Defining the Variables

🟢 Lite — Quick Review (1h–1d)

Gas Laws govern how pressure (P), volume (V), temperature (T), and moles (n) interact in gases. The master formula is PV = nRT, where R = 0.0821 L·atm/mol·K. Always convert temperature to Kelvin (K = °C + 273) — this single step prevents the most common errors. Key proportional laws: Boyle’s (P₁V₁ = P₂V₂ at constant T,n), Charles’s (V₁/T₁ = V₂/T₂ at constant P,n), Gay-Lussac’s (P₁/T₁ = P₂/T₂ at constant V,n). Dalton’s Law: Ptotal = Σ partial pressures. STP = 273 K and 1 atm. Unit checklist: 1 atm = 760 mmHg = 101.325 kPa. Convert mL → L, g → kg before substituting. UPCAT tip: ~15–25% of physical science questions test gas calculations; expect one PV = nRT problem and one proportional law problem per paper.

🟡 Standard — Regular Study (2d–2mo)

Defining the Variables

Pressure (P) is force per unit area exerted by gas molecules colliding with container walls, measured in atm, kPa, mmHg, or Pa. Volume (V) is the container’s internal capacity in liters (1 L = 1000 mL). Temperature (T) must always be in Kelvin for gas law calculations — never substitute Celsius directly into equations. Moles (n) equals mass divided by molar mass (n = m/M), where 1 mol = 6.022×10²³ particles (Avogadro’s number).

The Ideal Gas Equation

PV = nRT combines all four variables. R (the gas constant) equals 0.0821 L·atm/mol·K or 8.314 J/mol·K — choose based on your pressure unit. Rearranging gives P = nRT/V, V = nRT/P, and n = PV/RT. These forms let you solve for any missing variable when the other three are known.

Proportional Gas Laws

Boyle’s Law: P inversely proportional to V at constant T and n. P₁V₁ = P₂V₂.
Charles’s Law: V directly proportional to T at constant P and n. V₁/T₁ = V₂/T₂.
Gay-Lussac’s Law: P directly proportional to T at constant V and n. P₁/T₁ = P₂/T₂.
Avogadro’s Law: V directly proportional to n at constant P and T. V₁/n₁ = V₂/n₂.

STP Calculations

At STP (T = 273 K, P = 1 atm), 1 mol of any ideal gas occupies 22.4 L. Use this as a conversion shortcut: n = V/22.4 at STP.

Dalton’s Law of Partial Pressures

In a mixture, each gas exerts pressure independently: Pₐ = Xₐ × Ptotal, where Xₐ is the mole fraction. If a mixture contains 0.2 mol O₂ and 0.8 mol N₂ in 1 atm total pressure, P(O₂) = 0.2 × 1 = 0.2 atm.

UPCAT Exam Pattern

Expect one direct substitution problem using PV = nRT (given P, V, T → solve for n or identify the gas) and one proportional law problem (apply Boyle’s or Charles’s to find the final state). Watch for mixed-unit problems — conversions are deliberate traps.

🔴 Extended — Deep Study (3mo+)

Kinetic Molecular Theory Basis

Gas behavior stems from three assumptions: gas particles have negligible volume compared to the container, experience perfectly elastic collisions (no energy loss), and have no intermolecular forces except during collisions. These assumptions define an ideal gas — real gases like CO₂ and NH₃ deviate from ideal behavior at high pressure (particles are forced closer together) and low temperature (intermolecular attractions become significant). For UPCAT, the critical edge case: never assume ideal behavior at STP for heavy gases — calculate compressibility factor if required, though UPCAT typically assumes ideal conditions.

Graham’s Law of Effusion

The rate of gas effusion (escaping through a pinhole) is inversely proportional to the square root of its molar mass: rate₁/rate₂ = √(M₂/M₁). A balloon filled with He deflates faster than one filled with N₂ because He has lower molar mass. This law frequently appears in mixed gas comparison questions.

Combined Gas Law

When none of the variables (P, V, T) are held constant, use: P₁V₁/T₁ = P₂V₂/T₂. This single equation subsumes Boyle’s, Charles’s, and Gay-Lussac’s laws — apply it whenever two states are given and three variables change simultaneously.

Common Mistakes to Avoid

Kelvin conversion error: T(K) = T(°C) + 273 — forgetting this conversion when given Celsius temperatures produces answers off by 273 units.
Unit inconsistency: mixing L with mL, atm with kPa, or g with kg within the same calculation. Convert everything to consistent units before substituting into PV = nRT.
Mole fraction vs. mass fraction: Dalton’s law uses mole fraction (nᵢ/ntotal), not mass percentage. A 50 g mixture of O₂ (32 g/mol) and N₂ (28 g/mol) in equal masses has different mole and mass fractions.
Ideal gas assumption failure: real gases deviate most at high P and low T — if a problem states “gas at 5 atm and 100 K,” treating it as ideal introduces measurable error.

Worked Example

A 2.0 L container holds 0.5 mol of CO₂ at 300 K. Find the pressure in atm.

Using PV = nRT: P = nRT/V = (0.5 mol)(0.0821 L·atm/mol·K)(300 K) / 2.0 L = 12.315 / 2.0 = 6.16 atm.

Practice Prompts

A gas occupies 5.0 L at 2.0 atm and 400 K. What volume does it occupy at 1.0 atm and 200 K? (Answer: 10 L)
A mixture contains 0.3 mol He and 0.7 mol Ne at 2.0 atm total pressure. Calculate the partial pressure of He. (Answer: 0.6 atm)

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Defining the Variables