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The ideal gas law

One equation, PV = nRT, ties together everything about a gas. The math is easy; the mistakes are procedural — mixing up which variables move together, and forgetting that temperature has to be in kelvin.

§1

One law, four variables.

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The ideal gas law, PV = nRT, links a gas's pressure (P), volume (V), amount in moles (n), and temperature (T) through the gas constant R. Given any three, you can solve for the fourth.

The relationships are built in. At fixed temperature and amount, P and V are inversely related (squeeze the volume, raise the pressure). At fixed pressure and amount, V and T are directly related (heat it, it expands).

Temperature in the gas law must be in kelvin. Kelvin starts at absolute zero, so ratios of temperature only make sense in kelvin; using Celsius breaks every proportion.

UNIT 3 TOPIC 3.4 • IDEAL GAS LAW PV = nRT PLAYGROUND The ideal gas law relating pressure, volume, amount, and temperature. VARIABLES P Pressure (atm) V Volume (L) T Temperature (K) n Moles (mol) R Gas constant P V = n R T the ideal gas law at STP: nRT / P = 22.4 L STP READOUT P 1.00 atm V 22.4 L T 273 K n 1.00 mol R 0.08206 L·atm mol⁻¹·K⁻¹ P PRESSURE Force per unit area of gas particles. V VOLUME Space occupied by the gas. T TEMPERATURE Average kinetic energy of particles. n MOLES Number of gas particles present. BOYLE'S LAW P vs V · n, T constant INVERSE P ∝ 1/V P V CHARLES'S LAW V vs T · n, P constant DIRECT V ∝ T V T AVOGADRO'S LAW V vs n · P, T constant DIRECT V ∝ n V n CED ANCHOR · THE TAKEAWAY One equation unifies three relationships: Boyle (P–V), Charles (V–T), Avogadro (V–n). REMEMBER Use Kelvin; Pₐ = Xₐ · Ptotal AP Chemistry · Unit 3 · Properties of Substances & Mixtures
Fig. 3.4.1 The ideal gas law, PV = nRT, ties pressure, volume, amount, and temperature together. Rearrange for any variable, but keep temperature in kelvin and track which relationships are direct (V with T) and which are inverse (P with V).
§2

Using the gas law.

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Convert first, then reason the direction, then compute.

  1. Convert temperature to kelvin. Add 273 to a Celsius temperature. Do this before any ratio or substitution.
  2. Identify what is held constant. Which variables are fixed tells you the relationship between the changing ones.
  3. Reason direct or inverse. P and V are inverse (fixed T, n); V and T are direct (fixed P, n); P and T are direct (fixed V, n).
  4. Solve. Rearrange PV = nRT for the unknown, or use a ratio like P₁V₁/T₁ = P₂V₂/T₂ with kelvin temperatures.
§3

The pieces you'll meet.

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Keep the relationships and the units straight.

PV=nRT
Ideal gas law
Relates P, V, n, and T through the constant R.
kelvin
Kelvin
The temperature scale the gas law requires; K = °C + 273.
P & V
Boyle (inverse)
At fixed T and n, pressure and volume are inversely related.
V & T
Charles (direct)
At fixed P and n, volume and temperature are directly related.
P & T
Gay-Lussac (direct)
At fixed V and n, pressure and temperature are directly related.
R
Gas constant
The proportionality constant linking the four variables.
§4

Worked example: heating a gas at constant volume.

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Question. A sealed rigid container of gas is at 27 °C and 1.0 atm. It is heated to 327 °C. What is the new pressure?

Convert. 27 °C = 300 K and 327 °C = 600 K. (Always kelvin.)

Reason. Volume and amount are fixed, so pressure and temperature are directly related: P₁/T₁ = P₂/T₂.

Solve. P₂ = P₁ × (T₂/T₁) = 1.0 atm × (600/300) = 2.0 atm. Note the temperature doubled in kelvin (300 → 600), so the pressure doubled — a result you would miss if you used Celsius, where 27 → 327 is not a doubling.

§5

Mistakes that cost real points.

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Pitfall · 01

"Temperature can be plugged into the gas law in Celsius."

The gas law requires kelvin. Ratios of temperature are only meaningful from absolute zero, so 27 °C to 327 °C is a doubling in kelvin (300 to 600 K) but not in Celsius. Using Celsius gives wrong ratios and even nonsensical negatives.

Fix. Always convert to kelvin (K = °C + 273) before substituting or taking any temperature ratio.

Pitfall · 02

"Pressure and volume rise together."

At fixed temperature and amount, pressure and volume are inversely related: squeezing a gas into a smaller volume raises its pressure. Treating them as directly related reverses the physics of Boyle's law.

Fix. Recall that P and V trade off (inverse) at fixed T and n; only V and T (or P and T) move together.

Pitfall · 03

"If you double the temperature in Celsius, the volume doubles."

Volume is proportional to the kelvin temperature, not the Celsius temperature. Going from 20 °C to 40 °C is not a doubling — it is 293 K to 313 K, a tiny increase. The doubling rule only works in kelvin.

Fix. Apply the direct proportion using kelvin temperatures. Double the kelvin value to double the volume.

§6

Skill Check.

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