11 Gases and Kinetic Molecular Theory

How gases behave

A gas expands to fill its container and can be compressed readily because its particles are widely spaced and in constant motion. At the macroscopic scale, gas behavior is described by pressure $P$, volume $V$, temperature $T$, and amount $n$. Changing one of these properties can affect the others, depending on which conditions are held constant.

Relationships among gas properties

Each gas law describes a relationship while other conditions are held fixed:

• Boyle’s law: At constant temperature, pressure and volume are inversely related. If a gas is compressed to half its original volume, its pressure doubles: $P_1V_1=P_2V_2$.

• Charles’s law: At constant pressure, volume is proportional to absolute temperature: $\frac{V_1}{T_1}=\frac{V_2}{T_2}$. Heating a gas in a flexible balloon makes it expand.

• Amontons’s law (also called Gay-Lussac’s law): At constant volume, pressure is proportional to absolute temperature: $\frac{P_1}{T_1}=\frac{P_2}{T_2}$. Heating gas in a rigid container raises its pressure.

• Avogadro’s law: At constant pressure and temperature, volume is proportional to the amount of gas: $\frac{V_1}{n_1}=\frac{V_2}{n_2}$. Adding gas to a flexible container increases its volume.

Use kelvins for gas-law temperature calculations: $T(\mathrm{K})=T(^{\circ}\mathrm{C})+273.15$. Combining the pressure, volume, and temperature relationships gives the combined gas law for a fixed amount of gas:

Takeaway: Identify which properties remain constant before choosing a gas law, and convert temperatures to kelvins.

The ideal gas model

The Ideal gas equation combines pressure, volume, amount, and temperature in one relationship:

Here, $n$ is the amount in moles and $R$ is the ideal gas constant. Choose an $R$ value that matches the units in the problem. For example, use $0.08206\ \mathrm{L\,atm\,mol^{-1}\,K^{-1}}$ when pressure is in atmospheres and volume is in liters, or $8.314\ \mathrm{L\,kPa\,mol^{-1}\,K^{-1}}$ when pressure is in kilopascals and volume is in liters.

Example: Find the volume of $1.00\ \mathrm{mol}$ of an ideal gas at $1.00\ \mathrm{atm}$ and $273.15\ \mathrm{K}$. Rearrange the equation to solve for volume:

The ideal gas model is a useful approximation, especially at relatively low pressure and high temperature. Real gases can deviate from the model when particle volumes and intermolecular attractions become significant.

Takeaway: Use $PV=nRT$ when the gas’s pressure, volume, amount, or temperature is needed, and check that units are consistent.

Gas mixtures and Dalton’s law

For a mixture of nonreacting ideal gases, each component contributes a partial pressure: the pressure it would exert if it alone occupied the container at the same temperature. Dalton’s law states that total pressure is the sum of the component partial pressures:

The mole fraction of a component is its share of the mixture’s total amount:

A component’s partial pressure is its mole fraction multiplied by the total pressure:

Mole fractions sum to $1$.

Example: A mixture contains $1.00\ \mathrm{mol}$ of $\mathrm{N_2}$ and $3.00\ \mathrm{mol}$ of $\mathrm{O_2}$ at a total pressure of $2.00\ \mathrm{atm}$. Their mole fractions are $0.250$ and $0.750$, respectively. The partial pressures are $P_{\mathrm{N_2}}=0.500\ \mathrm{atm}$ and $P_{\mathrm{O_2}}=1.50\ \mathrm{atm}$; together they equal the total pressure of $2.00\ \mathrm{atm}$.

Takeaway: Find each component’s mole fraction, multiply it by the total pressure to get its partial pressure, and check that all partial pressures add to the total.

Explaining gas behavior at the particle level

Kinetic molecular theory explains gas behavior through a model of particles in motion. For an ideal gas, the model assumes:

1. Particles move continuously and randomly in straight lines between collisions.

2. Particle volumes are negligible compared with the container’s volume.

3. Pressure results from particles colliding with the container walls.

4. Particles exert no intermolecular attractions or repulsions, and collisions are elastic, conserving total kinetic energy.

5. The average translational kinetic energy of particles is proportional to absolute temperature.

The average translational kinetic energy is $\overline{KE}=\frac{3}{2}k_BT$ per particle, or $\overline{KE}=\frac{3}{2}RT$ per mole. At the same temperature, different gases have the same average translational kinetic energy, although lighter particles move faster on average.

The model connects particle motion to the gas laws. Raising temperature increases average particle speed and collision force. If volume is fixed, this raises pressure. If pressure is held constant, a heated gas expands so particles collide with the walls less frequently per unit area. Compressing a gas increases how often particles strike the walls, raising pressure.

Takeaway: The macroscopic gas laws reflect how particle motion, temperature, and collisions change together.