9.6 Gas Laws, Dalton's Partial Pressures & Ideal vs Real Gases

The Empirical Gas Laws

Quick Answer: The behavior of an ideal gas is captured in PV = nRT. The four named laws are special cases: Boyle (constant T, n), Charles (constant P, n), Avogadro (constant T, P), and the combined gas law. Pressure is inversely proportional to volume; volume is directly proportional to absolute temperature and to moles.

Named Laws

Always work in kelvin for any gas law: K = °C + 273.15. Forgetting this is the most common PA-CAT gas-law error.

The Ideal Gas Law

PV = nRT

• P in atm (or Pa with the matching R)
• V in liters
• n in moles
• T in kelvin
• R = 0.08206 L·atm/(mol·K) (also 8.314 J/(mol·K) = 8.314 Pa·m³/(mol·K))

Worked Ideal-Gas Example

A 2.50 L flask holds 0.150 mol of an ideal gas at 25 °C. What is the pressure in atm?

1. Convert T: 25 + 273.15 = 298.15 K.
2. Rearrange: P = nRT / V.
3. Plug in: P = (0.150)(0.08206)(298.15) / 2.50.
4. Numerator: 0.150 × 0.08206 = 0.012309; × 298.15 = 3.670.
5. P = 3.670 / 2.50 = 1.47 atm.

Worked Combined-Law Example

A sample of gas has V1 = 4.0 L at P1 = 1.5 atm and T1 = 300 K. Find V2 at P2 = 1.0 atm, T2 = 350 K.

V2 = V1 × (P1/P2) × (T2/T1) = 4.0 × (1.5/1.0) × (350/300) = 4.0 × 1.5 × 1.167 = 7.0 L

Molar Volume at STP

At STP (0 °C, 1 atm), one mole of an ideal gas occupies 22.4 L. At 25 °C, 1 atm (sometimes called SATP-adjacent), the molar volume is 24.5 L. These numbers let you convert between moles and gas volumes without doing the full PV = nRT calculation:

• 11.2 L of O2 at STP = 0.500 mol O2.
• 5.60 L of CO2 at STP = 0.250 mol CO2 → 0.250 × 44.0 = 11.0 g.

Dalton's Law of Partial Pressures

Total pressure of a gas mixture equals the sum of partial pressures each gas would exert alone:

Ptotal = P1 + P2 + P3 + ...

A gas's partial pressure = its mole fraction times the total pressure:

Pi = Xi × Ptotal,   where Xi = ni / ntotal

Worked Dalton Example

A 5.00 L container at 300 K holds 0.20 mol N2, 0.10 mol O2, and 0.05 mol CO2. Find each partial pressure and the total.

1. ntotal = 0.20 + 0.10 + 0.05 = 0.35 mol.
2. Ptotal = nRT/V = (0.35)(0.08206)(300)/5.00 = 8.617/5.00 = 1.72 atm.
3. Mole fractions: X(N2) = 0.20/0.35 = 0.571; X(O2) = 0.10/0.35 = 0.286; X(CO2) = 0.05/0.35 = 0.143.
4. Partial pressures: P(N2) = 0.571 × 1.72 = 0.983 atm; P(O2) = 0.286 × 1.72 = 0.493 atm; P(CO2) = 0.143 × 1.72 = 0.246 atm. Sum ≈ 1.72 atm ✓.

Partial Pressure of a Gas Collected Over Water

When a gas is collected by displacement of water, the collected gas is saturated with water vapor, so:

Pdry gas = Ptotal − Pwater vapor

The vapor pressure of water depends only on temperature (e.g., 23.8 mmHg at 25 °C). The PA-CAT may give you a problem where you must subtract this before applying the ideal gas law.

Kinetic Molecular Theory

The KMT assumptions define an ideal gas:

1. Particles are point masses (negligible volume).
2. Particles are in constant, random, straight-line motion.
3. Collisions are elastic (no kinetic energy lost).
4. There are no intermolecular forces except during collisions.
5. Average kinetic energy depends only on absolute temperature: KEavg = (3/2) kT per particle or (3/2) RT per mole, identical for all gases at the same T.

Consequences:

• At the same T, all gases have the same average KE; lighter molecules move faster on average (urms = √(3RT/M)).
• Pressure arises from collisions with the container walls.
• Temperature is a measure of average translational KE.

Real Gases and Deviations

Real gases deviate from ideal behavior when assumptions 1 and 4 fail:

• High pressure → molecules forced close together; their own volume becomes significant (V_real > V_ideal; repulsion at very short range).
• Low temperature → molecules move slowly enough for intermolecular attractions to matter (V_real < V_ideal; the gas is more compressible than ideal).

The van der Waals equation corrects both:

(P + a (n/V)²) (V − nb) = nRT

• a corrects for intermolecular attraction (effectively reduces P).
• b corrects for molecular volume (effectively reduces V).

When Do Real Gases Behave Most Ideally?

Low pressure, high temperature — molecules are far apart (volume negligible) and moving fast (attractions brief and weak). Helium and hydrogen behave most ideally because their intermolecular forces are weakest.

Tying It Together

The PA-CAT Bulletin of Information, rev. 20240815, lists Gases alongside Liquids/Solids/Gases as separate blueprint subtopics. Expect stoichiometry-style items that connect moles of gas to reaction coefficients (e.g., "how many liters of O2 at STP are needed to burn 2.0 mol of CH4?") — combine the gas law with the balanced equation.

Worked link: CH4 + 2 O2 → CO2 + 2 H2O says 1 mol CH4 needs 2 mol O2. Burning 2.0 mol CH4 → 4.0 mol O2 → 4.0 × 22.4 = 89.6 L O2 at STP.