Question

element x is a radioactive isotope such that every 32 years, its mass decreases by half. given that the initial mass of the sample reached 150 grams, to the nearest tenth of a year?

Answer 1

Answer:

the answer is 31.9 years

Step-by-step explanation:

initial mass,  $M_{0}$  = 150 gram

half life, $t_{half\hspace{0.1cm}life}$ = 32 years

new mass = half of initial mass = M = 0.5$M_{0}$

formula , M = $M_{0}$ $e^{\frac{\ln\frac{1}{2}}{t_{half} }\times x }$ where x is the number of years required.

therefore 0.5 $M_{0}$  = $M_{0}$ $e^{\frac{\ln\frac{1}{2}}{t_{half} }\times x } = M_{0} e^{-0.0217x}$

therefore -0.0217 x = $\ln{0.5} = -0.693$  

therefore x = $\frac{-0.693}{-0.0217}$ = 31.942