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

Step-by-step explanation:

`2A\rightarrow products`

According to mass action,

`\textrm{rate}=-(\Delta[\textrm A])/(2\Delta t)=k[\textrm A]^2`

Where, k is the rate constant

So,

`(d[A])/(dt)=-k[A]^2`

Integrating and applying limits,

`\int_([A_t])^([A_0])(d[A])/([A]^2)=-\int_(0)^(t)kdt`

we get:

`(1)/([A]) = (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]=(1)/(2)* [A_0]`

Applying in the equation as:

`t_(1/2)=(1)/(k[A_o])`