Half Life of First Order Reactions

Question

Why is the half life of a first order reaction directly proportional to the reaction rate for all points along the graph? (ie. when the concentration decreases by half, the reaction rate also decreases by half)

Answer 1

Hello Gnarls!

The half-life is just the time it takes to reduce the concentration by half of its initial value.

Let's assume we have a reaction of a compound A. Let's call the initial concentration [A]₀

When the reaction begins, the rate is given by (remember the first order reaction):

r₀ = k[A]₀ , where k is a constant.

When the concentration decreases by half, the rate also decreases by half:

r₁ = k[A]₀/2 = r₀/2

If I understood your doubt correctly, you didn't remember what a first order reaction meant. But, if you would like to know why the half-life of a first order reaction doesn't depend on the initial concentration, we will need some Calculus.

From the definition of rate of reaction:

r = -d[A]/dt (the derivate of concentration of A over time)

We know r is a first order reaction, so:

r = -d[A]/dt = k[A]

d[A]/[A] = -kdt

∫d[A]/[A] = ∫-kdt

ln([A]/[A]₀) = -k*t

From the definition of half-life:

ln(1/2) = -k*t_1/2

-ln(2) = -k*t_1/2

t_1/2 = ln(2)/k

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