General Chemistry I › Gases › full formula sheet
The ideal gas law
Pressure, volume, moles, and temperature — one equation to rule them all.
Notation on this page: P = pressure, V = volume, n = amount of gas in moles, T = absolute temperature (kelvin), R = the gas constant, whose value depends on your units.
Where it comes from
The problem the law solves is simple to state: a gas has four measurable variables — pressure, volume, temperature, and amount — and they all push on each other. Heat a sealed can and the pressure climbs. Compress a syringe and the temperature rises. Early scientists attacked this one variable at a time, freezing the others and measuring what happened. That gave three separate laws, each true but each blind:
The naive approach is to memorize all three and juggle them in sequence whenever more than one variable changes. Watch it break on a perfectly ordinary problem. A gas starts at P1 = 1.0 atm, V1 = 2.0 L, T1 = 300 K, and ends up compressed to V2 = 1.0 L and heated to T2 = 450 K. Find P2:
That is what PV = nRT really is: the three laws fused into one, so you never juggle again. And it has a physical picture that makes it obvious. Think of pressure as molecules slamming into the walls of their container:
- More molecules (n up) → more collisions per second → pressure rises. So P ∝ n.
- Hotter gas (T up) → molecules fly faster and hit harder → pressure rises. So P ∝ T.
- Smaller container (V down) → the same molecules strike the walls more often → pressure rises. So P ∝ 1/V.
Push pressure up with n and T, squeeze it down with V — that is P = nRT/V, the law in one sentence. Kinetic theory (Clausius, Maxwell, Boltzmann) later made this picture rigorous: pressure is molecular collisions, and temperature is average molecular kinetic energy.
Derivation
We combine the three empirical laws. Each proportionality comes with a condition — the variables it holds fixed — and the combining step must honor all three conditions at once.
Why is R the same for every gas? Because Avogadro’s law says gas identity never enters: at the same P, V, T you always get the same n, whether it is helium or xenon. So the calibration constant measured once is universal — one R for all ideal gases. (R = NA·kB: the Boltzmann constant per molecule, scaled up by Avogadro’s number to the mole.)
Kinetic-theory sketch (the microscopic “why”): molecules ricochet around a box, each collision pushing the wall. Pressure scales with the number of molecules N, their average kinetic energy (∝ T), and inversely with volume: P ∝ N·(avg KE)/V, so PV = NkBT = nRT. Full rigor — averaging the collisions over all directions, which produces the exact factor relating P to kinetic energy — needs statistical mechanics and is out of scope here; we state the result, which experiment confirms.
How to use it
The procedure, every time:
- List P, V, n, T and circle the unknown. Three knowns, one unknown — that is a gas-law problem.
- Check units FIRST — convert everything into one system before touching the formula. kPa ↔ atm (1 atm = 101.325 kPa), mL → L (÷1000), m³ → L (×1000), cm³ = mL.
- Convert °C to K. Always. K = °C + 273.15. The law is a proportion anchored at absolute zero; Celsius severs the anchor.
- Pick R to match your units (see below), then rearrange and solve.
- Sanity-check. At STP (0°C, 1 atm) one mole fills about 22.4 L — if your answer is wildly off that scale, recheck units. Also check that PV/nT has the units of your R.
Picking R
R is not two constants; it is one constant in two unit systems. Write the value with its units at the top of every gas problem, matched to the P and V you actually have.
When the law breaks down
“Ideal” means three fictions: point particles (molecules take up no space), no attractions between molecules, and perfectly elastic collisions. Real molecules have size and stickiness, so at high pressure (molecules crowded together) or low temperature (molecules slow enough to attract), real gases deviate. That is van der Waals territory:
Two power rearrangements
Molar mass from density. Since n = m/M, substitute into PV = nRT: PV = mRT/M. With density d = m/V:
Dalton’s law of partial pressures. In a mixture, each gas contributes its own n: Ptotal = PA + PB + … = (nA + nB + …)RT/V. The ideal gas law applied once per gas, then added.
Worked examples
Four problems, easiest first. In each one, read every step — the why of each move is the lesson.
Example 1 — the basic move: n = 2.0 mol, T = 300 K, V = 10.0 L. Find P.
- List and circle the unknown. P = ?, V = 10.0 L, n = 2.0 mol, T = 300 K. (T is already kelvin — nothing to convert.)
- Pick R by units. atm and liters → R = 0.08206 L·atm/(mol·K). (Why this one? The liters in R cancel the liters of V, the atm survives as the unit of P.)
- Rearrange: P = nRT/V.
- Plug in: P = (2.0)(0.08206)(300)/10.0 = 49.236/10.0 = 4.92 atm.
