10.2 Temperature Dependence of Rates: Arrhenius Equation, Collision Theory, and Transition State Theory

Temperature Dependence of Rates: Arrhenius Equation, Collision Theory, and Transition State Theory

Virtually all chemical reaction rates exhibit profound sensitivity to temperature. For most homogeneous reactions, a modest temperature rise of $10\text{ K}$ near room temperature increases the rate by a factor of 2 to 3. Understanding the microscopic origin of this exponential temperature dependence requires bridging empirical kinetics with statistical collision mechanics and the thermodynamic formulation of activated transition states.

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1. The Arrhenius Equation and Activation Energy

In 1889, Svante Arrhenius proposed that reactants must acquire a critical threshold of kinetic energy—the activation energy ($E_a$)—before chemical bonds can rupture and re-form into products:

$k = A e^{-E_a / RT}$

• $k$: Reaction rate constant.
• $A$: Pre-exponential factor (or frequency factor), possessing the exact same dimensional units as $k$. It represents the total frequency of molecular collisions occurring with proper steric orientation.
• $E_a$: Activation energy (typically in $\text{J}\cdot\text{mol}^{-1}$ or $\text{kJ}\cdot\text{mol}^{-1}$), representing the minimum kinetic energy colliding molecules must possess along the line of centers to achieve reaction.
• $e^{-E_a / RT}$: The Boltzmann factor, representing the statistical fraction of molecular collisions having kinetic energy equal to or exceeding $E_a$ at absolute temperature $T$.
• $R$: Universal gas constant ($8.31446\text{ J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}$).
• $T$: Absolute temperature in Kelvin ($\text{K}$). Always convert Celsius to Kelvin ($T = \text{ }^\circ\text{C} + 273.15$).

Differential Form of the Arrhenius Equation

Differentiating the natural logarithm of the Arrhenius equation with respect to temperature:

$\ln k = \ln A - \frac{E_a}{RT} \implies \frac{d\ln k}{dT} = \frac{E_a}{RT^2}$

This differential form mirrors the van 't Hoff equation of chemical thermodynamics ($\frac{d\ln K}{dT} = \frac{\Delta H^\circ}{RT^2}$), revealing that activation energy governs the temperature coefficient of rate constants just as reaction enthalpy governs the temperature coefficient of equilibrium constants.

Integrated Two-Point Arrhenius Equation

Assuming $E_a$ and $A$ remain constant over the temperature range from $T_1$ to $T_2$:

$\ln\left(\frac{k_2}{k_1}\right) = -\frac{E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) = \frac{E_a}{R} \left( \frac{T_2 - T_1}{T_1 T_2} \right)$

Converting to common logarithms ($\log_{10}$): $\log_{10}\left(\frac{k_2}{k_1}\right) = \frac{E_a}{2.3026 R} \left( \frac{T_2 - T_1}{T_1 T_2} \right)$

The Arrhenius Graphical Plot

Plotting $\ln k$ versus the reciprocal of absolute temperature ($1/T$, in $\text{K}^{-1}$):

• Slope ($m$): $m = -\frac{E_a}{R} \implies E_a = -R \times m$
• $y$-Intercept ($b$): $b = \ln A \implies A = e^b$

Worked Step-by-Step Problem: Two-Point Activation Energy

Scenario: The rate constant for the thermal gas-phase decomposition of ethyl iodide ($\text{C}_2\text{H}_5\text{I} \rightarrow \text{C}_2\text{H}_4 + \text{HI}$) is measured as $k_1 = 3.25 \times 10^{-4}\text{ s}^{-1}$ at $T_1 = 600.0\text{ K}$, and increases to $k_2 = 6.73 \times 10^{-2}\text{ s}^{-1}$ at $T_2 = 700.0\text{ K}$. Calculate the experimental activation energy $E_a$ and the pre-exponential factor $A$.

Step 1: Compute ratio of rate constants and reciprocal temperature difference: $\frac{k_2}{k_1} = \frac{6.73 \times 10^{-2}\text{ s}^{-1}}{3.25 \times 10^{-4}\text{ s}^{-1}} = 207.08$ $\ln\left(\frac{k_2}{k_1}\right) = \ln(207.08) = 5.3331$ $\frac{1}{T_1} - \frac{1}{T_2} = \frac{1}{600.0} - \frac{1}{700.0} = 1.6667 \times 10^{-3} - 1.4286 \times 10^{-3} = 2.3810 \times 10^{-4}\text{ K}^{-1}$

Step 2: Calculate $E_a$: $E_a = R \times \frac{\ln(k_2 / k_1)}{\frac{1}{T_1} - \frac{1}{T_2}} = (8.31446\text{ J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}) \times \frac{5.3331}{2.3810 \times 10^{-4}\text{ K}^{-1}}$ $E_a = (8.31446)(22398.6\text{ K}) = 186232\text{ J/mol} \approx 186.2\text{ kJ/mol}$

