Chemical Kinetics (ii)
The temperature dependence of reaction rates (Arrhenius equation)
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The rate constants of most reactions increase as the temperature is raised, this increase being usually within the range spanned by:
The relationship between the rate constant k and temperature is an empirical law, known as the Arrhenius equation:
where A is called the pre-exponential factor or the frequency factor, and Ea is the activation energy. Collectively, these two terms are known as the Arrhenius parameters and may be quoted from data books.
The 'log' form of the equation is particularly useful since a plot of ln k vs. 1/T is expected to be a straight line, with gradient -Ea / R and y-intercept or ln A.
This new equation permits the treatment of systems with highly temperature-dependent activation energy. It is a more general form of the Arrhenius equation as it allows Ea to be obtained from the slope (at the temperature of interest) of a plot of ln k against 1/T even if the Arrhenius plot is not a straight line. Non-Arrhenius behaviour is commonly a sign that quantum mechanical tunnelling is playing a significant role in the reaction.
Although until now we have regarded the Arrhenius parameters as purely
empirical quantities, it is worth anticipating that the activation energy
is the minimum kinetic energy that reactants must have in order to form
products. For example, in a gas-phase reaction there are numerous collisions
each second, but only a tiny proportion are sufficiently energetic to lead
to reaction. The fraction of collisions with a kinetic energy in excess
of an energy Ea is given by the Boltzmann distribution
as e-Ea/RT. Hence, the exponential
factor in the Arrhenius equation can be interpreted as the fraction of
collisions that have enough kinetic energy to lead to reaction. The pre-exponential
factor is a measure of the rate at which collisions occur irrespective
of their energy. Hence, the product of A and the exponential factor,
e-Ea/RT, gives the rate
of successful collisions. An analogous argument can also be made
for reactions in the liquid phase. All this is discussed at a later stage.