You are using a Geiger counter to measure the activity of a radioactive substance over the course of several minutes. If the reading of 400. counts has diminished to 100. counts after 90.3 minutes, what is the half-life of this substance? Express your answer with the appropriate units. View Available Hint (s) t1/2 t 1 / 2 t_1/2 = nothing

t₁/₂ = 45.1 min

Explanation:

The radioactive decay equation is given by

N/N₀ = e^-kt where N = counts after time t

N₀ = counts initially

k = decay constant

t = time elapsed

The question is what's the half-life of this substance, and we can solve it once we have k from the expression above since

t₁/₂ = 0.693/k

which is derived from that equation, but for the case N/N₀ is 0.5

Lets calculate k and t₁/₂:

N/N₀ = e^-kt (taking ln in the two sides of the equation)

ln (N/N₀) = ln e^-kt = - kt ⇒ k = - ln (N/N₀) / t

k = - ln (100/400) / 90.3 min = 0.01535 min⁻¹

t₁/₂ = 0.693/k = 0.693/0.01535 min⁻¹ = 45.1 min

We can check this answer since the time in the question is the double of this half-life and the data shows the material has decayed by a fourth: two half-lives.