3.5

Kinetic Molecular Theory (KMT) and Gas Properties

• Definition of Kinetic Molecular Theory: Kinetic Molecular Theory (KMT) relates the microscopic properties and motions of individual gas particles to the macroscopic physical properties of the gas sample.

• Postulate 1: Particle Volume: Gas particles are extremely small relative to the vast distances separating them. Consequently, the volume of individual gas particles is considered negligible ($\approx 0$). When measuring or discussing the volume of a gas, it refers entirely to the volume of the container enclosing the gas particles.

• Postulate 2: Particle Motion and Pressure: Gas particles are in constant, random, straight-line motion. They change direction only when they collide with each other or with the walls of the container. Collisions between gas particles and the interior walls of the container exert force per unit area, which generates gas pressure ($P$).

• Postulate 3: Elastic Collisions and Intermolecular Forces: Collisions between gas particles are perfectly elastic, meaning there is no net loss or gain of kinetic energy during collisions. In an ideal gas model, particles exert no attractive or repulsive forces on one another, meaning intermolecular forces (IMFs) are assumed to be zero ($\text{IMF} = 0$).

• Postulate 4: Average Kinetic Energy: The average kinetic energy ($KE_{\text{avg}}$) of gas particles is directly proportional to the absolute temperature of the gas in Kelvin ($\text{K}$). Average kinetic energy is fundamentally another term for absolute Kelvin temperature.

• Conditions for Ideal Gas Behavior: Ideal gas behavior is maximized under conditions of high temperature and low pressure.

  • High Temperature: Gas particles move rapidly with high kinetic energy, preventing weak attractive forces (IMFs) from pulling particles together during brief encounters.

  • Low Pressure: Gas particles occupy a large volume relative to their quantity, rendering particle-particle distances large and individual particle volumes negligible.

• Deviations in Real Gases:

  • Real gas particles possess actual physical volumes and experience intermolecular forces (such as London dispersion forces, dipole-dipole forces, and hydrogen bonding).

  • Larger gas molecules with expanded electron clouds exhibit greater polarizability, leading to stronger intermolecular forces and pronounced deviations from ideal behavior.

Mathematical Foundations of Kinetic Energy and Gas Laws

• Kinetic Energy Formula:

  • $KE = \frac{1}{2} m v^2$

  • $KE$ represents kinetic energy measured in Joules ($\text{J}$).

  • $m$ represents the mass of a particle measured in kilograms ($\text{kg}$).

  • $v$ represents particle velocity measured in meters per second ($\text{m/s}$).

• Mass vs. Velocity Relationship at Constant Temperature:

  • All gases at the exact same temperature share the exact same average kinetic energy ($KE_{\text{avg}}$).

  • Because $KE = \frac{1}{2} m v^2$, particle velocity depends inversely on particle mass at a given temperature.

  • Smaller, lighter gas particles move at significantly faster velocities ($v$) to achieve the same kinetic energy as larger, heavier particles.

  • Larger, heavier gas particles move at significantly slower velocities ($v$) at the same temperature.

• Gas Law Relationships derived from Combined Gas Law:

  • Direct Relationships (Stacked Fraction Form):

    • Volume and Temperature: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

    • Pressure and Temperature: $\frac{P_1}{T_1} = \frac{P_2}{T_2}$

    • Volume and Moles: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$

    • Direct relationships indicate that increasing one variable causes a proportional increase in the other.

  • Inverse Relationship (Linear Product Form):

    • Pressure and Volume: $P_1 V_1 = P_2 V_2$

    • An inverse relationship indicates that increasing container volume decreases collision frequency, thereby decreasing pressure.

Graham's Law of Effusion and Diffusion

• Definitions:

  • Diffusion: The process by which gas particles spread throughout a space or mix with another gas due to constant random motion.

  • Effusion: The passage of gas particles through a tiny opening or pinhole into an evacuated chamber.

• Graham's Law Equation:

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

  • $\text{Rate}_1$ and $\text{Rate}_2$ represent the rates of diffusion or effusion for Gas 1 and Gas 2.

  • $M_1$ and $M_2$ represent the molar masses of Gas 1 and Gas 2.

• Inverse Relationship to Particle Size/Mass: As particle size and molar mass increase, the rate of effusion and diffusion decreases. Larger, heavier particles travel slower and pass through small openings at a lower rate than smaller, lighter particles.

Maxwell-Boltzmann Distribution Curves

• Graph Overview and Axes:

  • The Maxwell-Boltzmann distribution plots the kinetic energy or speed of gas particles against the relative fraction or number of particles possessing that speed.

  • $x$\text{-axis}: Particle speed / velocity ($v$).

  • $y$\text{-axis}: Number or fraction of particles.

• Key Graph Features:

  • Asymmetrical Skew: The distribution curve skews to the left with a long rightward tail because particle speed cannot drop below zero ($0\,\text{m/s}$).

  • Peak of the Curve: Represents the most probable speed (the specific velocity possessed by the largest fraction of particles in the sample).

  • Average Speed: Positioned slightly to the right of the peak peak toward the tail end due to the asymmetrical rightward skew.

  • Area Under the Curve: Represents the total number of gas particles in the sample.

Graph Manipulation Scenarios and Temperature/Mass Effects

• Effect of Temperature Changes on the Distribution Curve:

  • Heating a Gas (Higher Temperature):

    • As temperature increases, particles gain kinetic energy and move faster on average.

    • The peak of the curve shifts to the right along the $x$\text{-axis}.

    • The height of the peak decreases and the curve broadens out (flattens).

    • If the total quantity of gas remains constant, the area under the curve remains identical.

  • Cooling a Gas (Lower Temperature):

    • As temperature decreases, average particle speed drops.

    • The peak of the curve shifts to the left toward lower speeds.

    • The height of the peak increases and the curve becomes narrower/skinnier.

• Effect of Particle Molar Mass at Constant Temperature:

  • Lighter Gas Particles (e.g., $H_2$ vs. $O_2$):

    • Lighter particles move faster at a given temperature.

    • The distribution curve shifts to the right and broadens out.

  • Heavier Gas Particles (e.g., $O_2$ vs. $H_2$):

    • Heavier particles move slower at a given temperature.

    • The distribution curve shifts to the left, becoming taller and narrower.

  • Example Comparison: In a mixture of Hydrogen gas ($H_2$) and Oxygen gas ($O_2$) at the same temperature, $H_2$ particles (molar mass $\approx 2.02\,\text{g/mol}$) travel at a significantly higher average speed than $O_2$ particles (molar mass $\approx 32.00\,\text{g/mol}$). The $H_2$ curve is shifted right and flattened relative to the tall, left-shifted $O_2$ curve.

• Effect of Quantity Changes and Complex Scenarios:

  • Removing Gas at Constant Temperature:

    • Particle speed distribution is unchanged because temperature remains constant.

    • The curve does not shift left or right.

    • Because total particle count is reduced, the peak height decreases and the overall area under the curve shrinks proportionally.

  • Adding Hotter Gas to an Existing Sample:

    • Adding a hotter gas introduces particles with higher velocities, shifting the overall distribution to the right.

    • Because total particle count increases, the overall area under the curve expands (taller/larger curve) to reflect the added gas volume.