Lesson 4: Charles's Law

4.1 Do all gases expand equally?

Gases expand when heated and contract when cooled. Do equal volumes of different gases expand by different amounts?

Experiment.

• Two flasks of exactly equal volume, each closed with a stopper through which passes a right-angle glass tube holding a small thread of mercury (2–3 cm long).
• Fill one with oxygen and the other with air or carbon dioxide.
• Place both in the same basin of water and add hot water.
• The two mercury threads move equal distances.

Conclusion. Equal volumes of different gases, under the same pressure, expand by equal amounts for the same rise of temperature; so their coefficient of volume expansion is the same.

Coefficient of volume expansion of a gas at constant pressure (α_v): the increase in volume of unit volume of the gas at 0 °C when its temperature rises by one degree Celsius at constant pressure.

4.2 Charles's apparatus

• A glass tube about 30 cm long and 1 mm wide, closed at one end, with a drop of mercury trapping a certain amount of dry air.
• The tube is fixed along with a thermometer to a graduated ruler inside a cylindrical glass jacket.

Procedure

1. Surround the tube with melting crushed ice until the trapped air is at 0 °C. Because the tube has a uniform cross-section, the length of the air column is a measure of its volume (V₀).
2. Remove the ice water and pass steam through the jacket until the air reaches 100 °C. Measure the length of the air column (volume V₁₀₀).
3. Plot volume against temperature (°C). A straight line is obtained. If it is extended backwards, it cuts the temperature axis at −273 °C.
4. The coefficient of volume expansion is

α_v = (V₁₀₀ − V₀) / (V₀ × 100)

For air (and for all gases) it is found to be 1/273 per °C (per kelvin).

4.3 Charles's law

Charles's law (first form):

• At constant pressure, the volume of a fixed mass of gas increases by 1/273 of its volume at 0 °C for every 1 °C rise in temperature.
• This fraction is the same for all gases.

Thus V_t = V₀ (1 + t/273), where t is in °C.

Second form (Kelvin scale). Using similar triangles in the graph (V against t, extended to −273 °C): V₁/T₁ = V₂/T₂, i.e.

V / T = constant (fixed mass, constant pressure), so V ∝ T

At constant pressure, the volume of a fixed mass of gas is directly proportional to its absolute (Kelvin) temperature.

4.4 Worked examples

Example 1. A gas has volume 300 cm³ at 0 °C. At 91 °C (constant pressure): V = 300 × (273 + 91)/273 = 300 × 364/273 = 400 cm³.

Example 2. 2 litres of a gas at 27 °C (300 K) is heated to 127 °C (400 K) at constant pressure. V₂ = 2 × 400/300 = 2.67 litres.

Example 3 (find the temperature). A gas of volume 500 cm³ at 7 °C (280 K) is heated at constant pressure until its volume becomes 750 cm³. T₂ = 280 × 750/500 = 420 K = 147 °C.

Example 4. Show the coefficient. A gas has 200 cm³ at 0 °C and 273.2 cm³ at 100 °C. α_v = (273.2 − 200)/(200 × 100) = 3.66 × 10⁻³ °C⁻¹ ≈ 1/273.