Question

The rate of decay of a chemical involved in a reaction that is second order (bimolecular) in one reactant A is given by: = k [AJ? -d[A/dt where k is the reaction rate coefficient. Derive an expression for the half-life of A in terms of k and the concentration of A at time t-0 (Ao)

Answer 1

Explanation:

$2A\rightarrow products$

According to mass action,

$\textrm{rate}=-\dfrac{\Delta[\textrm A]}{2\Delta t}=k[\textrm A]^2$

Where, k is the rate constant

So,

$\dfrac{d[A]}{dt}=-k[A]^2$

Integrating and applying limits,

$\int_{[A_t]}^{[A_0]}\frac{d[A]}{[A]^2}=-\int_{0}^{t}kdt$

we get:

$\dfrac{1}{[A]} = \dfrac{1}{[A]_0}+kt$

Where,

$[A_t]$ is the concentration at time t

$[A_0]$ is the initial concentration

Half life is the time when the concentration reduced to half.

So, $[A_t]=\frac{1}{2}\times [A_0]$

Applying in the equation as:

$t_{1/2}=\dfrac{1}{k[A_o]}$