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Ideal gases and the gas laws
Three empirical laws, found by squeezing and warming real gases, extrapolate to a shared absolute zero and merge into one equation of state. That equation gets written twice, once for moles and once for molecules.
Pick your board and the few notes written for the other boards quietly fold away, here and in the practice players. Nothing is deleted: every folded piece reopens on a tap.
Builds on Thermal energy transfer and specific heat capacity.
IN THIS TOPIC
- Say what an ideal gas is: one that obeys pV ∝ T at all pressures.
- Use the three empirical gas laws, each with its fixed quantity named.
- Explain how extrapolating those lines locates absolute zero.
- Apply pV = nRT and pV = NkT, choosing moles or molecules to suit the question.
- Calculate work done at constant pressure with pΔV, and convert between molar and molecular mass.
COMMON MISCONCEPTION
Zero degrees means zero heat.
Three empirical laws
The gas laws were found by experiment, squeezing and warming and measuring, which is what empirical means. Each of them holds one variable fixed. Boyle's law says that at constant temperature, pV is constant.
Charles's law says that at constant pressure, V/T is constant. The pressure law says that at constant volume, p/T is constant. Both of those demand kelvin, and in Celsius they are simply false. Required practical 8 investigates Boyle's and Charles's laws directly, and the mass of gas stays fixed throughout each one.
A gas thermometer works on exactly this principle, that any physical property changing steadily with temperature can be used to measure it. Three are standard, each reliable over its own range. The volume of a trapped gas. The resistance of a thermistor. The e.m.f. of a thermocouple, whose tiny junction makes it the pick for fast-changing or very high temperatures.
Outranking them all is the thermodynamic scale, which depends on no substance at all. It starts at absolute zero, and the size of its unit comes from physics rather than from any thermometer's habits. Since 2019 the kelvin has been fixed by giving the Boltzmann constant the exact value k = 1.380 649 × 10−23 J K−1, which pins one kelvin to a definite amount of particle energy. The triple point of water used to do that job and no longer does, so a textbook defining the kelvin by it is out of date.
Absolute zero
Plot volume against Celsius temperature for any gas and the line comes out straight. Extrapolate it backwards and it reaches zero volume at −273 °C. Remarkably, every gas's line meets the axis at the same place. The meeting is an extrapolation from the dilute-gas region where gases behave nearly ideally, since real gases condense long before the axis; that shared intercept defines absolute zero, 0 K, the temperature at which particle energy sits at its minimum. 0 °C is merely where water freezes, a parochial landmark 273 degrees above the real floor.
One equation of state
Set the three laws side by side and a single proportionality covers all of them, , with T the thermodynamic temperature in kelvin. A gas that obeys that at every pressure is what physics means by an ideal gas, and that sentence is the definition. Nothing real manages it perfectly. The ordinary gases of a laboratory come close over ordinary ranges, and they stray furthest when squeezed hard or cooled towards the temperature at which they would turn liquid, which is where the assumptions of the next lesson begin to fail.
Turn the proportionality into an equation and the constant records how much gas is there. That gives the ideal gas equation, written in two currencies. For n moles,
with R = 8.31 J K−1 mol−1, the molar gas constant. For N molecules,
with k = 1.38 × 10−23 J K−1, the Boltzmann constant.
The Avogadro constant NA = 6.02 × 1023 mol−1 is the exchange rate between the two, since N = nNA and k = R/NA. The same bridge links molar mass in kilograms per mole to molecular mass in kilograms per molecule, through M = NAm. Choosing the wrong currency is this unit's most reliable mark-loser.
One more tool. When a gas expands or is compressed at constant pressure, the work done is
which is not printed with the thermal equations (the booklet's only W = pΔV sits on the Engineering physics option page), and is how “work” from mechanics enters gas problems.
WORKED EXAMPLE
A tyre in the sun
A sealed tyre holds air at an absolute pressure of 250 kPa on a 20 °C morning. Find the pressure after the afternoon sun warms it to 50 °C, the volume barely changing.
