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.