Questions: Pre-exponential Factor and Collision Theory

Question 1

An experimentally measured pre-exponential factor A is 10,000 times smaller than the value predicted by collision theory for the same reaction. What is the most likely physical interpretation?

• A. The activation energy was incorrectly measured, causing a systematic error in A
• B. The reaction requires a very specific molecular orientation, so only a tiny fraction of geometrically possible collisions lead to reaction
• C. Collision theory overestimates A at low temperatures because the Maxwell-Boltzmann distribution shifts
• D. The collision cross-section is much larger than assumed, reducing the predicted collision frequency

Answer: (B)

When A_exp ≪ A_theory, the explanation is a severe steric requirement: the steric factor p = A_exp / A_theory ≪ 1. Only a tiny fraction of collisions have the correct geometric orientation for the reaction to proceed (e.g., an SN2 nucleophile must approach from the back side). p can be as small as 10⁻⁵ for highly orientation-specific reactions. The activation energy is determined separately from the slope of ln k vs 1/T and doesn't affect the comparison of A values.

Question 2

A reaction between an ion and a dipolar molecule is found to have A_exp approximately 100 times LARGER than the hard-sphere collision theory prediction. Which explanation is most consistent with this observation?

• A. There must be an error in the experimental measurement
• B. The steric factor must exceed 1, which is impossible since p is always between 0 and 1
• C. Long-range electrostatic attraction funnels reactants together more efficiently than hard-sphere geometry predicts
• D. The reaction proceeds through a unimolecular mechanism rather than a bimolecular collision

Answer: (C)

When A_exp > A_theory, collision theory has broken down — its hard-sphere model underestimates how often productive encounters occur. Ion-dipole or strong hydrogen-bond interactions extend the effective range of interaction well beyond what hard-sphere contact assumes, so reactants find each other more often than the geometric cross-section would predict. Option B (p > 1) sounds like a logical fix but reveals a misunderstanding: p is defined as the ratio A_exp / A_theory and can exceed 1, but a p > 1 signals that the model assumptions are wrong, not that more than all collisions react.

Question 3

The steric factor p can seldom exceed 1 because it represents the fraction of collisions with the correct orientation, and fractions can seldom be greater than 1.

• T. True
• F. False

Answer: False

While p is conceptually defined as a fraction of 'correctly oriented' collisions and should be between 0 and 1 in the simple collision theory framework, experimentally derived p values (computed as A_exp / A_theory) can exceed 1. This occurs when long-range attractive forces (ion-dipole, hydrogen bonding) cause reactants to encounter each other more frequently than hard-sphere geometry predicts. A p > 1 is a diagnostic signal that collision theory's hard-sphere model is inadequate for that reaction.

Question 4

The steric factor p and the activation energy Ea are independent parameters — a reaction can have a small steric factor (p ≪ 1) but still proceed rapidly at high temperatures.

• T. True
• F. False

Answer: True

The Arrhenius equation separates these two effects: k = A·exp(−Ea/RT), where A = p·σ·⟨v_rel⟩·Nₐ. The steric factor p reduces A (the pre-exponential factor) regardless of temperature, but the exponential term exp(−Ea/RT) captures the fraction of collisions with sufficient energy. A reaction with p = 10⁻⁵ (very specific orientation needed) but low Ea can still be fast at high temperatures because the energy barrier is small. Conversely, a reaction with p ≈ 1 but very high Ea will be slow at low temperatures. The two factors are physically distinct and thermodynamically independent.

Question 5

What physical phenomenon does the steric factor p represent, and how does the ratio A_experimental / A_theoretical diagnose the mechanism of a chemical reaction?

Model answer: The steric factor p represents the fraction of collisions that have the correct molecular orientation for reaction to occur. Even if two molecules collide with enough energy to overcome the activation barrier, the reaction only proceeds if the reacting functional groups are properly aligned. A reaction with p ≈ 1 (like K + Br₂) can react from almost any approach angle; an SN2 reaction with p ~ 10⁻⁵ requires precise back-side attack. The ratio A_exp / A_theory reveals mechanism: when the ratio ≈ 1, simple hard-sphere collision adequately describes the reaction; when it is much less than 1, severe geometric constraints operate; when it exceeds 1, long-range attractive forces or a complex mechanism (like pre-association) makes the hard-sphere model insufficient, motivating more sophisticated treatments like transition state theory.

This comparison is the practical payoff of collision theory. Transition state theory, which replaced the steric factor with a full statistical mechanical treatment of the activated complex, was developed precisely because large deviations in A_exp / A_theory signaled that the simple 'billiard ball collision' picture was missing important physics.