B.3 Gas laws

UNIT SYLLABUS

SL/HL 6 hours

The ideal gas is physics' most successful simplification: model molecules as point particles in random motion with no interactions except collisions, and the messy behaviour of $10^{23}$ particles collapses into one clean equation, $PV = nRT$. This unit connects the macroscopic (pressure, volume, temperature — the empirical gas laws) to the microscopic (kinetic theory, where pressure comes from molecular impacts and $P = \frac{1}{3}\rho v^2$). You will also learn when the model breaks: at high pressures and low temperatures, real molecules' size and mutual attractions start to matter.

Guiding Questions

? How does the motion of molecules explain the macroscopic properties of a gas?
? Under what conditions does the ideal gas model approximate real gases well?

Isotherms of an ideal gas on a p–V diagram: each curve shows PV = constant at one temperature, with higher temperatures lying further from the origin.

✓ What the IB expects you to master

Define pressure as $P = F/A$ for a force perpendicular to a surface.
Use the amount of substance $n = N/N_A$ .
Describe the ideal gas as a model system built from kinetic theory assumptions, used to approximate real gases.
Derive and use the empirical laws (constant $P$ , $V$ or $T$ ) combined as $\frac{PV}{T} = \text{constant}$ .
Apply the ideal gas equations $PV = Nk_BT$ and $PV = nRT$ .
Explain pressure microscopically — molecular momentum changes at the walls — leading to $P = \frac{1}{3}\rho v^2$ .
Relate internal energy of an ideal monatomic gas to temperature: $U = \frac{3}{2}Nk_BT = \frac{3}{2}nRT$ .
State the conditions (low pressure, low density, moderate temperature) under which real gases behave ideally.

1 Key Formulas

Pressure

P = \frac{F}{A}

Amount of substance

n = \frac{N}{N_{A}}

Ideal gas law

PV = nRT = Nk_{B}T

Combined gas law

\frac{P_{1}V_{1}}{T_{1}} = \frac{P_{2}V_{2}}{T_{2}}

Pressure (kinetic theory)

P = \tfrac{1}{3}\rho v^{2}

Internal energy (monatomic)

U = \tfrac{3}{2}Nk_{B}T = \tfrac{3}{2}nRT

2 Exam Preparation & Topic Explanations

Gas law problem technique

Before any algebra: convert temperatures to kelvin, decide what stays constant, and choose between the two-state form ( $P_1V_1/T_1 = P_2V_2/T_2$ ) and the single-state form ( $PV = nRT$ ). Two-state problems never need $R$ ; single-state problems usually do.

Graph literacy scores marks: isotherms on a p–V diagram, straight lines through the origin for $P$ – $T$ and $V$ – $T$ (kelvin only), and what "further from origin" means for temperature.

Pro Exam Strategy

Kelvin, always. A single Celsius slip invalidates the whole calculation.
The mass of gas is fixed unless the question says otherwise — a leaking container changes $n$ .
$P = \frac{1}{3}\rho v^2$ uses the mean square speed; take a square root at the end for $v_{rms}$ .
Extrapolating $V$ – $T$ lines to zero volume locates absolute zero — a classic data-analysis question.

3 MCQ Practice

Q1. A fixed mass of ideal gas is heated at constant volume. Its pressure rises because the molecules:

Expand and take up more space
Collide with the walls more often and with greater momentum change
Repel each other more strongly
Increase in mass

Q2. An ideal gas at 27 °C is heated at constant pressure until its volume doubles. Its new temperature is:

54 °C
600 K
327 °C
150 K

Q3. A real gas behaves most like an ideal gas at:

High pressure and low temperature
High pressure and high temperature
Low pressure and high temperature
Low pressure and low temperature

4 Short Answer Questions

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