Question

Suppose the half-life of a decaying radioactive isotope is 674 years. How long will it takefor the isotope to decay from 100 grams to 30 grams? Answer to the nearest hundredthof a year.accc

Answer 1

The half life formula is :

$N(t)=N_o(\frac{1}{2})^{\frac{t}{T}}$

where N(t) = remaining quantity after t years

No = Original Quantity

t = time in years

T = half life in years

From the problem, we have :

N(t) = 30 grams

No = 100 grams

T = 674

Solve for t :

$\begin{gathered} 30=100(\frac{1}{2})^{\frac{t}{674}} \\ \frac{30}{100}=(\frac{1}{2})^{\frac{t}{674}} \\ \frac{3}{10}=(\frac{1}{2})^{\frac{t}{674}} \end{gathered}$

Take ln of both sides :

$\begin{gathered} \ln (\frac{3}{10})=\ln (\frac{1}{2})^{\frac{t}{674}} \\ \ln (\frac{3}{10})=\frac{t}{674}\ln (\frac{1}{2}) \\ t=\frac{674\ln (\frac{3}{10})}{\ln (\frac{1}{2})} \\ t=1170.71 \end{gathered}$

The answer is t = 1170.71 years