Question 12.11: Using Decay Rates to Calculate a Half-Life A sample of ^41Ar......
Using Decay Rates to Calculate a Half-Life
A sample of {}^{41}Ar, a radioisotope used to measure the flow of gases from smokestacks, decays initially at a rate of 34,500 disintegrations/min, but the decay rate falls to 21,500 disintegrations/min after 75.0 min. What is the half-life of {}^{41}Ar?
STRATEGY
The half-life of a radioactive decay process is obtained by finding t_{1/2} in the equation
\ln \left(\frac{N_t}{N_0}\right)=(-\ln 2)\left(\frac{t}{t_{1 / 2}}\right)In the present instance, though, we are given decay rates at two different times rather than values of N_t and N_0. Nevertheless, for a first-order process like radioactive decay, in which rate = kN, the ratio of the decay rate at any time t to the decay rate at time t = 0 is the same as the ratio of N_t to N_0:
\frac{\text { Decay rate at time } t}{\text { Decay rate at time } t=0}=\frac{k N_t}{k N_0}=\frac{N_t}{N_0}Substituting the decay rates into the equation gives
\ln \left(\frac{21,500}{34,500}\right)=(-0.693)\left(\frac{75.0 \mathrm{~min}}{t_{1 / 2}}\right) \quad \text { or } \quad-0.473=\frac{-52.0 \mathrm{~min}}{t_{1 / 2}} \\ \text { so } \quad t_{1 / 2}=\frac{-52.0 \mathrm{~min}}{-0.473}=110 \mathrm{~min}The half-life of {}^{41}Ar is 110 min.