Gas Laws Deep Dive
- Use Dalton’s law and mole fractions to find partial pressures, including gas collected over water
- Apply Graham’s law to compare effusion and diffusion rates from molar masses
- Explain when and why real gases deviate from ideal behavior at high pressure or low temperature
Dalton’s law and gas collected over water
In a mixture, each gas pushes independently, and Dalton’s law says the total pressure is the sum of the partial pressures: P_total = P₁ + P₂ + …. Each partial pressure equals the gas’s mole fraction times the total: Pᵢ = Xᵢ × P_total. A classic lab application is collecting a gas over water: the gas bubbles up mixed with water vapor, so the measured total pressure includes the vapor pressure of water. To get the dry-gas pressure you subtract it: P_gas = P_total − P_water. The water’s vapor pressure depends only on temperature and is read from a table.
Graham’s law of effusion
Effusion is the escape of gas molecules through a tiny hole; diffusion is their spreading through space. Because all gases at the same temperature share the same average kinetic energy (½mv² is fixed by T), lighter molecules must move faster — velocity scales as 1/√(molar mass). Graham’s law captures this: the rate of effusion is inversely proportional to the square root of molar mass, rate ∝ 1/√M. Comparing two gases, rate₁/rate₂ = √(M₂/M₁). A gas with four times the molar mass effuses only half as fast, not a quarter as fast — the square root softens the ratio.
Why real gases deviate from ideal
The ideal gas law assumes molecules have zero volume and exert no forces on each other. Both assumptions fail under stress. At high pressure, molecules are crowded, so their own finite volume is no longer negligible and the real volume runs larger than PV = nRT predicts. At low temperature, molecules move slowly and lingering intermolecular attractions pull them together, so they strike the walls a little softer and the real pressure runs lower than predicted. A gas therefore behaves most ideally at high temperature and low pressure, where particles are fast, far apart, and effectively free. Gases with strong IMFs (like polar or hydrogen-bonding molecules) deviate the most.
Compare the effusion rates of helium (M = 4.00 g·mol⁻¹) and oxygen, O₂ (M = 32.0 g·mol⁻¹). How many times faster does helium effuse?
- 1.Graham’s law: rate_He / rate_O₂ = √(M_O₂ / M_He). Put the heavier gas on top under the root.
- 2.Substitute the molar masses: √(32.0 ÷ 4.00) = √8.00.
- 3.Take the square root: √8.00 = 2.83.
- 4.So helium effuses about 2.83 times faster than oxygen — sensible, since He is the lighter, faster molecule.
A student collects oxygen gas over water at 25 °C. The total pressure in the tube is 755 torr, and the vapor pressure of water at 25 °C is 24 torr. What is the partial pressure of the dry O₂?
- 1.The collected gas is a mixture of O₂ and water vapor, so Dalton’s law applies: P_total = P_O₂ + P_water.
- 2.Solve for the dry gas: P_O₂ = P_total − P_water.
- 3.Substitute: P_O₂ = 755 torr − 24 torr.
- 4.The water vapor accounts for 24 torr of the reading, leaving the rest for oxygen.
In Graham’s law the molar masses go under a square root, so a 4× mass difference gives only a 2× rate difference, and a 9× mass difference gives a 3× difference. Forgetting the root — treating rate as simply proportional to 1/M — is the most common error on this topic.
A rigid container holds 1.0 mol He, 2.0 mol Ne, and 1.0 mol Ar at a total pressure of 8.0 atm. What is the partial pressure of Ne?
Methane, CH₄ (M = 16 g·mol⁻¹), and sulfur dioxide, SO₂ (M = 64 g·mol⁻¹), effuse through the same pinhole. How many times faster does CH₄ effuse than SO₂?
Under which conditions would a real gas deviate MOST from ideal behavior?
For real-gas questions, tie the deviation to a cause: high pressure → molecular volume no longer negligible (real V larger); low temperature → intermolecular attractions reduce wall impacts (real P smaller). Naming the mechanism, not just "it deviates," is what earns the point.
Answer the 3 checkpoints as you read.
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