Boyle's law, Charles's law and the pressure law: WASSCE Physics

Pump air into a football and it gets firmer; heat an aerosol can and it may explode; squeeze a sealed syringe and the air inside resists more and more. Gases obey three simple laws linking their pressure, volume and temperature.

Learning objectives

By the end of this lesson you should be able to:

state Boyle's law, Charles's law and the pressure law
explain absolute zero and why the Kelvin scale is used in gas calculations
solve problems using $pV = \text{constant}$ , $V/T = \text{constant}$ and $p/T = \text{constant}$
interpret the graphs of the gas laws

The variables

A fixed mass of gas is described by its pressure $p$ , volume $V$ and absolute temperature $T$ (in kelvin). Each gas law keeps one variable constant and links the other two.

Boyle's law

Definition

Boyle's law: the volume of a fixed mass of gas is inversely proportional to its pressure, provided the temperature remains constant.

pV = \text{constant} \qquad p_1V_1 = p_2V_2

\begin{gathered}pV = \text{constant} \\[6pt] p_1V_1 = p_2V_2\end{gathered}

Worked example

A bicycle pump contains 200 cm³ of air at atmospheric pressure (100 kPa). The outlet is blocked and the air is compressed slowly (constant temperature) to 80 cm³. Find the new pressure.

p_2 = \dfrac{p_1V_1}{V_2} = \dfrac{100 \times 200}{80} = 250\ \text{kPa}

\begin{aligned}p_2 &= \dfrac{p_1V_1}{V_2} \\ &= \dfrac{100 \times 200}{80} \\ &= 250\ \text{kPa}\end{aligned}

Kinetic explanation: halving the volume crowds the same number of molecules into half the space, so they strike each unit area of the walls twice as often; the pressure doubles.

Charles's law

Definition

Charles's law: the volume of a fixed mass of gas is directly proportional to its absolute temperature, provided the pressure remains constant.

\dfrac{V}{T} = \text{constant} \qquad \dfrac{V_1}{T_1} = \dfrac{V_2}{T_2}

\begin{gathered}\begin{aligned}\dfrac{V}{T} &= \text{constant}\end{aligned} \\[6pt] \dfrac{V_1}{T_1} = \dfrac{V_2}{T_2}\end{gathered}

Worked example

A balloon holds 3.0 litres of air at 27 °C. What is its volume at 77 °C, at the same pressure?

Convert to kelvin: $T_1 = 300$ K, $T_2 = 350$ K.

V_2 = V_1\dfrac{T_2}{T_1} = 3.0 \times \dfrac{350}{300} = 3.5\ \text{litres}

\begin{aligned}V_2 &= V_1\dfrac{T_2}{T_1} \\ &= 3.0 \times \dfrac{350}{300} \\ &= 3.5\ \text{litres}\end{aligned}

Common mistake

Always use kelvin in gas law calculations. Using Celsius gives nonsense: doubling from 20 °C to 40 °C does not double the volume, because the absolute temperature only rises from 293 K to 313 K.

The pressure law

Definition

Pressure law (Gay-Lussac's law): the pressure of a fixed mass of gas is directly proportional to its absolute temperature, provided the volume remains constant.

\dfrac{p}{T} = \text{constant} \qquad \dfrac{p_1}{T_1} = \dfrac{p_2}{T_2}

\begin{gathered}\begin{aligned}\dfrac{p}{T} &= \text{constant}\end{aligned} \\[6pt] \dfrac{p_1}{T_1} = \dfrac{p_2}{T_2}\end{gathered}

Worked example

A car tyre has a pressure of 200 kPa at 27 °C in the morning. After a long drive in the sun its temperature is 57 °C. Find the new pressure (volume constant).

p_2 = 200 \times \dfrac{330}{300} = 220\ \text{kPa}

\begin{aligned}p_2 &= 200 \times \dfrac{330}{300} \\ &= 220\ \text{kPa}\end{aligned}

Absolute zero

Three graphs for a fixed mass of gas: pressure against volume is a curve; pressure against one over volume is a straight line through the origin; volume against temperature in Celsius is a straight line that meets the temperature axis at minus 273 degrees Celsius

Figure 1. Graphs of the gas laws. WASSCE Tutor diagram.

If the volume–temperature (or pressure–temperature) graph of any gas is extended backwards, it meets the temperature axis at about −273 °C. At this temperature the volume (or pressure) would be zero.

Definition

Absolute zero (0 K, −273 °C) is the lowest possible temperature, at which the molecules of a substance have their minimum kinetic energy. The Kelvin scale starts at absolute zero: $T = \theta + 273$ .

Graphs to recognise

Law	Graph	Shape
Boyle	$p$ against $V$	curve (hyperbola)
Boyle	$p$ against $\dfrac{1}{V}$	straight line through the origin
Boyle	$pV$ against $p$	horizontal line
Charles	$V$ against $T$ (K)	straight line through the origin
Charles	$V$ against θ (°C)	straight line meeting the θ-axis at −273 °C
Pressure law	$p$ against $T$ (K)	straight line through the origin

Key points

Boyle (constant $T$ ): $p_1V_1 = p_2V_2$ .
Charles (constant $p$ ): $V_1/T_1 = V_2/T_2$ .
Pressure law (constant $V$ ): $p_1/T_1 = p_2/T_2$ .
Use kelvin: $T = \theta + 273$ . Absolute zero is −273 °C.

WASSCE examination focus

WAEC tests this

State each gas law with its condition (fixed mass, and what is kept constant), solve problems (converting to kelvin), sketch or identify the graphs, and explain a law using the kinetic theory. Objective items often show a graph and ask which law it represents.

Quick check

Which quantity is kept constant in Boyle's law?
A gas at 100 kPa occupies 6 litres. What volume does it occupy at 300 kPa (same temperature)?
Convert −73 °C to kelvin.

Answers: 1. Temperature (and the mass of gas). 2. 2 litres. 3. 200 K.

Boyle's law, Charles's law and the pressure law

Learning objectives

The variables

Boyle's law

Charles's law

The pressure law

Absolute zero

Graphs to recognise

Key points

WASSCE examination focus

Quick check

Keep going with gas laws

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