# SN2 - Second-order Nucleophilic Substitution

## Activation Energy of $SN2$ Reactions

Due to the existence of , reactions usually proceed more rapidly when the temperature is increased. Furthermore, most reactions do not occur at extremely low temperatures for the same exact reason.

Hint
As a general rule, the reaction rate of many reactions that proceed at about room temperature is redoubled through increasing the temperature by 10 °C.

Even if a sample has an evenly distributed temperature, their molecules do not all possess the same kinetic energy. The speed and thus the kinetic energy of the molecules are widely scattered. The distribution of the molecules' kinetic energy at a given temperature was mathematically described by Boltzmann (Boltzmann distribution law). In the illustration below, the Boltzmann distribution is depicted according to two different temperatures, whereat T1 < T2. Fig.1
Boltzmann distribution of the molecules' kinetic energy for two temperatures.

The number of molecular collisions of molecules that possess the sufficient energy to surpass the activation energy barrier is directly proportional to the area under the Boltzmann distribution curve to the right of the activation energy $Ea$. At temperature T1, this area is coloured blue-green. In contrast, the corresponding area at the higher temperature T2 is represented by the whole blue-green and purple region. Obviously, the number of collisions that lead to a successful conquest of the activation energy barrier is higher at a higher temperature. As a result, the reaction rate rises considerably. Even a small increase in temperature causes a relatively large increase in the reaction rate. This fact is mathematically described by the Arrhenius equation, which explains the influence of the temperature and the activation energy $Ea$ on the rate constant k and thus on the reaction rate:

k= k0 e-A/T,

with A = $Ea$/R. Due to their definition, $Ea$ and T (absolute temperature) are always positive. Therefore, according to the Arrhenius equation, the reaction rate is the higher the lower the activation energy $Ea$ is, as $Ea$ is part of the negative exponent's numerator. In contrast, the reaction rate increases accordingly when the temperature is raised, as T is part of the negative exponent's denominator.

Another general rule to take into account is that a reaction proceeds noticeably at room temperature when its activation energy $Ea$ is smaller than 85 kJ/mol. If the activation energy is (considerably) higher than 85 kJ/mol, the reactants must be heated in order to obtain a practicable reaction rate.

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