key ideas — 15.1 The mole
Amount of substance is an SI base quantity, with base unit the mole (mol).
🧑🏫 Teacher's explanation
Think of a 'mole' exactly like a 'dozen'. If I say I have a dozen eggs, you know I have 12 eggs. In chemistry and physics, atoms are so incredibly tiny that counting them one by one is impossible. Instead, we group them into massive 'bundles' called moles. One mole is just a massive bundle containing 6.02 × 10^23 particles.
📖 Definitions
🔍 Worked example — Easy
Calculate the number of molecules in 0.50 moles of Nitrogen gas (N_2).
🔍 Worked example — Standard
A sample of Helium contains 1.204 × 10^24 atoms. Calculate the number of moles in the sample.
🔍 Worked example — Exam-style
A container holds 4.48 × 10^24 molecules of Oxygen (O_2). Calculate the mass of the gas if the molar mass of O_2 is 32.0 g mol^-1.
⚠️ Common errors
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🎯 Exam question
Key ideas — 15.2 The equation of state
An ideal gas is a gas that obeys
🧑🏫 Teacher's explanation
Imagine a room filled with millions of tiny, bouncy ping-pong balls. If you heat the room, the balls move faster and hit the walls harder, increasing pressure. If you make the room bigger, they hit the walls less often, decreasing pressure. This relationship—where pressure, volume, and temperature are all linked—is the essence of an ideal gas.
📖 Definitions
🔍 Worked example — Exam-style
An ideal gas is kept at constant pressure. Initially, the volume is V_1 = 2.0 m^3 at T_1 = 273 K. If the volume is compressed to V_2 = 0.50 m^3, calculate the new temperature T_2.
⚠️ Common errors
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🎯 Exam question
Key ideas — 15.3 (a) Assumptions of the kinetic theory
The kinetic theory models an ideal gas with these assumptions:
🧑🏫 Teacher's explanation
Imagine a crowded ballroom filled with thousands of tiny, hyperactive bouncy balls. To make the math manageable, we pretend these balls are so small that they don't actually take up any space, and they don't attract or repel each other unless they are actually hitting one another. We also assume that when they hit the walls or each other, they bounce off perfectly without losing any energy.
📖 Definitions
🔍 Worked example — Easy
State the five basic assumptions of the kinetic theory of gases.
🔍 Worked example — Standard
A gas contains N molecules, each of mass m. The mean-square speed of the molecules is langle c^2 rangle. Show that the pressure p is given by p = 1/3 Nmlangle c^2 rangle/V.
🔍 Worked example — Exam-style
A sample of an ideal gas has a mean-square speed langle c^2 rangle = 4.5 × 10^-22 m^2s^-2. Calculate the average translational kinetic energy of a molecule if the temperature is 300 K.
⚠️ Common errors
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🎯 Exam question
Key ideas — 15.3 (b) Pressure from molecular motion: $pV = \tfrac13 Nm\langle c^2\rangle$
Derivation (cube of side l, N molecules of mass m):
🧑🏫 Teacher's explanation
Imagine a room filled with thousands of tiny, hyperactive ping-pong balls bouncing off the walls. Even though each individual collision is tiny, the sheer number of impacts occurring every second creates a steady, constant push against the walls. This 'push' is what we experience as gas pressure.
📖 Definitions
🔍 Worked example — Easy
A sample of an ideal gas contains 2.0 × 10^23 molecules, each with a mass of 4.8 × 10^-26 kg. The mean-square speed of the molecules is 3.0 × 10^5 \text{m}^2\text{s}^{-2}$. Calculate the pressure exerted by the gas in a volume of $0.50$ \text{m}^3$.
🔍 Worked example — Standard
Using the relationship between pV = 1/3Nmlangle c^2 rangle and pV = NkT, deduce the expression for the average translational kinetic energy of a molecule, langle E_k rangle.
🔍 Worked example — Exam-style
A gas has a density rho = 1.20 \text{kg m}^{-3} and an r.m.s. speed of 500 \text{m s}^{-1}$. Calculate the pressure of the gas.
