lecture slides chapter 9

Properties of Gases: The Air We Breathe

General Properties of Gases

Gases are a unique state of matter characterized by their ability to expand and fill any container. Unlike liquids and solids, gases are highly compressible and their properties are influenced by several variables.

• Compressibility: Gases can be compressed; liquids and solids cannot.

• Volume and Pressure: The volume of a gas is inversely proportional to its pressure.

• Volume and Temperature: The volume of a gas is directly proportional to its temperature.

• Volume and Quantity: The volume is directly proportional to the amount (quantity) of gas present.

• Miscibility: Gases are generally miscible, meaning they can mix in any proportion.

• Diffusion and Molar Mass: The rate at which gases diffuse is inversely proportional to their molar mass.

• Expansion: Gases expand to occupy the entire volume of their container.

Diffusion, Effusion, and Kinetic Molecular Theory of Gases

Kinetic Molecular Theory (KMT)

The behavior of gases can be explained using the kinetic molecular theory, which is based on several key assumptions:

• Negligible Volume: Gas molecules have tiny volumes compared to the total volume the gas occupies.

• No Intermolecular Forces: Gas particles do not interact with each other.

• Elastic Collisions: Collisions between gas molecules are perfectly elastic (no energy is lost).

• Kinetic Energy and Temperature: The average kinetic energy of gas particles is proportional to the absolute temperature.

Effusion and Diffusion

Effusion is the process by which gas particles escape through tiny holes in a container. Diffusion is the mixing of one gas into another.

• Effusion Rate: The rate of effusion is inversely proportional to the square root of the molar mass of the gas.

Graham's Law of Effusion:

• The rate of effusion ($r$) of a gas is given by:

$r \propto \frac{1}{\sqrt{\text{density}}} \propto \frac{1}{\sqrt{\text{molar mass}}}$

Kinetic Molecular Theory of Gases: Root-Mean-Square Speed

Root-Mean-Square Speed ($u_{rms}$)

The root-mean-square speed is a measure of the average speed of gas particles, which is related to their kinetic energy.

• Definition: $u_{rms}$ is the speed of a particle whose kinetic energy is exactly the same as the average kinetic energy of all the particles in the sample.

• Distribution: Not all particles move at the same speed; there is a distribution of speeds in any sample of gas.

Kinetic Energy of Gases

Average Kinetic Energy

The average kinetic energy of a gas particle is directly proportional to the temperature of the gas.

• Formula for Kinetic Energy:

$KE_{avg} = \frac{1}{2} m u_{rms}^2$

$KE_{avg} = \frac{3}{2} k_B T$

• Variables: $m$ = mass of particle, $u_{rms}$ = root-mean-square speed, $k_B$ = Boltzmann constant, $T$ = absolute temperature (Kelvin).

• Implication: As temperature increases, the average kinetic energy and the speed of gas particles increase.

Speed Distribution Among Gases

Different gases at the same temperature have different speed distributions due to differences in molar mass. Lighter gases (e.g., He) move faster on average than heavier gases (e.g., O2).

• Example: At the same temperature, helium atoms move faster than oxygen molecules.

Additional info: The root-mean-square speed can be calculated using the formula: $u_{rms} = \sqrt{\frac{3RT}{M}}$ where $R$ is the gas constant, $T$ is temperature in Kelvin, and $M$ is the molar mass in kg/mol.