- Sanity-check: 2 mol at 300 K in 10 L — roughly 5 atm is sensible (compare: 1 mol at STP fills 22.4 L at 1 atm; doubling the moles and halving the volume pushes P up several-fold). ✓
Example 2 — tire pressure on a hot day: 220 kPa at 20°C. Parked in the sun it reaches 60°C; volume is constant. Find P2.
- Convert to kelvin FIRST. T1 = 20 + 273.15 = 293 K, T2 = 60 + 273.15 = 333 K. (Why first? Because the next step is a ratio, and ratios are where Celsius lies.)
- Spot the shortcut. n and V are fixed, so PV = nRT collapses to P/T = constant → P1/T1 = P2/T2. No R needed — it cancels.
- Solve: P2 = P1(T2/T1) = 220 × 333/293 = ≈ 250 kPa.
- Sanity-check: temperature rose ~14% in absolute terms (333/293 ≈ 1.14), so pressure should rise ~14%: 220 × 1.14 ≈ 250. ✓
Example 3 — molar mass from density: a gas has density d = 1.96 g/L at P = 1.00 atm, T = 273 K. Find M.
- Start from the rearrangement M = dRT/P. (Why? PV = nRT with n = m/M gives PV = mRT/M; divide by V and M pops out as dRT/P. Density is mass per volume, so the volume cancels.)
- Pick R by units. atm and L → R = 0.08206 L·atm/(mol·K).
- Plug in: M = (1.96)(0.08206)(273)/1.00 = 43.91/1.00 = ≈ 43.9 g/mol.
- Interpret: 43.9 g/mol sits right next to CO2 (44.01 g/mol). One density measurement plus the gas law identified the gas. ✓
Example 4 — judgment call: the Celsius trap, wrong then right. P = 2.0 atm, V = 3.0 L, T = 20°C. Find n.
Path A — the trap (plug 20°C in directly).
- n = PV/RT = (2.0)(3.0)/(0.08206 × 20) = 6.0/1.641 = 3.66 mol.
- The verdict: 3.66 mol in 3.0 L at 2 atm? At STP, 3.66 mol would fill 82 L — cramming it into 3 L at just 2 atm is absurd. The answer smells wrong because it is wrong.
Path B — the right way (kelvin first).
- T = 20 + 273.15 = 293.15 K.
- n = (2.0)(3.0)/(0.08206 × 293.15) = 6.0/24.056 = 0.249 mol. ✓
The lesson: the wrong answer is ~15× too big (3.66 vs 0.249). Plugging Celsius doesn’t make the answer “a little off” — it manufactures a catastrophe, because 20°C is not 20 on the absolute scale, it is 293. Kelvin or it didn’t happen.
Memorization tips
- Say it as a word: “piv-nert” — PV nRT rolls off the tongue once you say it a few times.
- Kelvin or it didn’t happen: tattoo “+273.15” on the inside of your eyelids. A Celsius temperature in PV = nRT breaks every ratio.
- R is a chameleon: write the R value with its units at the top of every gas problem, matched to the P and V units you have. atm+L → 0.08206; Pa+m³ → 8.314.
- Anchor on 22.4 L: one mole at STP ≈ 22.4 L. If a problem gives you 44.8 L at STP, you already know it’s ~2 mol before computing — and it catches factor-of-10 unit disasters.
- Read it as a story: P = nRT/V. Pressure is pushed up by more molecules and higher temperature, squeezed down by smaller volume. The story is the formula.
- The laws combine by multiplication: V ∝ nT/P — one ∝ per law, multiplied. If you ever forget the combined form, rebuild it from the three laws in thirty seconds.
Final challenge
Five mixed questions — computations, judgment calls, and the traps, all in one. Score 5/5 and the ideal gas law is yours.
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How to learn a formula here
- Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
- Work the examples with the answers covered, then uncover one step at a time and compare.
- Finish with the final challenge — five mixed questions including the classic traps.
- Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.
Frequently asked questions
What is the ideal gas law?
PV = nRT relates a gas’s pressure, volume, moles, and absolute temperature. It combines Boyle’s, Charles’s, and Avogadro’s laws into one equation.
Why must temperature be in Kelvin?
The law needs absolute temperature because volume is proportional to T measured from absolute zero. Using Celsius breaks every ratio and gives nonsense answers.
Which R value should I use?
Match R to your units: 8.314 J/(mol·K) for pressure in pascals and volume in cubic meters, or 0.08206 L·atm/(mol·K) for pressure in atm and volume in liters.
When does the ideal gas law fail?
At high pressure or low temperature, where molecules’ own volume and intermolecular attractions matter. Real gases then deviate, and the van der Waals equation does better.
Where does the ideal gas law come from?
Empirically, it unites Boyle’s (1662), Charles’s (1787), and Avogadro’s (1811) laws. Kinetic theory (Clausius, Maxwell, Boltzmann) later derived it from molecular motion.
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