Gas Laws Worked Examples: Ideal, Combined, and PV=nRT
Boyle, Charles, the combined gas law, the ideal gas law PV=nRT, Dalton, and Graham — with the unit traps that sink students and four fully worked examples.
Learning Objectives
- ✓Apply the ideal gas law and combined gas law with consistent units.
- ✓Use Dalton’s law of partial pressures and Graham’s law of effusion.
- ✓Avoid the Kelvin and gas-constant unit errors that cause most wrong answers.
1. Direct Answer: The Gas Laws at a Glance
The ideal gas law, PV = nRT, ties together pressure (P), volume (V), moles (n), and temperature (T), with R the gas constant. Choose R to match your pressure units: R = 0.08206 L·atm/(mol·K) when pressure is in atm and volume in liters, or R = 8.314 J/(mol·K) for SI units. Temperature must ALWAYS be in Kelvin (K = °C + 273.15) — this is the single most common mistake. The individual laws are special cases: Boyle’s (P₁V₁ = P₂V₂ at constant T, n), Charles’s (V₁/T₁ = V₂/T₂ at constant P, n), and Gay-Lussac’s (P₁/T₁ = P₂/T₂ at constant V, n). The combined gas law, P₁V₁/T₁ = P₂V₂/T₂, handles a fixed amount of gas changing all three. At STP (0°C, 1 atm) one mole of any ideal gas occupies 22.4 L.
Key Points
- •PV = nRT; pick R to match units (0.08206 L·atm/mol·K or 8.314 J/mol·K).
- •Temperature must be in Kelvin (°C + 273.15) for every gas law.
- •At STP one mole of ideal gas = 22.4 L.
2. The Combined Gas Law and Its Special Cases
When a FIXED amount of gas changes condition, use the combined gas law P₁V₁/T₁ = P₂V₂/T₂. Each simpler law is just the combined law with one variable held constant: hold T constant and it collapses to Boyle’s P₁V₁ = P₂V₂ (pressure and volume are inversely related — squeeze the volume, pressure rises); hold P constant and you get Charles’s V₁/T₁ = V₂/T₂ (heat a balloon, it expands); hold V constant and you get Gay-Lussac’s P₁/T₁ = P₂/T₂ (heat a sealed rigid container, pressure climbs). You rarely need to memorize all four names if you can rearrange the combined law and cancel whatever is constant. The catch, again, is Kelvin: any ratio involving temperature is nonsense in Celsius because Celsius has an arbitrary zero.
Key Points
- •Combined law: P₁V₁/T₁ = P₂V₂/T₂ for a fixed amount of gas.
- •Boyle (constant T), Charles (constant P), Gay-Lussac (constant V) are special cases.
- •Cancel whatever variable is held constant rather than memorizing each law.
3. Worked Example 1: Solving PV=nRT for Moles
How many moles of gas occupy 5.00 L at 2.00 atm and 27°C? First convert temperature: T = 27 + 273.15 = 300.15 K (use 300 K). Choose R = 0.08206 L·atm/(mol·K) to match atm and L. Rearrange: n = PV/(RT) = (2.00 atm × 5.00 L) / (0.08206 × 300 K) = 10.0 / 24.62 = 0.406 mol. Check the units: atm·L cancels against the L·atm in R, and K cancels K, leaving mol. If you had left T at 27 in the denominator you would have computed n = 10.0/(0.08206×27) = 4.51 mol — more than ten times too large — which is exactly the Kelvin error in action. The sanity check: 0.406 mol at roughly STP-ish conditions giving about 5 L is reasonable, since a mole is ~22 L at STP and we have higher pressure here.
Key Points
- •Convert °C to Kelvin BEFORE substituting.
- •n = PV/(RT) with R = 0.08206 gives 0.406 mol.
- •Skipping the Kelvin conversion inflates the answer by an order of magnitude.
4. Worked Example 2: Combined Gas Law
A gas occupies 2.0 L at 1.0 atm and 273 K. What volume does it occupy at 3.0 atm and 546 K? Use P₁V₁/T₁ = P₂V₂/T₂ and solve for V₂: V₂ = V₁ × (P₁/P₂) × (T₂/T₁) = 2.0 L × (1.0/3.0) × (546/273) = 2.0 × 0.333 × 2.0 = 1.33 L. The reasoning is intuitive: tripling the pressure tends to shrink the volume to one-third, while doubling the absolute temperature tends to double it, and the two effects partly cancel to give 1.33 L. Notice both temperatures are in Kelvin and only the RATIOS matter, so you never need R for a combined-law problem — it cancels. This is why the combined law is faster than PV=nRT when the amount of gas does not change.
