How do you derive an expression for the half-life of a second-order reaction?

Question

It's an exam question

Answer 1

Ok a second order reaction has a rate of reaction proportional to the square of the amount of reactants present.

so dX/dt = - k X^2 --------------(1)

From (1)

-k dt = dX / X^2
Integrating both sides:

-kt = -1/X + c

c = 1/X* (Where X* is the initial amount)

therefore:
-kt = 1/X* - 1/X

For half-life: X = X*/2

Therefore: -kt = 1/X* - 2/X*

t = 1/kX*
This is half-life

Answer 2

Second Order Half Life

Answer 3

Your chemistry text should have an explanation of how to derive that expression. I'd check there.