🧂Physical Chemistry II Unit 1 Review

Collision theory and transition state theory are two frameworks for explaining why reactions happen at the rates they do. Collision theory focuses on the physical requirements for a reaction: molecules must collide with enough energy and the right orientation. Transition state theory (TST) goes deeper, modeling the fleeting high-energy configuration that forms at the top of the energy barrier and using it to derive quantitative rate constants. Together, these theories connect molecular-level behavior to the macroscopic rate laws you've already seen in earlier sections.

Collision Theory: Principles and Rate Laws

Basic Principles

Collision theory starts from a simple premise: reactant molecules must physically collide before they can react. But not every collision leads to products. Two conditions must be met:

Sufficient energy — The colliding molecules need enough kinetic energy to overcome the activation energy barrier, $E_a$ . Collisions below this threshold just bounce off.
Proper orientation — Even with enough energy, the molecules must approach each other in a geometry that allows the relevant bonds to break and form. A collision where the reactive sites face away from each other won't produce anything.

The reaction rate therefore depends on:

Collision frequency — How often reactant molecules collide per unit time. Higher concentrations pack more molecules into the same volume, increasing collision frequency and raising the rate.
The fraction of effective collisions — Only a small subset of all collisions satisfy both the energy and orientation requirements.

This gives the collision theory rate expression:

$k = Z \cdot \rho \cdot e^{-E_a / RT}$

where $Z$ is the collision frequency, $\rho$ is the steric factor (a number between 0 and 1 that accounts for orientation requirements), $E_a$ is the activation energy, $R$ is the gas constant, and $T$ is the absolute temperature. The steric factor is often much less than 1, reflecting the fact that most orientations are unproductive.

Factors Affecting Reaction Rates

Temperature: Increasing temperature raises the average kinetic energy of molecules. The Boltzmann distribution shifts so that a larger fraction of collisions exceed $E_a$ . This is why even modest temperature increases can dramatically speed up reactions, as captured quantitatively by the Arrhenius equation:

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

where $A$ is the pre-exponential factor (related to collision frequency and orientation).

Catalysts: A catalyst provides an alternative reaction pathway with a lower activation energy. At the same temperature, a larger fraction of collisions now have enough energy to react. The catalyst is not consumed and does not change the thermodynamics of the reaction, only the kinetics.

Transition State: Concept and Significance

Basic Principles, Activation Energy and Temperature Dependence | Chemistry [Master]

Definition and Characteristics

The transition state (also called the activated complex) is the highest-energy configuration along the reaction coordinate. At this point, old bonds are partially broken and new bonds are partially formed. It sits at the saddle point on the potential energy surface.

A few key features distinguish the transition state:

It is not a stable intermediate. Its lifetime is on the order of $10^{-13}$ seconds (roughly one molecular vibration period), so it cannot be isolated or directly observed.
It represents a single point on the energy surface, not a local minimum. Any small perturbation sends it forward to products or backward to reactants.
The energy difference between the reactants and the transition state defines the activation energy, $E_a$ .

Importance in Chemical Reactions

The structure of the transition state controls both the rate and the selectivity of a reaction. Its geometry and electronic configuration determine which reaction pathway is followed and what stereochemical outcome results. For example, the distinction between $S_N1$ and $S_N2$ mechanisms comes down to whether the transition state involves one molecule (unimolecular) or two (bimolecular) at the rate-determining step.

Because the transition state cannot be observed directly, its properties are inferred from kinetic data, isotope effects, and computational chemistry. Understanding its structure is one of the central goals of mechanistic chemistry.

Transition State Theory: Calculating Rate Constants

Quantitative Approach

TST provides a statistical mechanical framework for calculating rate constants. The central assumption is that the reactants and the transition state are in a quasi-equilibrium. This means you can use thermodynamic quantities to describe the "equilibrium" between reactants and the activated complex, even though the transition state is not a true chemical species.

The quasi-equilibrium constant for forming the transition state is:

$K^{\ddagger} = \frac{[\text{TS}]^{\ddagger}}{[\text{reactants}]}$

The rate of the reaction is then proportional to the concentration of the transition state complex multiplied by the frequency at which it crosses the barrier and decomposes into products.

