Temperature Dependence: Arrhenius Equation & Collision Theory

Temperature profoundly impacts reaction rates. For most chemical reactions, a $10^\circ$ rise in temperature doubles the reaction rate. This guide breaks down the quantitative relationship provided by Svante Arrhenius.

1. The Arrhenius Equation

The mathematical relationship between the rate constant ($k$), temperature ($T$), and activation energy ($E_a$) is given by:

Exponential Form

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

$A$: Frequency Factor (Pre-exponential factor), $E_a$: Activation Energy, $R$: Gas Constant

Logarithmic Form (Single Temp)

$$ \ln k = \ln A - \frac{E_a}{RT} $$

Two Temperature Form

$$ \log \frac{k_2}{k_1} = \frac{E_a}{2.303 R} \left[ \frac{T_2 - T_1}{T_1 T_2} \right] $$

Exam Cheat Sheet: Graphical Analysis

When plotting $\ln k$ (y-axis) against $1/T$ (x-axis):

Slope ($m$): Equal to $-E_a/R$. (Negative slope indicates rate decreases as $1/T$ increases, i.e., $T$ decreases).
Intercept ($c$): Equal to $\ln A$.
Concept: Higher $E_a$ leads to a steeper slope, meaning the reaction is more sensitive to temperature changes.

Temperature dependence of rate of reaction chart

Figure: Boltzmann Distribution Curve showing effect of Temperature on Activation Energy

2. Collision Theory & Activation Energy

According to collision theory, not all molecular collisions result in a chemical reaction. Only those collisions are effective that possess:

Criteria for Effective Collisions

Energy Barrier: Reacting molecules must possess energy $\ge$ Threshold Energy.
Orientation Barrier: Molecules must collide with proper orientation to break old bonds and form new ones.
Activation Energy ($E_a$): The extra energy required by reactant molecules to reach the threshold energy.
Threshold Energy = Average K.E. of reactants + Activation Energy

3. Maxwell-Boltzmann Distribution

This statistical distribution explains why a small temperature increase causes a large increase in rate. A $10^\circ$ rise doesn't just increase the average kinetic energy slightly; it effectively doubles the fraction of molecules having energy greater than $E_a$.

Effect of Catalyst

A catalyst provides an alternate pathway with lower Activation Energy ($E_a'$).
It does not change the enthalpy ($\Delta H$) or Gibbs free energy ($\Delta G$) of the reaction.
It does not shift the equilibrium constant ($K_{eq}$), but helps attain equilibrium faster.

Mind Map of Temperature dependence of the rate of reaction