TL;DR:
The ideal gas law
PV = nRTlinks pressure, volume, amount and temperature. Always use temperature in kelvin.Boyle's law (constant T and n):
P₁V₁ = P₂V₂. Charles's law (constant P and n):V₁/T₁ = V₂/T₂.A 2.00 L balloon cooled from 25 °C to −18 °C shrinks to about 1.71 L.
Put an inflated balloon in a freezer and in half an hour it looks soft and small. Take it out and it swells back. Nothing leaked and nothing popped. The gas inside simply took up less room when it was cold, and the gas laws tell you exactly how much less.
What the equation says
The ideal gas law is PV = nRT. P is pressure, V is volume, n is the amount of gas in moles, T is the absolute temperature in kelvin and R is the gas constant. With pressure in atmospheres and volume in litres, R = 0.08206 L·atm/(mol·K). With SI units, R = 8.314 J/(mol·K).
An ideal gas is a model: tiny particles, no attractions, only collisions with each other and the walls. Real gases follow it closely at ordinary pressure and temperatures well above where they would condense.
Why kelvin, not Celsius
The gas law needs a temperature scale that starts at true zero, where particle motion would stop. That is the kelvin scale: T(K) = T(°C) + 273 (more exactly 273.15). If you used Celsius, a gas at 0 °C would seem to have no pressure or volume at all, which is nonsense.
The older laws are special cases
Boyle's law: hold T and n fixed, so
PVis constant andP₁V₁ = P₂V₂. Squeeze a gas to half its volume and the pressure doubles.Charles's law: hold P and n fixed, so
V/Tis constant andV₁/T₁ = V₂/T₂. Volume is proportional to kelvin temperature.Gay-Lussac's law: hold V and n fixed, so
P₁/T₁ = P₂/T₂.Combined: with n fixed,
P₁V₁/T₁ = P₂V₂/T₂.
You never need to memorise them separately. Write PV = nRT, decide what stays constant, and the relation falls out.
Worked example: the balloon in the freezer
A balloon holds 2.00 L of air at 25 °C. You put it in a freezer at −18 °C. Assume the pressure stays equal to the room's and no gas escapes. What is the new volume?
Convert to kelvin:
T₁ = 25 + 273 = 298 KandT₂ = −18 + 273 = 255 K.Pressure and amount are constant, so use Charles's law:
V₁/T₁ = V₂/T₂.Solve:
V₂ = V₁ × T₂/T₁ = 2.00 × 255/298.Compute:
255 ÷ 298 = 0.856, soV₂ = 1.71 L.
Check the direction: colder gas takes up less room, and 1.71 L is less than 2.00 L. Check with the equation. At 1.00 atm, n = PV/RT = (1.00 × 2.00)/(0.08206 × 298) = 2.00/24.45 = 0.0818 mol. Now use the new state: V = nRT/P = 0.0818 × 0.08206 × 255 / 1.00 = 1.71 L. The same.
What if you had used Celsius by mistake? 2.00 × (−18)/25 = −1.44 L, a negative volume. That impossible result is a good warning sign that you forgot to convert.
A Boyle's law example
A syringe holds 30.0 mL of air at 1.00 atm. You push the plunger until the volume is 10.0 mL, keeping the temperature steady. From P₁V₁ = P₂V₂: P₂ = 1.00 × 30.0 / 10.0 = 3.00 atm. A third of the volume gives three times the pressure.
Sealed containers and heat: Heating a sealed container raises the pressure. Never heat or burn aerosol cans, sealed bottles or closed jars, and never seal dry ice in a bottle, because they can burst and cause serious injury. The balloon in a freezer is a safe demonstration; heat is not.
There is one more handy fact: at the same temperature and pressure, equal volumes of any ideal gas hold equal numbers of particles. Put n = 1.00 mol at 273 K and 1.00 atm into the law and V = nRT/P = 1.00 × 0.08206 × 273 / 1.00 = 22.4 L. That is the molar volume of an ideal gas under those conditions, a figure you will see in many textbooks.
Remember that pressure here is the total pressure of the gas on the walls. If a container holds several gases, each contributes its own share, and the shares add. For the ideal gas law, only the total number of moles of gas matters.
How to check your answer
Kelvin. Every temperature in the equation is in kelvin.
Direction. Cooling should shrink the volume, compressing should raise the pressure.
Round trip. Compute n from the first state and use it in the second.
Units. If R is in L·atm, pressure must be in atm and volume in L.
Mistakes that cost marks
Using Celsius.
Choosing the wrong value of R for the pressure unit.
Writing the ratio upside down, so a cooled gas expands.
Using Boyle's law when the temperature changes too.
Forgetting that a rubber balloon pushes back, so its pressure is a little above the room's.
Where you will use it next
Gas laws feed into stoichiometry with gases and into thermodynamics. The energy side of heating and cooling appears in endothermic and exothermic reactions, and the way concentration and pressure shift reactions is in Le Chatelier's principle with examples.
Frequently asked questions
Why does the balloon expand again when warmed?
Warmer particles move faster and push harder on the balloon wall, so it stretches until the pressure inside matches the outside again.
Does air in a tire follow the ideal gas law?
Roughly, yes. The pressure rises as the tire warms, as in pressure in fluids and tires.
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