how to solve functional equation?

How do I solve this Half-Life Equation?

• My professor loves teaching us new calculations and then putting the entire opposite of that method on exams or homework. I have absolutely no idea how to solve this equation so any help would be appreciated. Use the integrated rate equation for a first order reaction. At the "half-life", t = t 1/2(subscript) and [A]t = [A]0(subscript)/2. Solve this integrated rate equation for the half life. You must show all your algebraic manipulations.

• Answer:

  The integrated rate law for a first-order reaction can be written a couple of different ways, but one form of it is: ln[A]/[A]o = - kt If t = t1/2, then [A] = 1/2[A]o. So, ln 1/2[A]o/[A]o = - k (t1/2) Cancelling [A]o, the equation becomes: ln (1/2) = - k t1/2 t 1/2 = -0.693 / - k t1/2 = 0.693/k