Step 3: Calculate pre-exponential factor $A$ using $T_1 = 600.0\text{ K}$ data: $\ln A = \ln k_1 + \frac{E_a}{R T_1} = \ln(3.25 \times 10^{-4}) + \frac{186232}{(8.31446)(600.0)} = -8.0318 + 37.3312 = 29.2994$ $A = e^{29.2994} = 5.30 \times 10^{12}\text{ s}^{-1}$

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2. Collision Theory of Bimolecular Gas Reactions

Classical collision theory (developed by Trautz and Lewis) provides a mechanical microscopic model for bimolecular gas-phase reactions of hard spherical molecules ($A + B \rightarrow \text{Products}$):

$v = Z_{AB} \cdot f \cdot P$

1. Collision Frequency ($Z_{AB}$)

The collision density represents the total number of bimolecular collisions per unit volume per second: $Z_{AB} = \sigma_{AB} \left( \frac{8 k_B T}{\pi \mu} \right)^{1/2} N_A^2 [A][B]$

• Collision Cross-Section ($\sigma_{AB}$): $\sigma_{AB} = \pi d_{AB}^2 = \pi \left(\frac{d_A + d_B}{2}\right)^2$, where $d_A$ and $d_B$ are hard-sphere molecular diameters.
• Reduced Mass ($\mu$): $\mu = \frac{m_A m_B}{m_A + m_B}$.
• Mean Relative Speed: $\bar{c}_{rel} = \left( \frac{8 k_B T}{\pi \mu} \right)^{1/2}$.

2. Energy Factor ($f$)

The fraction of collisions possessing relative kinetic energy along the line of centers exceeding threshold energy $E_0$ is given by the Maxwell-Boltzmann distribution: $f = e^{-E_0 / RT}$

3. Steric Factor ($P$) and the Harpoon Mechanism

Experimentally measured rate constants are frequently smaller (or dramatically larger) than collision frequency predictions. The steric factor ($P$) corrects for the geometric probability that colliding molecules collide with proper mutual three-dimensional orientation:

$k = P Z_{AB}' e^{-E_0 / RT}$ where $Z_{AB}' = Z_{AB} / ([A][B])$.

• Normal Range ($P < 1$): For simple spherical atoms (e.g., $2\text{I} \rightarrow \text{I}_2$), $P \approx 1$. For complex organic molecules (e.g., Diels-Alder cycloadditions), $P \approx 10^{-2}$ to $10^{-6}$, because reactive functional groups must align with tight spatial precision.
• Anomalously Large Values ($P \gg 1$, $P \approx 10$ to $100$): Observed in gas-phase reactions between alkali metal atoms and halogen molecules ($K + \text{Br}_2 \rightarrow \text{KBr} + \text{Br}$). This phenomenon is governed by the harpoon mechanism: at relatively large separation distances ($\sim 0.5-0.8\text{ nm}$), the valence electron of potassium "harpoons" across space to the halogen via field-assisted electron transfer: $\text{K} + \text{Br}_2 \longrightarrow \text{K}^+ + \text{Br}_2^- \longrightarrow \text{KBr} + \text{Br}$ The resulting Coulombic electrostatic attraction ($-\frac{e^2}{4\pi\varepsilon_0 r}$) reels the ions together from distances far exceeding hard-sphere contact radii, yielding effective collision cross-sections and steric factors orders of magnitude greater than classical kinetic theory predictions.

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3. Transition State Theory (Eyring-Polanyi Theory)

Developed by Henry Eyring, Michael Polanyi, and Eugene Wigner (1935), Transition State Theory (TST) models chemical reactions through the continuous topology of a Potential Energy Surface (PES). As reactant molecules approach, their potential energy rises along the minimum-energy reaction coordinate to reach an energetic saddle point known as the activated complex or transition state ($[AB]^\ddagger$).

Fundamental Postulates of TST

1. Reactants are in a state of virtual thermodynamic quasi-equilibrium with the activated complex: $A + B \rightleftharpoons [AB]^\ddagger \longrightarrow \text{Products}$ $K^\ddagger = \frac{[AB]^\ddagger}{[A][B]}$
2. The activated complex possesses all ordinary molecular degrees of freedom except for one special vibrational mode along the reaction coordinate, which has been converted into translational motion over the barrier top.
3. The universal frequency of barrier passage is $\nu^\ddagger = \frac{k_B T}{h}$, where $k_B$ is Boltzmann's constant and $h$ is Planck's constant.

The Eyring Rate Equation

$\text{Rate} = \kappa \nu^\ddagger [AB]^\ddagger = \kappa \left( \frac{k_B T}{h} \right) K^\ddagger [A][B]$ $k = \kappa \frac{k_B T}{h} K^\ddagger$ where $\kappa$ is the transmission coefficient (the quantum probability that an activated complex passing the barrier summit proceeds forward to products without reflecting backward; for most adiabatic reactions, $\kappa \approx 1$).