Sealed and rigid means n and V are both fixed, so pV = nRT collapses to p proportional to T. Kelvin first, 293 K rising to 323 K.
p2 = 250 × 323/293 = 276 kPa.
Feed in 20 and 50 instead of 293 and 323 and you get a preposterous 625 kPa. The kelvin conversion decides the difference between a warm tyre and an exploded one.
GUIDED PRACTICE
A balloon in the cold
A balloon holds 3.0 litres at 300 K indoors and is carried out to 273 K, the pressure staying atmospheric. Decide which quantities are fixed, then find the new volume.
Show the working
Fixed n and p leave V proportional to T, so V2 = 3.0 × 273/300 = 2.7 litres.
The balloon visibly sags in the cold, and the equation of state says by how much. Roughly a tenth off the absolute temperature costs roughly a tenth of the volume.
INDEPENDENT PRACTICE
The density of the air in this room
Use pV = nRT to estimate the density of air at 101 kPa and 293 K. (Molar mass of air ≈ 0.029 kg mol−1.)
Show the working
One cubic metre holds n = pV/RT = 101 000/(8.31 × 293) = 41.5 mol.
Mass = 41.5 × 0.029 = 1.20 kg, so ρ ≈ 1.2 kg m−3. That is the famous number behind every drag and estimation question, derived here instead of remembered.
ASSESSMENT FOCUS
- “What is an ideal gas?” has a one-line answer, and CIE asks for it in as many words: a gas that obeys pV ∝ T at all pressures, with T the thermodynamic temperature. Every calculation on this page assumes it, so it is worth a line whichever paper you sit.
- Kelvin, always, with T(K) = θ(°C) + 273. One Celsius temperature left inside a gas-law calculation poisons the answer.
- Choose the currency before the constant. Moles go with R and molecules go with k, and mixing n with k is the commonest slip here.
- Extrapolation questions want a sentence. Every gas's extrapolated line meets the temperature axis at the same point, −273 °C, and that point defines absolute zero; real gases condense before reaching it, so the meeting point is drawn, not observed.
- Two things get quoted with W = pΔV. Learn it, because the thermal section of the data sheet does not print it, and state the condition of constant pressure alongside it.
CHECK YOURSELF
A cylinder of volume 0.020 m3 holds gas at 250 kPa and 300 K. How many moles does it contain, and roughly how many molecules is that?
Show a hint
One equation for n, then cross the bridge.
Show the answer
n = pV/RT = (250 × 103 × 0.020)/(8.31 × 300) = 2.0 mol.
Crossing the bridge, N = nNA = 2.0 × 6.02 × 1023 = 1.2 × 1024 molecules.
Sense check. About four grams' worth of hydrogen, or fifty-six of nitrogen, a perfectly ordinary cylinderful, and already a trillion trillion particles.
Kelvin in every gas law, no exceptions.
Moles ride with R; molecules ride with k.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their mark schemes on the ideal gases and the gas laws questions page.
WHERE TO GO NEXT
- Required practical 8: boyle's law and charles's law puts this topic in the lab, and the written papers ask about it.
- Gradients and areas under graphs is the maths this lesson leans on, worked through from GCSE.
- Rearranging equations is the maths this lesson leans on, worked through from GCSE.
CHECK YOUR PROGRESS
Rate how confident you feel with each objective for this lesson. Ratings are saved in this browser, on this device, unless you sign in.
- Say what an ideal gas is: one that obeys pV ∝ T at all pressures.
- Use the three empirical gas laws, each with its fixed quantity named.
- Explain how extrapolating those lines locates absolute zero.
- Apply pV = nRT and pV = NkT, choosing moles or molecules to suit the question.
- Calculate work done at constant pressure with pΔV, and convert between molar and molecular mass.
Open the full revision checklist to track your progress across the whole unit.