Question

How is Half-life related to initial concentration for a second order reaction?

Answer 1

For a second-order reaction, the half-life is inversely proportional to the initial concentration of the reactant.

For a second-order reaction 

Rate = k[A]²

The integrated rate law is

`1/[A] = 1/[A]_0 + kt`

At half-life t = t_1/2​, [A] = `[A]_0/2`

Substitute into the integrated rate law:

`1/([A]_0//2) = 1/[A]_0 + kt_(1//2)`

⇒ `2/[A]_0 = 1/[A]_0 + kt_(1//2)`

⇒ `kt_(1//2) = 1/[A]_0`

⇒ `t_(1//2) = 1/(k[A]_0)`

For a second order reaction `t_(1//2) prop 1/[A]_0`

This means:

As initial concentration increases, half-life decreases.

As initial concentration decreases, half-life increases.