Key Points
- •V₂ = V₁ × (P₁/P₂) × (T₂/T₁); both T in Kelvin.
- •Pressure and volume inversely related; volume and absolute temperature directly related.
- •No R needed — ratios cancel it for fixed-amount problems.
5. Dalton, Graham, and Mixtures of Gases
DALTON’S LAW: in a mixture, the total pressure equals the sum of partial pressures, P_total = P₁ + P₂ + …, and each gas’s partial pressure equals its mole fraction times the total: P_i = x_i × P_total. This is essential for gas-collection-over-water problems, where you subtract the water vapor pressure to get the dry-gas pressure. GRAHAM’S LAW of effusion: lighter gases effuse faster, with rate ∝ 1/√M, so rate₁/rate₂ = √(M₂/M₁). Helium (M = 4) effuses about three times faster than CO₂ (M = 44) because √(44/4) ≈ 3.3. Real gases deviate from ideal behavior at HIGH pressure and LOW temperature, where molecular volume and intermolecular attractions matter — the van der Waals equation corrects for both. Under ordinary lab conditions the ideal gas law is an excellent approximation.
Key Points
- •Dalton: P_total = ΣP_i and P_i = x_i × P_total (subtract water vapor in gas-over-water problems).
- •Graham: rate ∝ 1/√M, so rate₁/rate₂ = √(M₂/M₁).
- •Real gases deviate at high pressure and low temperature (van der Waals correction).
6. Solving Gas Law Problems in ChemistryIQ
Snap a photo of any gas law problem and ChemistryIQ identifies which law applies, converts temperatures to Kelvin automatically, selects the correct value of R for your units, and solves step by step — flagging the Kelvin and unit-consistency mistakes before they cost you the answer. It handles combined-law, partial-pressure, and effusion problems with the full setup shown. This content is for educational purposes only.
Key Points
- •Automatic Kelvin conversion and R-constant selection.
- •Identifies the applicable law and shows the full setup.
- •Catches unit-consistency errors before the final answer.
High-Yield Facts
- ★PV = nRT; R = 0.08206 L·atm/mol·K (atm, L) or 8.314 J/mol·K (SI).
- ★Temperature is ALWAYS in Kelvin (°C + 273.15) in every gas law.
- ★Combined gas law P₁V₁/T₁ = P₂V₂/T₂ needs no R — ratios cancel it.
- ★Dalton: P_i = x_i × P_total; STP molar volume = 22.4 L/mol.
- ★Graham: rate ∝ 1/√M; real gases deviate at high P and low T.
Practice Questions
1. A 4.0 L gas at 2.0 atm is compressed to 1.0 L at constant temperature. What is the new pressure?
2. How many liters does 0.50 mol of an ideal gas occupy at STP?
3. Which effuses faster and by how much: H₂ or O₂?
FAQs
Common questions about this topic
Because the gas laws describe direct or inverse proportionalities anchored at absolute zero, and only the Kelvin scale has its zero at absolute zero. Celsius places zero at the freezing point of water arbitrarily, so a ratio like T₂/T₁ in Celsius produces meaningless or even negative values. Converting with K = °C + 273.15 is mandatory for Charles’s, Gay-Lussac’s, combined, and ideal gas calculations — it is the most common source of wrong answers.
Use PV=nRT when you need an absolute quantity — the moles, mass, or density of a gas at one set of conditions, or when the amount of gas is unknown. Use the combined gas law P₁V₁/T₁ = P₂V₂/T₂ when a fixed amount of gas changes from one set of conditions to another, because the ratios cancel R and the number of moles. If the problem describes a single state, reach for PV=nRT; if it describes a change between two states, the combined law is faster.
Match R to your units. Use R = 0.08206 L·atm/(mol·K) when pressure is in atmospheres and volume in liters — the most common gen-chem setup. Use R = 8.314 J/(mol·K) for SI units (pascals, cubic meters) and for energy-related calculations. There is also R = 62.36 L·Torr/(mol·K) for pressures in Torr or mmHg. Picking the wrong R is a units error that produces an answer off by a constant factor.
At HIGH pressure and LOW temperature. Under those conditions the molecules are crowded close together, so their finite volume and the attractive forces between them — both ignored by the ideal gas law — become significant. The van der Waals equation adds correction terms for molecular volume (b) and intermolecular attraction (a) to account for this. At ordinary temperatures and pressures, gases behave nearly ideally and PV=nRT works well.
Snap a photo of the problem and ChemistryIQ identifies the relevant law, converts temperatures to Kelvin, selects the correct R for your units, and solves step by step while flagging unit-consistency mistakes. It handles combined-law, Dalton partial-pressure, and Graham effusion problems with the full setup shown. This content is for educational purposes only.