A sample of radioactive material is initially found to have

A sample of radioactive material is initially found to have an activity of 115.0 decays/min. After 4 d 23 h, its activity is measured to be 84.5 decays/min.

(a) Calculate the half-life of the material.

__________________________days

(b) How long (from the initial time) will it take for the sample to reach an activity level of 10.0 decays/min?

____________________________days

Solution

a) The activity is given by,

R = dN/dt = N₀e^t = R₀e^t

We are given the activity at two different times the first of which can be taken as t₁=0, while the second one is

t₂ = 4days,23hours = 119hours. At t₁ we then have

R₁ = R₀e^t₁ = R₀

At t₂ we have,               R₂ = R₀e^t₂

Taking the ratio of R₂ to R₁

we have

R2/R1 = et2,

84.5/115 = e⁽¹¹⁹⁾

ln(84.5/115)= 119

0.3082 = 119

= 2.59×10³hrs¹

The half-life in terms of the decay constant is given by

t_1/2=0.693/ = 0.693/2.59×10³hrs¹= 267.57hours

b) The activity is given by,   R = R₀e^t

R₀ is just the initial activity (remember that we can start the clock at any time) which is 115 decays/minute.

We then have, R = R₀e^t

                                              10 = 115 e^{-(2.59×10-3)t}

                               8.696x10^-2 = e^{-(2.59×10-3)t}

                         ln(8.696x10-2) = -(2.59×10-3)t

                                      -2.4423 = -(2.59×10-3)t

                                                  t = 942.97 hours