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Figure %: Boltzmann distributions for T1 greater than T2

Combining the above considerations, we state the following relationship between the rate constant and the activation energy, called the Arrhenius equation:

The variable k is the rate constant, which is dependent on the frequency of the collisions f, orientation factor p, activation energy Ea, and temperature T. From the expression for the Arrhenius equation you should note that a small increase in activation energy leads to a large decrease in rate constant. Furthermore, temperature has a similarly exponential effect on the rate constant. An experimental rule of thumb is that a 10oC increase in temperature leads to a doubling of the rate constant.

One application of the Arrhenius equation that is useful is the determination of the activation energy for a reaction. Taking the natural log of the Arrhenius equation gives a linear equation:

A graph of ln k versus 1 / T should give a straight line whose slope is - Ea / R. By measuring the rate constant at a range of different temperatures, you can construct a graph to determine the activation energy of a reaction.

Catalysis

A catalyst speeds up a reaction without being explicit in the overall balanced equation. It does this by providing an alternate mechanism for the reaction that has a lower activation barrier than does the uncatalyzed pathway. Compare the catalytic and regular mechanisms for the hydrogenation of ethylene to ethane and their associated reaction coordinate diagrams in :

Figure %: Mechanisms of ethylene hydrogenation

As you can see, the catalyst changes the mechanism of the reaction and lowers the activation energy. The catalyst, because it does not appear in the overall balanced equation has absolutely no effect on the thermodynamics of the reaction.

There are two types of catalysts--heterogeneous catalysts and homogeneous catalysts. There is no fundamental difference in how these catalysts work. The difference lies in whether the catalyst is in the same phase (solid, liquid, or gas) as the reagents. A homogeneous catalyst is in the same phase as the reactants while a heterogeneous catalyst is not. An enzyme is a biological homogeneous catalyst.

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