Lecture 21

Mixtures of Gases: Dalton's Law of Partial Pressures

Partial Pressure and Dalton's Law

When gases are mixed in a container, each gas exerts its own pressure independently of the others. This individual pressure is called the partial pressure. Dalton's Law of Partial Pressures states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of each individual gas.

• Partial Pressure: The pressure exerted by a single gas in a mixture.

• Dalton's Law of Partial Pressures: The total pressure of a mixture of gases is the sum of the pressures each gas would exert if it were alone in the container.

Equation:

$P_{\text{total}} = P_1 + P_2 + P_3 + \ldots$

Mole Fraction and Partial Pressure

The mole fraction of a component in a mixture is the ratio of the number of moles of that component to the total number of moles of all components. The partial pressure of a gas can be calculated using its mole fraction and the total pressure.

• Mole Fraction (X): A dimensionless quantity expressing the ratio of moles of one component to the total moles in the mixture.

• Partial Pressure Formula: $P_A = X_A \times P_{\text{total}}$

Example: If a mixture contains 0.015 mole fraction CO2, 0.18 mole fraction O2, and 0.805 mole fraction Ar, and the total pressure is 745 mm Hg, the partial pressure of each gas is calculated by multiplying its mole fraction by the total pressure.

• Partial pressure of CO2: $P_{CO_2} = 0.015 \times 745 = 11.18$ mm Hg

• Partial pressure of O2: $P_{O_2} = 0.18 \times 745 = 134.1$ mm Hg

• Partial pressure of Ar: $P_{Ar} = 0.805 \times 745 = 599.7$ mm Hg

Practice: If CO2 is removed, recalculate the total pressure and the new mole fraction of O2.

Kinetic Molecular Theory

Five Basic Postulates

The kinetic molecular theory explains the behavior of gases based on the motion of their molecules. It is built on five fundamental postulates:

1. Volume of gas molecules is negligible: $V_{\text{gas}} \ll V_{\text{container}}$

2. Gas molecules are in constant, random motion.

3. Collisions are perfectly elastic: Energy can be transferred between molecules, but the total energy remains unchanged.

4. No intermolecular forces: Gas molecules neither attract nor repel each other.

5. Average kinetic energy is proportional to temperature: $\text{KE}_{\text{avg}} \propto T$ (in Kelvin). All gases at the same temperature have the same average kinetic energy.

Distribution of Molecular Speeds

Not all gas molecules move at the same speed. The kinetic molecular theory predicts a distribution of molecular speeds, described by the Maxwell-Boltzmann distribution. This distribution shows how molecular speeds vary at different temperatures.

• Root Mean Square Speed ($u_{rms}$): The square root of the average of the squares of the molecular speeds.

• Maxwell-Boltzmann Distribution: As temperature increases, the distribution broadens and shifts to higher speeds.

Equation for Root Mean Square Speed:

$u_{rms} = \sqrt{\frac{3RT}{M}}$

Where $R$ is the gas constant, $T$ is temperature in Kelvin, and $M$ is molar mass in kg/mol.

• At higher temperatures, molecules move faster and the spread of speeds increases.

• At the same temperature, heavier molecules move more slowly on average than lighter molecules.

Example: The distribution of molecular speeds for O2 and H2 at the same temperature shows H2 molecules move faster due to their lower molar mass.

Mean Free Path

The mean free path is the average distance a gas molecule travels between collisions with other molecules.

• Depends on pressure, temperature, and size of molecules.

Diffusion and Effusion: Graham's Laws

Definitions

• Diffusion: The gradual mixing of molecules of one gas with those of another due to random molecular motion.

• Effusion: The process by which gas molecules escape through a small opening under pressure.

Graham's Law of Diffusion and Effusion

Graham's Law states that the rate of diffusion or effusion of a gas is inversely proportional to the square root of its molar mass.

• Diffusion/Effusion Rate Formula:

$\frac{\text{Rate}_1}{\text{Rate}_2} = \sqrt{\frac{M_2}{M_1}}$

• Where $M_1$ and $M_2$ are the molar masses of gases 1 and 2, respectively.

• The rate of effusion is inversely proportional to the time required for effusion.

• Faster effusion means less time to effuse.

Example: If an unknown gas effuses at a rate that is 0.468 times that of O2 at the same temperature, its molar mass can be found using Graham's Law:

$\frac{\text{Rate}_{\text{unknown}}}{\text{Rate}_{O_2}} = 0.468 = \sqrt{\frac{M_{O_2}}{M_{\text{unknown}}}}$

Solve for $M_{\text{unknown}}$:

$M_{\text{unknown}} = \frac{M_{O_2}}{(0.468)^2}$

Real Gases: The van der Waals Equation

Deviations from Ideal Gas Behavior

Real gases deviate from ideal behavior at high pressures and low temperatures due to intermolecular forces and the finite volume of molecules. The van der Waals equation corrects for these deviations.

• van der Waals Equation:

$\left(P + a\frac{n^2}{V^2}\right)(V - nb) = nRT$

• $a$ corrects for intermolecular attractions.

• $b$ corrects for the finite volume of molecules.

Example: Use the van der Waals equation to calculate the pressure of a real gas given $a$, $b$, $n$, $V$, and $T$.

*Additional info: Some definitions and equations have been expanded for clarity and completeness. Example calculations and context have been added to make the notes self-contained.*