IB Physics SL
Topic B.3 — The Behaviour of Gases
Paper 1 & 2
PV = nRT · PV = NkBT
~7 min read
The Ideal Gas Equation
Boyle’s, Charles’s and Gay-Lussac’s laws each freeze one variable and describe how the other two move together. Merge all three into one relationship, add in exactly how much gas you have, and you get an equation you can actually calculate with.
📘 What you need to know
- The ideal gas equation comes in two equivalent forms: PV = nRT and PV = NkBT.
- R = 8.31 J K⁻¹ mol⁻¹ is the ideal gas constant — it combines Boyle’s, Charles’s and Gay-Lussac’s laws into one macroscopic constant.
- kB = 1.38 × 10⁻²³ J K⁻¹ is the Boltzmann constant — the same idea, but scaled down to a single molecule.
- The two constants are linked by kB = R ÷ NA, where NA is the Avogadro constant.
- An ideal gas is defined as one that obeys this equation at every pressure, volume and temperature.
From Three Laws to One Constant
Boyle’s, Charles’s and Gay-Lussac’s laws each apply under different conditions — each one keeps a different variable fixed. Combining them shows that PV ÷ T doesn’t just stay constant in each individual case; it stays constant for a fixed amount of gas under any conditions. That single constant of proportionality is the ideal gas constant.
🔗 The three laws, side by side
- Boyle’s law — relationship PV = constant — held fixed: temperature, amount of gas
- Charles’s law — relationship V ∝ T — held fixed: pressure, amount of gas
- Gay-Lussac’s (pressure) law — relationship P ∝ T — held fixed: volume, amount of gas
An ideal gas is defined as one that obeys the resulting equation at every pressure, volume and temperature — real gases only approximate this behaviour under the right conditions.
The Ideal Gas Equation
Bringing in the amount of substance from moles turns the proportionality into a full equation:
Ideal gas equation
PV = nRT
where P is pressure (Pa), V is volume (m³), n is the number of moles (mol), R = 8.31 J K⁻¹ mol⁻¹ is the ideal gas constant, and T is the thermodynamic temperature (K).
Counting Individual Particles — The Boltzmann Constant
The ideal gas constant works at the scale of moles — convenient for lab-sized quantities of gas, but it says nothing about a single molecule. The Boltzmann constant is the microscopic equivalent, linked to R through the Avogadro constant:
Boltzmann constant
kB = R ÷ NA
Since the number of moles is related to the number of particles by n = N ÷ NA, substituting this into the ideal gas equation gives an equivalent version written in terms of individual particles rather than moles:
Ideal gas equation (particle form)
PV = NkBT
where N is the number of individual molecules and kB = 1.38 × 10⁻²³ J K⁻¹. Which version to use just depends on whether a question gives you an amount in moles or a number of particles — both describe exactly the same physics.
Quick recap: R is the macroscopic constant that goes with moles (n); kB is the microscopic constant that goes with individual particles (N). They’re linked by kB = R ÷ NA, and PV = nRT and PV = NkBT describe exactly the same equation.
WE 1A weather balloon is filled with 0.045 m³ of helium at a pressure of 1.05 × 10⁵ Pa and a temperature of 15 °C. Calculate the number of moles of helium in the balloon.
Step 1 — Convert temperature to kelvin
T = 15 + 273 = 288 K
Step 2 — Rearrange the ideal gas equation
PV = nRT → n = PV ÷ (RT)
Step 3 — Substitute
n = [(1.05 × 10⁵)(0.045)] ÷ [(8.31)(288)]
n ≈ 1.97 mol (3 s.f.)
R comes straight from the data booklet — no need to memorise it.
WE 2A canister holds nitrogen gas at 2.4 × 10⁵ Pa, occupying 3.0 × 10⁻³ m³, at a temperature of 22 °C. Calculate the number of gas molecules present.
Step 1 — Convert temperature to kelvin
T = 22 + 273 = 295 K
Step 2 — Rearrange the particle form of the equation
PV = NkBT → N = PV ÷ (kBT)
Step 3 — Substitute
N = [(2.4 × 10⁵)(3.0 × 10⁻³)] ÷ [(1.38 × 10⁻²³)(295)]
N ≈ 1.77 × 10²³ molecules (3 s.f.)
Notice this used kB, not R — because the question asked for a number of molecules, not a number of moles.
💡 Top tips
- Both R and kB are given in the data booklet — you don’t need to memorise their values, just which one goes with which form of the equation.
- Match the constant to the quantity you’re given: moles → use R; number of particles → use kB.
- Convert temperature to kelvin before substituting — every version of the ideal gas equation requires thermodynamic temperature.
⚠ Common mistakes
- Mixing up R and kB — using the Boltzmann constant with a number of moles, or the gas constant with a number of particles, gives an answer that’s out by a factor of the Avogadro constant.
- Forgetting to convert °C to kelvin before substituting.
- Using a volume in litres or cm³ instead of m³ — the ideal gas equation needs SI units throughout.
Up next: Kinetic Theory of Gases — where we step back from the equation itself and look at the molecular assumptions it’s actually built on.
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