Basic Principles, Collision Theory | Chemistry

The Eyring Equation

This reasoning leads to the Eyring equation, the central result of TST:

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

where:

$k_B$ = Boltzmann constant ( $1.381 \times 10^{-23}$ J/K)
$T$ = absolute temperature (K)
$h$ = Planck's constant ( $6.626 \times 10^{-34}$ J·s)
$\Delta G^{\ddagger}$ = Gibbs free energy of activation
$R$ = gas constant (8.314 J/(mol·K))

The prefactor $k_B T / h$ has units of frequency ( $\text{s}^{-1}$ ) and represents the rate at which the activated complex crosses the barrier. At 298 K, this factor is approximately $6.2 \times 10^{12} \text{ s}^{-1}$ .

The Gibbs free energy of activation decomposes into enthalpic and entropic contributions:

$\Delta G^{\ddagger} = \Delta H^{\ddagger} - T\Delta S^{\ddagger}$

This decomposition is powerful because it separates two physically distinct effects:

$\Delta H^{\ddagger}$ (enthalpy of activation) reflects the energy required to reach the transition state, including bond stretching and partial bond breaking.
$\Delta S^{\ddagger}$ (entropy of activation) reflects the change in molecular order on going from reactants to the transition state. A highly negative $\Delta S^{\ddagger}$ means the transition state is more ordered than the reactants (common in bimolecular reactions where two free molecules must come together). A near-zero or positive $\Delta S^{\ddagger}$ suggests a loose, dissociative transition state.

Determining Activation Parameters Experimentally

To extract $\Delta H^{\ddagger}$ and $\Delta S^{\ddagger}$ from experimental data:

Measure the rate constant $k$ at several different temperatures.
Plot $\ln(k/T)$ vs. $1/T$ (an Eyring plot).
The slope of the resulting line equals $-\Delta H^{\ddagger}/R$ .
The y-intercept equals $\ln(k_B/h) + \Delta S^{\ddagger}/R$ .

This is analogous to an Arrhenius plot ( $\ln k$ vs. $1/T$ ), but the Eyring plot directly yields the thermodynamic activation parameters rather than just $E_a$ and $A$ .

Collision Theory vs. Transition State Theory

Limitations of Collision Theory

Collision theory is intuitive and useful for qualitative reasoning, but it has real shortcomings:

Orientation effects are handled crudely. The steric factor $\rho$ is essentially a fudge factor. It must be determined empirically and doesn't give you structural insight into why certain orientations work.
No thermodynamic decomposition. Collision theory lumps everything into $E_a$ and $A$ . It doesn't separate enthalpic and entropic contributions, so it can't explain why two reactions with similar activation energies might have very different rates.
Best suited to simple gas-phase reactions. For reactions in solution or involving complex molecules, collision theory becomes increasingly inadequate.

Advantages of Transition State Theory

TST addresses these limitations in several ways:

Entropic effects are explicit. Through $\Delta S^{\ddagger}$ , TST accounts for how molecular complexity, solvent reorganization, and loss of translational/rotational freedom affect the rate. This matters enormously for bimolecular reactions in solution and for enzymatic catalysis.
Structural insight. By modeling the transition state geometry, TST connects rate constants to molecular structure and can predict how structural modifications will change the rate.
Quantitative rate constants. The Eyring equation provides a direct route from thermodynamic activation parameters to $k$ , applicable to elementary reaction steps.

Limitations of TST

TST is not perfect either. Its key assumptions can break down:

The quasi-equilibrium assumption requires that the reactants and transition state maintain a Boltzmann distribution. This fails for very fast reactions or under non-equilibrium conditions.
No recrossing correction. TST assumes every trajectory that reaches the transition state proceeds to products. In reality, some trajectories recross the barrier and return to reactants, causing TST to overestimate the rate. The transmission coefficient $\kappa$ (typically $\leq 1$ ) is sometimes introduced to correct for this: $k = \kappa \cdot \frac{k_B T}{h} \cdot e^{-\Delta G^{\ddagger}/RT}$
Quantum tunneling is neglected. For reactions involving light atoms (especially hydrogen transfer), tunneling through the barrier can be significant, and TST underestimates the rate unless a tunneling correction is applied.

🧂Physical Chemistry II Unit 1 Review