How is the rate equation derived for a multi-step reaction?

The rate equation for a multi-step reaction is derived by considering the slowest step, also known as the rate-determining step.

In a multi-step reaction, the overall reaction is broken down into a series of simpler reactions, each of which can be described by its own rate equation. However, the rate of the overall reaction is determined by the slowest of these steps, which is known as the rate-determining step. This is because the overall reaction cannot proceed faster than this slowest step, much like how the speed of a convoy is determined by the slowest vehicle.

To derive the rate equation for a multi-step reaction, we first need to identify the rate-determining step. This is typically the step with the highest activation energy, as this energy barrier determines how quickly the reaction can proceed. Once we have identified the rate-determining step, we can write down its rate equation. This equation describes how the rate of the reaction depends on the concentrations of the reactants involved in this step.

However, the rate equation for the rate-determining step might include species that are not present in the overall reaction. These are known as reaction intermediates. To derive the rate equation for the overall reaction, we need to eliminate these intermediates. This is done by using the rate equations for the other steps in the reaction to express the concentrations of the intermediates in terms of the concentrations of the reactants and products in the overall reaction.

Once we have done this, we can substitute these expressions into the rate equation for the rate-determining step. This gives us the rate equation for the overall reaction, which describes how the rate of the reaction depends on the concentrations of the reactants and products. This equation is a powerful tool for predicting the behaviour of the reaction under different conditions, and for understanding the mechanisms by which the reaction occurs.