Module 8.3: Reaction Rates and Temperature

8.3.1 Temperature and Rate

I For a reaction to occur, the energy of reactants must be equal or higher than the activation energy

1. as temperature increases, the rates of all reactions increase.

   1. Both wanted and unwanted reactions occur faster as $\textrm{T}$ increases.

2. Arrhenius equation: An equation that defines temperature dependence on rate constant $\textrm{ln} (\textrm{k}_{r}) = \textrm{ln} (\textrm{A}) - \frac{\textrm{E}_{a}}{\textrm{RT}}$ (Eq. 137)

   1. $\textrm{A}$ and $\textrm{E}_{a}$ are known as the Arrhenius parameters; they are found experimentally

   2. $\textrm{A}$ is known as the frequency factor (rate at which collisions occur)

   3. $\textrm{E}_{a}$ is known as the activation energy

   4. there is another form of this equation, that is used more commonly: $\textrm{k}_{r} = \textrm{Ae}^{-\frac{\textrm{E}_{a}}{\textrm{RT}}}$

3. From the Arrhenius equation, we can see that for the rate constant,

   1. higher activation energy results in stronger dependence of temperature.

   2. lower activation energy results in smaller dependence of temperature.

4. the Arrhenius equation can be used when two temperature and two rate constants are given without the need of the Arrhenius parameters: $\textrm{ln}\frac{\textrm{k}_{r2}}{\textrm{k}_{r1}} = \frac{\textrm{E}_{A}}{\textrm{R}} (\frac{1}{\textrm{T}_{1}} - \frac{1}{\textrm{T}_{2}})$ (Eq. 139)

5. The pre-exponential factor can be calculated by the following: $\textrm{A} = \sigma (\frac{8\textrm{kT}}{\pi \mu})^{\frac{1}{2}} \textrm{N}_{A}$ (Eq. 140)

   1. $\mu$ is known as “reduced mass”, $\mu = \frac{\textrm{m}_{A}\textrm{m}_{B}}{\textrm{m}_{A}+\textrm{m}_{B}}$

8.3.2 Collision Theory

II Collision theory: when gaseous reactants collide, the product is formed only if the reactants collide with a certain minimum kinetic energy & correct orientation

1. reaction profile: illustrates the potential energy of reactants as they approach one another

   1. the potential energy of the molecules rises when they contact (repulsion); old bonds start to break and new bonds start to form.

   2. the potential energy reaches maximum (the activation energy) when the reactants are highly distorted (old bonds mostly broken, new bonds partly formed)

   3. The reactants must collide with enough kinetic energy to overcome the increase in potential energy

   4. as new bonds fully form and the product molecules separate, the potential energy decreases (Ref. Figure 88)

   5. The reaction coordinate indicates how far the reaction has progressed



2. The rate of collision (collision frequency) between two molecules (A and B) is proportional to both concentrations of A and B.

3. The Maxwell-Boltzmann distribution can be used to calculate the fraction of molecules that exceed a certain kinetic energy ($\nu_{min}$) at a certain temperature

   1. The Boltzmann distribution states that $\textrm{N}_{i} \propto \textrm{e}^{-\frac{\textrm{E}_{a}}{\textrm{RT}}}$ (Ref. 5.2.2)

   2. Therefore, the collision frequency is $\nu \propto [\textrm{A}][\textrm{B}] \textrm{e}^{-\frac{\textrm{E}_{a}}{\textrm{RT}}}$

   3. This relationship implies that $\textrm{k}_{r} \propto \textrm{e}^{-\frac{\textrm{E}_{a}}{\textrm{RT}}}$

8.3.3 Transition State Theory

III The transition-state theory is an extension of collision theory that can be applied in general

1. An activated complex is the physical state that corresponds to the peak potential energy

   1. if there is not enough kinetic energy, the activated complex collapses back into reactants

   2. if there is enough kinetic energy, the cluster of atoms becomes product