5.3 Kinetic Molecular Theory & Maxwell-Boltzmann Distribution

5.3 Kinetic Molecular Theory & Maxwell-Boltzmann Distribution

The Kinetic Molecular Theory (KMT), developed by Clausius, Maxwell, and Boltzmann, connects the macroscopic gas laws ($PV = nRT$) to microscopic molecular mechanics.

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Five Core Postulates of KMT

1. Negligible Particle Volume: Gas particles (atoms or molecules) are point masses whose collective physical volume is negligible compared to the total container volume. Most of a gas is empty space.
2. Continuous, Random Motion: Particles move continuously in straight lines in random directions, obeying Newton's laws until colliding with walls or other particles.
3. Perfectly Elastic Collisions: Collisions between particles and with container walls are completely elastic. While individual particles exchange kinetic energy, total kinetic energy is conserved at constant temperature.
4. No Intermolecular Forces: Gas particles experience no attractive or repulsive intermolecular forces; they interact solely through instantaneous physical collisions.
5. Kinetic Energy Proportional to Absolute Temperature: The average translational kinetic energy ($\overline{KE}$) of gas particles depends strictly on absolute thermodynamic temperature ($T$) in Kelvin, regardless of gas identity or mass: $\overline{KE} = \frac{1}{2} m \overline{u^2} = \frac{3}{2} k_B T = \frac{3}{2}\left(\frac{R}{N_A}\right)T$ Per mole of gas: $\overline{KE}_{\text{molar}} = \frac{3}{2} RT$

At a given temperature, all gases possess identical average translational kinetic energy. A heavy krypton atom ($\mathcal{M} = 83.8\text{ g/mol}$) and a light helium atom ($\mathcal{M} = 4.00\text{ g/mol}$) at $300\text{ K}$ have the same $\overline{KE} = \frac{3}{2}RT$. Because $\overline{KE} = \frac{1}{2}m\overline{u^2}$, lighter molecules must travel with higher average velocities.

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Microscopic Derivation of Gas Pressure

Gas pressure results from the cumulative force of frequent, elastic molecular collisions against container walls.

Consider a particle of mass $m$ with velocity $u_x$ colliding elastically with a perpendicular container wall. Momentum reverses from $+mu_x$ to $-mu_x$, transferring momentum: $\Delta p_x = 2 m u_x$ In a container of length $L$, the particle collides with the wall every $\Delta t = 2L / u_x$ seconds. By Newton's second law, average force is: $F_1 = \frac{\Delta p_x}{\Delta t} = \frac{2 m u_x}{2L / u_x} = \frac{m u_x^2}{L}$ Summing over all $N$ particles moving isotropically in three dimensions ($\overline{u_x^2} = \frac{1}{3}\overline{u^2}$) and dividing by surface area ($A = L^2$): $P = \frac{F_{\text{total}}}{A} = \frac{1}{3}\frac{N m \overline{u^2}}{V}$ Multiplying by $V$ and substituting $\frac{1}{2}N_A m \overline{u^2} = \frac{3}{2}RT$ yields: $PV = nRT$

• Boyle's Law: Halving the volume at constant temperature doubles the particle density ($N/V$), which doubles the wall collision frequency and therefore doubles the pressure.
• Gay-Lussac's Law: Heating gas increases molecular speeds ($\overline{u^2}$), making collisions more frequent and energetic, raising pressure.

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Characteristic Molecular Speeds

Thermal collisions generate a distribution of molecular speeds summarized by three metrics:

1. Most Probable Speed ($u_{mp}$): Speed corresponding to the peak of the distribution: $u_{mp} = \sqrt{\frac{2RT}{\mathcal{M}}}$
2. Average (Mean) Speed ($u_{avg}$): Arithmetic mean of all molecular speeds: $u_{avg} = \sqrt{\frac{8RT}{\pi \mathcal{M}}} \approx \sqrt{\frac{2.55 RT}{\mathcal{M}}}$
3. Root-Mean-Square Speed ($u_{rms}$): Speed directly related to average kinetic energy: $u_{rms} = \sqrt{\overline{u^2}} = \sqrt{\frac{3RT}{\mathcal{M}}}$

Relative Hierarchy and Speed Formulas

Because $\sqrt{2} < \sqrt{8/\pi} < \sqrt{3}$: $u_{mp} < u_{avg} < u_{rms} \quad (1.000 : 1.128 : 1.225)$

Calculation Protocol for $u_{rms}$

In $u_{rms} = \sqrt{3RT/\mathcal{M}}$:

• Use $R = 8.3145\text{ J}/(\text{mol}\cdot\text{K}) = 8.3145\text{ kg}\cdot\text{m}^2/(\text{s}^2\cdot\text{mol}\cdot\text{K})$.
• Express molar mass $\mathcal{M}$ in kilograms per mole ($\text{kg/mol}$) (e.g., $0.02802\text{ kg/mol}$ for $\text{N}_2$). This yields velocities in standard meters per second ($\text{m/s}$).

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The Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution defines the speed distribution of ideal gas particles at thermal equilibrium: $f(u) = 4\pi \left(\frac{\mathcal{M}}{2\pi RT}\right)^{3/2} u^2 \exp\left(-\frac{\mathcal{M}u^2}{2RT}\right)$

Key Features

• Asymmetric Curve: Starts at the origin ($f(0) = 0$), rises due to $u^2$, peaks at $u_{mp}$, and decays with a long high-speed exponential tail.
• Constant Area: Total area under the curve is normalized to $1.0$ ($100\%$ of particles).

Temperature and Molar Mass Effects

Connection to Chemical Kinetics and Activation Energy

The Maxwell-Boltzmann speed distribution provides the physical foundation for collision theory in chemical reaction kinetics. In any reaction, only collisions possessing kinetic energy greater than or equal to the activation energy ($E_a$) can overcome electrostatic repulsions and break existing chemical bonds. Because the high-energy exponential tail expands dramatically with temperature, even a modest temperature rise of $10\text{ K}$ substantially increases the fraction of molecules with $KE \ge E_a$, causing reaction rates to accelerate noticeably.

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Worked Example: Calculating $u_{rms}$

Compare $u_{rms}$ for helium ($\text{He}, 4.003\text{ g/mol} = 0.004003\text{ kg/mol}$) and xenon ($\text{Xe}, 131.29\text{ g/mol} = 0.13129\text{ kg/mol}$) at $298.15\text{ K}$.

$u_{rms}(\text{He}) = \sqrt{\frac{3(8.3145)(298.15)}{0.004003}} = \sqrt{\frac{7436.9}{0.004003}} = 1363\text{ m/s}$ $u_{rms}(\text{Xe}) = \sqrt{\frac{3(8.3145)(298.15)}{0.13129}} = \sqrt{\frac{7436.9}{0.13129}} = 238\text{ m/s}$ Helium moves $\sqrt{131.29 / 4.003} = 5.73$ times faster than xenon at the same temperature.