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F. Gases and Atmospheric Chemistry · Gases and Atmospheric Chemistry

Gay-Lussac's Law and the Combined Gas Law

F3.5 F3.4 F2.3About 70 minutes
Learning goals
  • I can state Gay-Lussac's law and explain it with the kinetic molecular theory.
  • I can solve problems in which pressure and temperature change at constant volume.
  • I can use the combined gas law when pressure, volume and temperature all change.
  • I can obtain Boyle's, Charles's and Gay-Lussac's laws from the combined gas law.

Words to know

Gay-Lussac's law
At constant volume and amount, pressure is directly proportional to Kelvin temperature: P₁/T₁ = P₂/T₂.
combined gas law
P₁V₁/T₁ = P₂V₂/T₂, for a fixed amount of gas when P, V and T may all change.
rigid container
A container whose volume cannot change, such as a steel tank or a can.

Why the warning label?

Look at a can of whipped cream or spray paint. The label warns: Do not puncture or incinerate. Do not store above 50 °C. The can is rigid, so its volume can't change. What happens to the gas inside when it gets hot? This lesson completes the set of simple gas laws and then combines them into one powerful equation.

Gay-Lussac's law (pressure and temperature)

For a fixed amount of gas at constant volume, the pressure is directly proportional to the Kelvin temperature.

P₁ ÷ T₁ = P₂ ÷ T₂ (constant V and n)

Joseph Louis Gay-Lussac studied this relationship around 1802. (Some books also call it Amontons's law, after Guillaume Amontons, who noticed it about a century earlier.)

KMT explanation: in a rigid container the particles cannot spread out. Heating them raises their average kinetic energy, so they hit the walls harder and more often. The pressure rises.

P = 200 kParigid can, 300 KP = 400 kParigid can, 600 Kheatsame volume, same particles: faster particles hit harder and more often
In a rigid container, doubling the Kelvin temperature doubles the pressure. Longer arrows show faster-moving particles.
01002003004000100200300Temperature (K)Pressure (kPa)101.3 kPa at 273 K138 kPa at 373 K
Pressure versus Kelvin temperature for a gas at constant volume is a straight line through the origin (dashed part extrapolated). This sample is at 101.3 kPa at 0 °C.
Worked example

Example 1: A spray can in a campfire

An aerosol can contains gas at 350 kPa at 25 °C. If it were thrown into a campfire and reached 300 °C, what would the pressure become?

  1. T₁ = 25 + 273 = 298 K; T₂ = 300 + 273 = 573 K; P₁ = 350 kPa; V is constant.
  2. Predict: hotter → higher pressure.
  3. P₂ = P₁ × T₂ ÷ T₁ = 350 kPa × 573 K ÷ 298 K = 673 kPa
  4. The pressure nearly doubles, enough to burst the can violently.

About 673 kPa. That's why the label says never to incinerate.

The combined gas law

Often P, V and T all change at once: a weather balloon rises into colder, thinner air; gas collected in a warm lab must be compared with STP. Boyle's, Charles's and Gay-Lussac's laws combine into one equation:

P₁V₁ ÷ T₁ = P₂V₂ ÷ T₂ (constant n)

You don't need to memorize the three separate laws if you know this one: just cross out whatever stays constant.

  • T constant → P₁V₁ = P₂V₂ (Boyle)
  • P constant → V₁/T₁ = V₂/T₂ (Charles)
  • V constant → P₁/T₁ = P₂/T₂ (Gay-Lussac)
Combined gas lawP₁V₁/T₁ = P₂V₂/T₂(n constant)Boyle's lawP₁V₁ = P₂V₂T constantCharles's lawV₁/T₁ = V₂/T₂P constantGay-Lussac's lawP₁/T₁ = P₂/T₂V constant
The combined gas law contains the three simple gas laws. Cross out the variable that stays constant to get each one.
Worked example

Example 2: Converting a gas volume to STP

A chemist collects 5.00 L of gas at 25 °C and 100 kPa. What volume would it occupy at STP (0 °C and 101.325 kPa)?

  1. State 1: P₁ = 100 kPa, V₁ = 5.00 L, T₁ = 298 K. State 2: P₂ = 101.325 kPa, T₂ = 273 K, V₂ = ?
  2. Rearrange: V₂ = V₁ × (P₁ ÷ P₂) × (T₂ ÷ T₁)
  3. Think of it as two corrections. The pressure goes up slightly → multiply by 100/101.325 (less than 1). The temperature goes down → multiply by 273/298 (less than 1).
  4. V₂ = 5.00 L × (100 ÷ 101.325) × (273 ÷ 298) = 4.52 L

V₂ = 4.52 L at STP

Worked example

Example 3: A weather balloon rises

A weather balloon is launched with 6.0 m³ of helium at 100 kPa and 15 °C. At an altitude of about 10 km, the pressure is about 26 kPa and the temperature is about −50 °C. Estimate the balloon's volume there.

