Boyle's, Charles's and Pressure Laws

Historically, the ideal gas equation pV = nRT did not spring into existence fully formed. It grew out of three simpler experimental relationships discovered over the course of the 17th, 18th and 19th centuries. Each describes the behaviour of a fixed amount of gas when one of the three variables — pressure, volume or temperature — is held constant. Together, they encapsulate the observed macroscopic behaviour of gases across a huge range of conditions.

Module 5.1.3 of the OCR A-Level Physics A specification (H556) requires you to know these three experimental gas laws, to use them in calculations, and to see how they combine into the single ideal gas equation that we develop in the next lesson.

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Setting the Scene

All three laws apply to a fixed amount of gas — the same number of molecules throughout the experiment — and work only for ideal (or near-ideal) behaviour. Real gases deviate from the laws at high pressure (where the molecules are packed closely enough for their own size to matter) and at low temperature (where intermolecular forces become significant). Under normal laboratory conditions — a few atmospheres, temperatures from about 100 K up to several hundred — the approximation is excellent.

The three variables are:

• Pressure p, measured in pascals (Pa = N m⁻²). Atmospheric pressure at sea level is roughly 1.01 × 10⁵ Pa ≈ 100 kPa.
• Volume V, measured in cubic metres (m³). One litre = 10⁻³ m³; one cm³ = 10⁻⁶ m³.
• Temperature T, measured in kelvin (not Celsius). This is crucial. All the gas laws here fail if you use Celsius.

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Boyle's Law (constant T)

Robert Boyle, in 1662, published experimental results showing that at constant temperature, the volume of a fixed mass of gas is inversely proportional to its pressure.

Boyle's Law: For a fixed mass of ideal gas at constant temperature, p ∝ 1/V, or equivalently pV = constant.

If you have an initial state (p_1, V_1) and a final state (p_2, V_2) at the same temperature, then

p_1 V_1 = p_2 V_2

Graphical forms

• A plot of p against V is a hyperbola (a curve that asymptotically approaches both axes).
• A plot of p against 1/V is a straight line through the origin with slope equal to pV = nRT.
• A plot of pV against V (or against p) is a horizontal line.

The hyperbola at constant temperature is called an isotherm; higher temperatures correspond to isotherms further from the origin.

Physical picture

Boyle's law is easy to understand at the molecular level: if you halve the volume available to a gas, you double the frequency with which molecules strike the walls of the container, so the pressure doubles. You will derive this quantitatively in Lesson 9.

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Worked Example 1: Boyle's Law

A bicycle pump contains 0.20 L of air at atmospheric pressure (1.01 × 10⁵ Pa). If the piston is pushed in until the volume is 0.050 L (the valve stays shut), what is the new pressure? Assume the temperature does not change.

p_1 V_1 = p_2 V_2
p_2 = p_1 V_1 / V_2
    = (1.01 × 10⁵)(0.20) / (0.050)
    = (1.01 × 10⁵)(4)
    = 4.04 × 10⁵ Pa
    ≈ 4.0 × 10⁵ Pa

The volume was reduced by a factor of 4, so the pressure rose by the same factor, to about 4 atmospheres. (Of course, in reality a bicycle pump does heat up slightly during compression; Boyle's law strictly applies only if the compression is slow enough for heat to dissipate, keeping T constant.)

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Charles's Law (constant p)

Jacques Charles, around 1787 (the results were published much later by Gay-Lussac), discovered that at constant pressure, the volume of a fixed mass of gas is directly proportional to its absolute temperature.

Charles's Law: For a fixed mass of ideal gas at constant pressure, V ∝ T, or equivalently V/T = constant.

Thus for two states of the gas at the same pressure,

V_1 / T_1 = V_2 / T_2

Absolute zero — an empirical consequence

Charles found that the volume of a gas at constant pressure decreases linearly as the temperature falls, and if you extrapolate the straight line back far enough, it crosses zero volume at around -273 °C. This is the first experimental hint of absolute zero: the temperature at which the volume of an ideal gas would (if it stayed gaseous) become zero.

Of course, real gases liquefy long before this — nitrogen, for example, liquefies at -196 °C. But the extrapolation is still valid as an ideal-gas prediction and gives exactly the right value of absolute zero as the zero of the Kelvin scale.

Graphical forms

• A plot of V against T (in kelvin) is a straight line through the origin with slope nR/p.
• A plot of V against θ (in Celsius) is a straight line with a positive intercept on the volume axis; if extrapolated backwards, it crosses the temperature axis at about -273 °C.

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Worked Example 2: Charles's Law

A helium balloon has a volume of 2.0 L at 20 °C (293 K). It is left in a warm car where the temperature rises to 45 °C (318 K). Assuming the pressure stays constant and the balloon does not burst, find the new volume.

V_1 / T_1 = V_2 / T_2
V_2 = V_1 T_2 / T_1
    = (2.0)(318) / 293
    ≈ 2.17 L

So the balloon swells by about 8.5% — a clear illustration of why balloons left in hot cars can pop.

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The Pressure Law (constant V)

Guillaume Amontons in 1702, and later more precisely Joseph Louis Gay-Lussac in 1809, found that at constant volume, the pressure of a fixed mass of gas is directly proportional to its absolute temperature.

Pressure Law (Gay-Lussac's Law): For a fixed mass of ideal gas at constant volume, p ∝ T, or equivalently p/T = constant.

Thus for two states at the same volume,

p_1 / T_1 = p_2 / T_2

This law underlies the operation of the constant-volume gas thermometer, which is the most accurate standard thermometer known and is used to define the temperatures of fixed points on the international temperature scale.

Graphical forms

• A plot of p against T (in kelvin) is a straight line through the origin with slope nR/V.

Just as with Charles's law, the extrapolation to p = 0 gives T = 0 K = -273.15 °C, providing another experimental path to absolute zero.

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Worked Example 3: Pressure Law

A car tyre is inflated to p_1 = 2.4 × 10⁵ Pa on a cool morning at 5 °C (278 K). After some driving, the tyre warms up to 50 °C (323 K). Assuming the tyre volume does not change significantly, what is the new pressure?

p_1 / T_1 = p_2 / T_2
p_2 = p_1 T_2 / T_1
    = (2.4 × 10⁵)(323) / 278
    ≈ 2.79 × 10⁵ Pa

So the pressure rises by about 16%. Tyre pressures should always be measured when the tyres are cold, before driving, for this reason.

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Combining the Three Laws