Question

Uranium-238 decays very slowly.its half-life is 4.47 billion years.

Answer 1

Answer:

12.53 % of a sample of Uranium-238 will remain after 13.4 billion years.

Explanation:

The half life of the uranium-238 = $t_{\frac{1}{2}}$=4.47 billion years

All the radioactive reaction are of first order kinetics. The rate constant and t half of the reaction are related as:

$k=\frac{0.693}{t_{\frac{1}{2}}}=\frac{0.693}{4.47 \text{billion years}}=0.1550 (\text{billion years})^{-1}$

$k=\frac{2.303}{t}\log\frac{[A_o]}{[A]}$

where,  

k = rate constant = $0.1550 (\text{billion years})^{-1}$

t = time taken during radio decay  = 4.47 billion years

$[A_o]$ = initial amount of the reactant = $1.45\times 10^{-6}mol/L$

[A] = amount left left after time t.

$\log \frac{[A]}{[A_o]}=-\frac{kt}{2.303}$

$\frac{[A]}{[A_o]}=0.1253=\frac{12.53}{100}=12.53\%$

12.53 % of a sample of Uranium-238 will remain after 13.4 billion years.

Answer 2

13.4 billion years is 3 times of the half-life, 4.47 billion years. So the Uranium-238 will go through three times of half decay. So the remain percentage will be 50%*50%*50%=12.5%.