⚠️ Common errors
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🎯 Exam question
Key ideas — 15.3 (c) Root-mean-square speed
The root-mean-square speed is
🧑🏫 Teacher's explanation
Imagine a crowd of people running randomly in a large hall. Some are sprinting, some are strolling, and some are barely moving. If you want to describe the 'typical' speed of a person in that crowd, you can't just take a simple average of their velocities, because for every person running North, there is likely someone running South, which would make the average velocity zero! Instead, we square the speeds to make them all positive, average them, and then take the square root to bring the units back to m/s. This value is the root-mean-square speed, c_r.m.s..
📖 Definitions
🔍 Worked example — Easy
A sample of helium gas has a mean-square speed langle c^2 rangle = 4.0 × 10^22 m^2s^-2. Calculate the root-mean-square speed c_r.m.s..
🔍 Worked example — Standard
Calculate the average translational kinetic energy of a molecule of Nitrogen at a temperature of 300 K.
🔍 Worked example — Exam-style
A gas occupies a volume V at pressure p. It contains N molecules, each of mass m. Show that the root-mean-square speed is c_r.m.s. = √(3pV/Nm).
⚠️ Common errors
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🎯 Exam question
Key ideas — 15.3 (d) Temperature and molecular kinetic energy
Compare pV = tfrac13 Nmlangle c^2rangle with pV = NkT:
🧑🏫 Teacher's explanation
Imagine a room filled with thousands of tiny, hyperactive bouncy balls. These balls are constantly zooming around and slamming into the walls. Each time a ball hits a wall, it gives the wall a tiny 'kick'. Pressure is simply the sum of all these billions of tiny kicks spread over the surface area of the walls.
📖 Definitions
🔍 Worked example — Easy
Calculate the average translational kinetic energy of a single molecule of Helium at a temperature of 300 K. (Use k = 1.38 × 10^-23 J K^-1)
🔍 Worked example — Standard
A sample of Nitrogen gas has an average translational kinetic energy per molecule of 7.5 × 10^-21 J. Calculate the absolute temperature of the gas.
🔍 Worked example — Exam-style
An ideal gas is contained in a vessel. The pressure p and volume V are related to the number of molecules N and their mass m by pV = 1/3Nmlangle c^2 \rangle. If the temperature is doubled, what happens to the root-mean-square speed c_rms of the molecules? (Assume mass m remains constant)
⚠️ Common errors
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🎯 Exam question
(a) Show that the average translational kinetic energy of a molecule is 3/2kT, given that pV = 1/3Nmlangle c^2 \rangle and pV = NkT. [2] (b) Calculate the root-mean-square speed of a molecule of Neon (m = 3.35 × 10^-26 kg) at a temperature of 273 K. [2] [4 marks]
Worked example
Example 1: equation of state. Find the number of moles and molecules in a 0.020 m³ cylinder of gas at 2.0 × 10^5 Pa and 300 K.
🔍 Worked example — Easy
Find the number of moles n and the number of molecules N in a 0.020 m^3 cylinder of gas at a pressure of 2.0 × 10^5 Pa and a temperature of 300\ ext K. Use R = 8.31 J K^-1 mol^-1 and N_A = 6.02 × 10^23 mol^-1.
🔍 Worked example — Standard
Calculate the root-mean-square speed c_r.m.s. of helium atoms at 300 K. The mass of one helium atom is m = 6.64 × 10^-27 kg. Use k = 1.38 × 10^-23 J K^-1.
🔍 Worked example — Exam-style
A gas is held at a constant volume of 0.50 m^3 and a temperature of 27 ^circC. It is heated until the pressure increases from 1.0 × 10^5 Pa to 2.5 × 10^5 Pa. Calculate the final temperature in Celsius.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) Any four from the list in 15.3 (a). (2) nRT = NkT and N = nN_A, so nR = nN_Ak, giving k = R/N_A. (3) Mean = 400 m s⁻¹; langle c^2rangle = (4 + 9 + 16 + 25 + 36) × 10^4/5 = 1.8 × 10^5; c_r.m.s. = 424 m s⁻¹. (4) tfrac32 × 1.38 × 10^-23 × 400 = 8.3 × 10^-21 J. (5) c_r.m.s. ∝ √(T), so it doubles.