AP Chemistry 3.3: Gas Laws and Kinetic Molecular Theory

• Gases can be described using four major variables: pressure (P), volume (V), amount in moles (n), and absolute temperature (T) .

  • These variables are connected by the ideal gas law :

  • PV = nRT

  • P = pressure.

  • V = volume.

  • n = number of moles.

  • R = ideal-gas constant.

  • T = absolute temperature in kelvin .

• Never substitute a Celsius temperature directly into the ideal gas law.

Convert using:

• K = °C + 273.15

  • Example: 25°C = 298.15 K, usually reported as approximately 298 K depending on precision.

A common gas constant is:

• R = 0.08206 L·atm·mol⁻¹·K⁻¹

  • when pressure is in atm and volume in liters.

Another common value is:

• R = 8.314 J·mol⁻¹·K⁻¹

  • when working with compatible SI/energy units.

• Always make sure your units match the chosen value of R.

  • Khan Academy's current Unit 3 ideal-gas section includes direct PV=nRT problems, volume-change problems, gas mixtures, and partial pressures.

• To solve for moles:

  • n = PV/(RT)

  • Example: P = 1.00 atm, V = 10.0 L, T = 300 K.

  • n = (1.00 × 10.0)/(0.08206 × 300)

  • n ≈ 0.406 mol

• If the amount of gas stays constant, the relationship between two states can be written:

  • P₁V₁/T₁ = P₂V₂/T₂

  • This combines the common pressure-volume and temperature-volume/pressure relationships.

• At constant temperature and amount, pressure and volume are inversely related:

  • P ∝ 1/V

  • If the volume decreases, gas particles collide with the container walls more frequently, so pressure increases.

• At constant pressure and amount:

  • V ∝ T

  • Increasing Kelvin temperature increases particle speed and collision frequency/force. To keep pressure constant, volume must increase.

• At constant volume and amount:

  • P ∝ T

  • Increasing temperature makes collisions with the container walls more frequent and energetic, causing pressure to rise.

  • Be careful: these proportionalities require the other relevant variables to remain constant.

• Dalton's law of partial pressures describes gas mixtures:

  • Ptotal = P₁ + P₂ + P₃ + ...

• Each gas in a mixture exerts its own partial pressure .

  • For an ideal gas mixture, each gas behaves approximately as if the other gases were not present.

• Mole fraction is:

  • Xᵢ = nᵢ / ntotal

  • Partial pressure can then be calculated:

  • Pᵢ = XᵢPtotal

  • Example: a gas mixture has 2 mol N₂ and 1 mol O₂.

  • Total = 3 mol.

  • XN₂ = 2/3.

  • XO₂ = 1/3.

  • If total pressure is 900 torr, then PN₂ = 600 torr and PO₂ = 300 torr.

• If a gas is collected over water, total measured pressure can include both the desired gas and water vapor:

  • Ptotal = Pgas + PH₂O

  • Therefore the pressure of the dry collected gas can be found by subtracting the water-vapor pressure when that information is provided.

• Common errors include using Celsius, mismatching R units, forgetting to convert mL → L when necessary, or using the wrong partial-pressure relationship.

• Kinetic molecular theory (KMT) provides a particle-level model that explains ideal-gas behavior.

• The model assumes gas particles are extremely small relative to the distances separating them.

  • Their individual volumes are treated as negligible compared with the container volume .

• Ideal gas particles move continuously and randomly.

  • They travel in straight lines between collisions.

• Collisions between ideal gas particles and with container walls are considered elastic .

  • Elastic means total kinetic energy is conserved during collisions.

  • Individual particles can exchange kinetic energy during collisions, but the total energy of the system is conserved under the ideal model.

• Ideal gas particles are assumed to have no significant intermolecular attractions or repulsions except during brief collisions.

• Gas pressure results from particles colliding with the walls of their container.

  • More frequent and/or more energetic collisions generally produce greater pressure.

• The average translational kinetic energy of gas particles depends only on absolute temperature.

  • Therefore, two different gases at the same Kelvin temperature have the same average translational kinetic energy .

  • This is very important: at the same temperature, Xe and H₂ have the same average translational KE.

  • They do not have the same average speed.

• Kinetic energy is related to mass and speed:

  • KE = ½mv²

  • If two particles have the same average KE but one has much less mass, the lighter particle must generally move faster.

  • Thus, at the same temperature:

  • lighter gas → greater characteristic molecular speed

  • heavier gas → lower characteristic molecular speed

• A common expression for root-mean-square speed is:

  • uᵣₘₛ = √(3RT/M)

  • where M is the molar mass expressed in units compatible with R.

The relationship tells you:

• uᵣₘₛ ∝ √T

  • and:

  • uᵣₘₛ ∝ 1/√M

  • Increasing temperature increases molecular speed.

  • Increasing molar mass at the same temperature decreases molecular speed.

• A Maxwell-Boltzmann distribution shows that particles in a gas do not all have identical speeds.

  • Instead, speeds are distributed across a range.

  • At higher temperature, the speed distribution shifts toward higher speeds, becomes broader, and its peak generally becomes lower.

  • At lower temperature, the distribution is more concentrated toward lower speeds.

  • At the same temperature, lighter gases have distributions shifted toward higher speeds than heavier gases.

  • Khan Academy's Unit 3 KMT section includes KMT assumptions, its connection to the gas laws, and Maxwell-Boltzmann distributions.

• KMT explains Boyle's law: when volume decreases at constant T and n, particles strike the walls more often, increasing pressure.

• KMT explains pressure-temperature behavior: increasing T raises average KE, producing more energetic collisions and increasing pressure when volume is fixed.

• It explains volume-temperature behavior: at constant pressure, increasing T requires increased volume so collision conditions can balance the increased particle speeds.

• The ideal gas law describes a model . Real gases do not behave perfectly ideally under all conditions.

• Two major ideal-gas assumptions eventually become inaccurate:

  • particles have negligible volume;

  • particles do not attract or repel one another.

  • Real gas particles do occupy physical space .

  • Real particles also experience intermolecular attractions and short-range repulsions.

• Gases behave most ideally at high temperature and low pressure .

  • At high temperature , particles have relatively high kinetic energy.

  • Their motion is large compared with the effects of intermolecular attraction, making attractions less important.

  • At low pressure , gas particles are far apart.

  • Their actual volumes become very small compared with the total container volume.

  • Attractions also become less significant because particles spend less time close together.

Therefore:

• high T + low P = most ideal behavior

Gases generally deviate most strongly from ideal behavior at:

• low T + high P

• At low temperature, particles move more slowly, so intermolecular attractions have a larger effect on their motion.

• At high pressure, particles are forced much closer together.

  • Their actual size now takes up a non-negligible fraction of the container volume.

  • Intermolecular attractions and repulsions become more important.

• Real-gas attractions can cause measured pressure to be lower than the simplest ideal prediction in some conditions because particles being pulled toward one another may strike the walls less strongly/frequently than the ideal model predicts.

• At very high compression, the finite volume of the particles and short-range repulsions can become extremely important.

• Substances with stronger intermolecular attractions tend to show larger deviations from ideal-gas behavior under comparable conditions.

• If AP asks which gas behaves most ideally , look for the gas with weaker intermolecular attractions under high-T/low-P conditions.

  • If asked which gas behaves least ideally , stronger IMFs plus low-T/high-P conditions are clues.

  • Khan Academy's current Unit 3 includes both an introduction to real gases and real-gas deviations from ideal behavior.