4.7 Collision Theory, Transition States & Empirical Rate Laws

4.7 Collision Theory, Transition States & Empirical Rate Laws

Chemical kinetics investigates the rates of chemical reactions, the molecular pathways (mechanisms) by which reactants convert to products, and the energy barriers separating them. Understanding rate laws, integrated rate equations, and catalyst action is vital for interpreting enzyme kinetics, drug clearance, and metabolic control.

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1. Collision Theory, Transition State Theory, and Energy Coordinate Diagrams

Collision Theory

For a chemical reaction to occur, reactant molecules must collide. According to collision theory, the rate of a reaction is proportional to the rate of effective collisions, which requires two criteria:

1. Kinetic Energy Criterion: Colliding molecules must possess kinetic energy greater than or equal to the activation energy ($E_a$). Collisions with insufficient kinetic energy bounce apart elastically without reacting.
2. Steric Orientation Criterion: Molecules must collide with proper spatial orientation so that reactive functional groups align to permit orbital overlap and bond rearrangement.

$$\text{Reaction Rate} = Z \cdot p \cdot e^{-E_a / RT}$$

Where $Z$ is collision frequency, $p$ is the steric orientation probability factor, and the exponential term $e^{-E_a / RT}$ represents the fraction of collisions possessing energy $\ge E_a$.

Transition State Theory (Activated Complex)

As reacting molecules collide, their electron clouds distort, initial bonds stretch and weaken, and new bonds begin to form. At the point of maximum potential energy along the reaction pathway, the system forms a fleeting, unstable molecular species called the transition state (or activated complex, denoted by $\ddagger$):

• The transition state represents a high-energy potential barrier that cannot be isolated or trapped; its typical lifetime is on the order of femtoseconds ($10^{-13}\text{ to } 10^{-14}\text{ s}$).
• Activation Energy ($E_a$): The difference in potential energy between the reactants and the transition state ($E_a = E_\ddagger - E_{\text{reactants}}$). For reversible reactions, the reverse activation energy is $E_{a,\text{rev}} = E_\ddagger - E_{\text{products}}$, so: $$\Delta H_{\text{rxn}} = E_{a,\text{fwd}} - E_{a,\text{rev}}$$
• Intermediates versus Transition States: An intermediate resides in a local potential energy minimum between two consecutive transition states. Intermediates have fully formed chemical bonds and can frequently be detected or isolated, whereas transition states reside at local energy maxima.

Kinetic versus Thermodynamic Reaction Products

When a chemical reaction can yield two different products through competing pathways (e.g., electrophilic addition to conjugated dienes):

• Kinetic Product:
  • Forms through the transition state with the lower activation energy ($E_{a,\text{kin}} < E_{a,\text{thermo}}$).
  • Forms faster at lower temperatures because a larger fraction of molecules have energy exceeding the lower barrier.
  • Less thermodynamically stable (higher final free energy).
  • Dominates under conditions of low temperature, short reaction times, and irreversible pathways.
• Thermodynamic Product:
  • Forms through a transition state with a higher activation energy ($E_{a,\text{thermo}} > E_{a,\text{kin}}$).
  • Forms slower, but produces the more stable product (lower final free energy, more negative $\Delta G$).
  • Dominates under conditions of high temperature, extended reaction times, and reversible equilibrium, where molecules possess sufficient energy to overcome the higher barrier and equilibrate toward the global free energy minimum.

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2. Reaction Rates, Rate Laws, and the Initial Rates Method

Defining Reaction Rate

For a general reaction: $a\text{A} + b\text{B} \longrightarrow c\text{C} + d\text{D}$, the instantaneous rate of reaction is defined by the rate of disappearance of reactants or appearance of products, normalized by stoichiometric coefficients:

$$\text{Rate} = -\frac{1}{a} \frac{\Delta [\text{A}]}{\Delta t} = -\frac{1}{b} \frac{\Delta [\text{B}]}{\Delta t} = +\frac{1}{c} \frac{\Delta [\text{C}]}{\Delta t} = +\frac{1}{d} \frac{\Delta [\text{D}]}{\Delta t}$$

• Units of reaction rate: Molarity per second ($\text{M/s}$ or $\text{mol}/(\text{L}\cdot\text{s})$).
• Negative signs reflect the consumption of reactants; positive signs reflect the production of products.

Differential Rate Laws

A rate law mathematically relates the reaction rate to the molar concentrations of reactants raised to empirical powers:

$$\text{Rate} = k [\text{A}]^m [\text{B}]^n$$

Where:

• $k$ is the empirical rate constant (temperature-dependent).
• $m$ is the partial reaction order with respect to reactant $\text{A}$.
• $n$ is the partial reaction order with respect to reactant $\text{B}$.
• The overall reaction order is the sum of partial orders: $\text{Order}_{\text{overall}} = m + n$.

Units of the Rate Constant ($k$)

Because the reaction rate is always in units of $\text{M}\cdot\text{s}^{-1}$, the dimensional units of $k$ depend on the overall reaction order ($o = m + n$):

$$\text{Units of } k = \text{M}^{1 - o} \cdot \text{s}^{-1} = \left(\frac{\text{mol}}{\text{L}}\right)^{1 - o} \cdot \text{s}^{-1}$$

• Zero Order ($o = 0$): $\text{M}\cdot\text{s}^{-1}$ (or $\text{mol}/(\text{L}\cdot\text{s})$)
• First Order ($o = 1$): $\text{s}^{-1}$
• Second Order ($o = 2$): $\text{M}^{-1}\cdot\text{s}^{-1}$ (or $\text{L}/(\text{mol}\cdot\text{s})$)
• Third Order ($o = 3$): $\text{M}^{-2}\cdot\text{s}^{-1}$ (or $\text{L}^2/(\text{mol}^2\cdot\text{s})$)

The Initial Rates Method

In the method of initial rates, multiple experimental trials are conducted where the initial concentration of one reactant is varied systematically while all other reactant concentrations are held constant:

$$\frac{\text{Rate}_2}{\text{Rate}_1} = \left(\frac{[\text{A}]_2}{[\text{A}]_1}\right)^m$$

If doubling $[\text{A}]$ leaves the rate unchanged $\implies m = 0$ (zero order). If doubling $[\text{A}]$ doubles the rate $\implies m = 1$ (first order). If doubling $[\text{A}]$ quadruples the rate ($2^2 = 4$) $\implies m = 2$ (second order).

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Reaction Order, Rate Law & Units of the Rate Constant

Because the rate itself always carries units of M·s⁻¹, the units of $k$ change with overall order — a fast way to check that an experimentally derived rate law is self-consistent.