Thermodynamic Formulation of Transition State Theory

From classical thermodynamics, the standard Gibbs free energy of activation is $\Delta G^\ddagger = -RT \ln K^\ddagger \iff K^\ddagger = e^{-\Delta G^\ddagger / RT}$. Because $\Delta G^\ddagger = \Delta H^\ddagger - T\Delta S^\ddagger$:

$k = \frac{k_B T}{h} e^{-\Delta G^\ddagger / RT} = \frac{k_B T}{h} e^{\Delta S^\ddagger / R} e^{-\Delta H^\ddagger / RT}$

• $\Delta G^\ddagger$: Standard Gibbs free energy of activation.
• $\Delta H^\ddagger$: Standard enthalpy of activation.
• $\Delta S^\ddagger$: Standard entropy of activation.

Interpreting Activation Entropy ($\Delta S^\ddagger$)

• $\Delta S^\ddagger < 0$ (Negative Activation Entropy): The activated complex is substantially more ordered, rigid, or constrained than the separated reactants. This is diagnostic of associative mechanisms, bimolecular additions, cyclizations, or reactions involving extensive electrostriction (where forming ionic charges freeze solvent dipoles into rigid hydration shells).
• $\Delta S^\ddagger > 0$ (Positive Activation Entropy): The activated complex is less constrained and more disordered than reactants. This is diagnostic of dissociative mechanisms, unimolecular bond fission, or charge neutralization steps that release bound solvent molecules into bulk solution.

Interconverting Arrhenius and Eyring Parameters

Applying the thermodynamic definition $E_a \equiv RT^2 \frac{d\ln k}{dT}$ to the Eyring equation:

At $298.15\text{ K}$, $RT = (8.3145\text{ J/mol}\cdot\text{K})(298.15\text{ K}) = 2.479\text{ kJ/mol}$, while the universal Eyring frequency factor is: $\frac{k_B T}{h} = \frac{(1.38065 \times 10^{-23}\text{ J/K})(298.15\text{ K})}{6.62607 \times 10^{-34}\text{ J}\cdot\text{s}} = 6.2125 \times 10^{12}\text{ s}^{-1}$

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4. Kinetic Isotope Effect (KIE)

The Kinetic Isotope Effect (KIE) is the change in the rate of a chemical reaction when an atom in a reactant is replaced by one of its heavier isotopes. Most commonly, hydrogen ($^1\text{H}$, protium) is replaced by deuterium ($^2\text{H}$ or $\text{D}$):

$\text{KIE} \equiv \frac{k_H}{k_D}$

Quantum Mechanical Origin: Zero-Point Energy (ZPE)

In quantum mechanics, a vibrating chemical bond behaves as a quantum harmonic oscillator. Even at absolute zero ($0\text{ K}$), a bond possesses a non-zero Zero-Point Energy (ZPE): $E_0 = \frac{1}{2} h \nu = \frac{1}{2} \hbar \sqrt{\frac{k_{force}}{\mu}}$ Because isotopic substitution alters nuclear mass without altering the electronic potential energy surface (Born-Oppenheimer approximation), the force constant $k_{force}$ is identical for $\text{C-H}$ and $\text{C-D}$ bonds. However, the reduced mass of $\text{C-D}$ is nearly twice that of $\text{C-H}$: $\mu_{\text{C-H}} \approx \frac{12 \times 1}{12 + 1} = 0.923\text{ u}, \quad \mu_{\text{C-D}} \approx \frac{12 \times 2}{12 + 2} = 1.714\text{ u}$ $\nu_{\text{C-D}} \approx \frac{1}{\sqrt{1.857}} \nu_{\text{C-H}} \approx 0.734 \,\nu_{\text{C-H}}$

• The vibrational stretching frequency of a typical $\text{C-H}$ bond is $\sim 2900-3000\text{ cm}^{-1}$ ($\text{ZPE} \approx 17.5\text{ kJ/mol}$), whereas for $\text{C-D}$ it is $\sim 2100-2200\text{ cm}^{-1}$ ($\text{ZPE} \approx 12.8\text{ kJ/mol}$).
• The $\text{C-D}$ bond sits deeper in its potential energy well by $\Delta \text{ZPE} \approx 4.7\text{ kJ/mol}$.
• In a transition state where the $\text{C-H}$ (or $\text{C-D}$) bond is completely broken, this stretching vibrational mode is lost in both isotopomers. Consequently, breaking the $\text{C-D}$ bond requires an activation energy higher by $\Delta \text{ZPE}$: $E_{a, D} - E_{a, H} \approx \Delta \text{ZPE} \approx 4.7-5.0\text{ kJ/mol}$ $\frac{k_H}{k_D} = \exp\left( \frac{E_{a, D} - E_{a, H}}{RT} \right) \approx \exp\left( \frac{4800\text{ J/mol}}{(8.314)(298.15)} \right) = e^{1.936} \approx 6.9$

Primary vs. Secondary KIE