6.2 Transition state theory fundamentals

Transition State Theory Fundamentals

Transition State Theory (TST) explains how chemical reactions proceed through a high-energy intermediate called the activated complex. Rather than just asking "did the molecules collide hard enough?" (as collision theory does), TST zooms in on the structure and energy of that fleeting intermediate to predict reaction rates. This framework is central to understanding why some reactions are fast and others are slow.

Activated Complex in Reactions

The activated complex is a high-energy, unstable species that forms when reactant molecules collide with sufficient energy and proper orientation. It sits at the highest point on the reaction coordinate diagram, representing the maximum potential energy along the pathway from reactants to products.

Think of it as the "halfway house" of a reaction. At this point, old bonds are partially broken and new bonds are partially formed. The molecule is neither fully reactant nor fully product.

• The activated complex determines the reaction rate because it represents the hardest energy barrier to cross.
• Once formed, it can either decompose forward into products (like $\text{H}_2\text{O}$ or $\text{NH}_3$, depending on the reaction) or fall back apart into the original reactants.
• The rate of the overall reaction depends on both the concentration of activated complexes that form and how quickly they decompose into products.

Transition State Theory Postulates

TST rests on a few core assumptions that distinguish it from simpler models:

1. Quasi-equilibrium assumption. Reactants and the activated complex exist in a rapid, reversible equilibrium. This means the concentration of the activated complex is directly proportional to the concentrations of the reactants, and you can use equilibrium expressions to describe it.

2. One-way crossing. The activated complex passes through the transition state only once. It doesn't oscillate back and forth at the energy peak. Once it reaches the transition state, it either proceeds to products or reverts to reactants in a single pass.

3. Rate depends on decomposition. The overall rate of product formation is governed by how fast the activated complex breaks down into products, not just how often molecules collide.

Rate Constant Calculation Methods

The Arrhenius equation connects the rate constant $k$ to activation energy and temperature:

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

where:

• $k$ = rate constant
• $A$ = pre-exponential (frequency) factor, representing how often molecules collide with the correct orientation
• $E_a$ = activation energy (in J/mol)
• $R$ = universal gas constant ($8.314 \text{ J mol}^{-1} \text{K}^{-1}$)
• $T$ = temperature in Kelvin

To calculate $k$:

1. Obtain $E_a$ and $A$ from experimental data or literature values.
2. Make sure $T$ is in Kelvin and $E_a$ is in J/mol (not kJ/mol) so units are consistent with $R$.
3. Plug all values into the Arrhenius equation and solve.

Note that the Arrhenius equation originally comes from empirical observation, but TST provides a theoretical basis for it. The full TST expression (the Eyring equation) uses thermodynamic quantities like $\Delta G^\ddagger$, but at this level, the Arrhenius form is what you need to work with.

Activation Energy and Reaction Rates

Activation energy ($E_a$) is the minimum energy reactants must possess to form the activated complex. It's the height of the energy barrier on the reaction coordinate diagram.

The relationship between $E_a$ and reaction rate is exponential, not linear. That exponential sits in the Arrhenius equation as $e^{-E_a/RT}$, which means:

• A higher $E_a$ means a much smaller fraction of molecules have enough energy to react, so the rate drops sharply.
• A lower $E_a$ means more molecules can clear the barrier, and the reaction speeds up.

Because the relationship is exponential, even modest changes in $E_a$ have dramatic effects. For example, at room temperature, increasing $E_a$ by about 5.7 kJ/mol cuts the rate roughly in half. This is why catalysts are so powerful: by lowering $E_a$ even slightly, they can increase reaction rates by orders of magnitude.