Gases: Partial Pressures, Kinetic Molecular Theory, Diffusion, Effusion, and Real Gases

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

• Mole Fraction ($X_A$): The ratio of the number of moles of component A to the total number of moles in the mixture. It is dimensionless.

• Relationship between partial pressure and mole fraction: $P_A = X_A \times P_{\text{total}}$

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 can be calculated as:

• $P_{CO_2} = 0.015 \times 745 = 11.2$ mm Hg

• $P_{O_2} = 0.18 \times 745 = 134.1$ mm Hg

• $P_{Ar} = 0.805 \times 745 = 599.2$ 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 at the molecular level. It is based on several key assumptions:

• 1) $V_{\text{gas}} \ll V_{\text{container}}$ (The volume of gas molecules is much smaller than the volume of the container.)

• 2) Gas molecules are in constant, random motion.

• 3) Collisions between gas molecules are perfectly elastic; energy may be transferred but the total energy remains unchanged.

• 4) Gas molecules do not attract or repel each other.

• 5) The average kinetic energy of gas molecules is proportional to the absolute temperature (in Kelvin): $\text{KE}_{\text{avg}} \propto T$

Any gas at the same temperature will 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 speeds, described by the Maxwell-Boltzmann distribution. As temperature increases, the distribution broadens and the average speed increases.

• Root Mean Square Speed ($u_{rms}$): The square root of the average of the squares of the molecular speeds. $u_{rms} = \sqrt{\frac{3RT}{M}}$ where $R$ is the gas constant, $T$ is temperature in Kelvin, and $M$ is molar mass.

• At higher temperatures, molecules move faster and the distribution flattens.

• For two gases at the same temperature, the gas with the higher molar mass will have a lower average speed.

Example: The distribution of molecular speeds for O2 and H2 at the same temperature shows H2 molecules move much faster on average.

Mean Free Path

Mean Free Path: The average distance a gas molecule travels between collisions.

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

Diffusion and Effusion

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 hole into a vacuum.

Graham's Law of Diffusion and Effusion

Graham's Law describes the relationship between the rates of diffusion or effusion and the molar masses of gases.

• The rate of diffusion or effusion of a gas is inversely proportional to the square root of its molar mass.

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

• For effusion, the time taken is inversely proportional to the rate: $\text{Rate} \propto \frac{1}{\text{Time}}$

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

Non-Ideal Gas Behavior

Real gases deviate from ideal behavior at high pressures and low temperatures. The van der Waals equation introduces corrections for intermolecular forces and the finite volume of gas molecules:

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

• Where $a$ corrects for intermolecular attractions and $b$ corrects for the volume occupied by gas molecules.

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

*Additional info: Some missing context and equations have been filled in based on standard General Chemistry curriculum and textbook conventions.*