  1. P₁ = 100 kPa, V₁ = 6.0 m³, T₁ = 288 K; P₂ = 26 kPa, T₂ = 223 K
  2. Pressure correction: 100/26 (much bigger) → volume increases. Temperature correction: 223/288 (smaller) → volume decreases a little.
  3. V₂ = 6.0 × (100 ÷ 26) × (223 ÷ 288) = 17.9 m³

About 18 m³, three times bigger. The drop in pressure wins over the drop in temperature. Weather balloons keep expanding as they rise until they eventually burst.

Hands-on activity

Track a tire through the seasons

You need: a grown-up, a tire pressure gauge, a bicycle or car tire, and a thermometer (or a weather app).

  1. On a warm afternoon, the grown-up measures the tire pressure (gauge reading) and you record the air temperature.
  2. Repeat on a cold morning (ideally at least 10 degrees colder), before the car or bike has been used, so the tire is at air temperature.
  3. Convert both gauge readings to absolute pressure by adding about 101 kPa (or 14.7 psi).
  4. Use Gay-Lussac's law to predict the second absolute pressure from the first. Compare with your measurement.

What to look for: if a tire reads 220 kPa (gauge) at 20 °C, its absolute pressure is 321 kPa. At −20 °C, Gay-Lussac's law predicts 321 × 253 ÷ 293 = 277 kPa absolute, or about 176 kPa on the gauge: a drop of more than 40 kPa. That's why tire-pressure warning lights often come on during the first cold snap of autumn. A tire is not perfectly rigid, so your result may be slightly different.

Just for fun

Every autumn, Canadian service stations get a rush of drivers whose tire-pressure warning lights came on during the first cold snap. The tires usually aren't leaking. They're just obeying Gay-Lussac's law. Gases are very law-abiding: they only bend the rules a little at high pressure and low temperature.

Your turn

1

State Gay-Lussac's law in words.

2

A gas in a rigid container is at 150 kPa and 25 °C. It is heated to 100 °C. Find the new pressure.

3

A rigid tank of gas at 200 kPa and 300 K is cooled to 0 °C. What is the new pressure?

4

A 2.0 L sample of gas at 150 kPa and 300 K changes to 100 kPa and 400 K. Find the new volume.

5

Convert 500 mL of gas at 98.5 kPa and 21 °C to STP.

6

Which gas law applies? (a) A balloon is squeezed at constant temperature. (b) A sealed glass jar is warmed in the sun. (c) A weather balloon rises.

More practice: warm-up

1

Write Gay-Lussac's law as an equation.

2

Write the combined gas law.

3

In a rigid container, which variable stays constant?

4

If the Kelvin temperature of a gas in a rigid tank doubles, what happens to its pressure?

5

Which simple law do you get from the combined gas law when T is constant?

More practice: on your own

1

A rigid can holds gas at 125 kPa and 25 °C. It is left in a hot car at 50 °C. Find the new pressure.

2

A rigid steel tank holds gas at 400 kPa and 100 °C. What is its pressure after it cools to 25 °C?

3

You breathe in 1.5 L of air at STP. In your lungs it is at 37 °C and 95.0 kPa. What volume does it occupy there?

4

Show how Charles's law can be obtained from the combined gas law.

5

Explain with the KMT why an aerosol can must not be stored above 50 °C.

Challenge

1

A tire gauge reads 220 kPa at 20 °C. Atmospheric pressure is 101 kPa. What will the gauge read at −20 °C, assuming the tire's volume stays constant?

2

A 30.0 L sample of air is at 20 °C and 101.3 kPa. A cold front moves in: the temperature drops to −10 °C and the pressure falls to 90.0 kPa. Find the new volume and explain why it hardly changed.

3

A gas sample's pressure is tripled and its Kelvin temperature is doubled. What happens to its volume?

Check your understanding

Quiz

1.

Gay-Lussac's law applies when which variables are held constant?

2.

A rigid container of gas at 100 kPa and 300 K is heated to 450 K. The new pressure is:

3.

Which equation is the combined gas law?

4.

A gas's Kelvin temperature doubles while its pressure also doubles. Its volume:

5.

Why do car tire-pressure warnings often appear on the first cold morning of autumn?

6.

To change a gas volume measured at 25 °C and 100 kPa to STP, you would multiply by:

For parents and teachers

Big idea (F3.5, F3.4, F2.3): Gay-Lussac's law (P₁/T₁ = P₂/T₂) and the combined gas law (P₁V₁/T₁ = P₂V₂/T₂), which contains all three simple gas laws.

How to help: encourage the "correction factor" method: for each variable that changes, ask whether it should make the answer bigger or smaller, and set up the ratio so it does. This catches most rearranging errors.

Common difficulties:

  • Using gauge pressure. Tire and tank gauges show pressure above atmospheric.
  • Forgetting kelvins (again!).
  • Solving the combined law in one messy step. Writing the two correction factors separately is clearer.

Safety: a grown-up should handle the tire gauge on a car. Never heat a sealed container; aerosol cans can burst.