Show that for a zero order reaction, half life is directly proportional to the initial concentration of the reactant, but in a first order reaction, the half life of the reaction is independent of the initial concentration of reactant

The half-life is a timescale in which each half-life represents the reduction of the initial population to 50% of its original state. We can represent the relationship by the following equation.

 

Using the integrated form of the rate law, we can develop a relationship between zero-order reactions and the half-life.

Substitute

 

Solve for time,  

 

So, for zero-order reactions, the half-life depends on the initial concentration of reactant and the rate constant.

 

 

Rate law for first order reaction:

 is the reaction rate and  is the reaction rate coefficient. In this example, the units of  are 1/s. The units can vary with other types of reactions.

The half-life is a timescale by which the initial population is decreased by half of its orignal value, t_1/2.. We can represent the relationship by the following equation.

 

Using the integrated form of the rate law, we can develop a relationship between first-order reactions and the half-life.

 

Substitute

 

Therefore, for first-order reactions, the half-life is independent of the initial concentration of reactant.