Answer: 7.50 grams of radioactive isotope will remain
Explanation:- Radioactive decay follows first order kinetics
Half-life of sample of lanthanum-138 = 105 billion years
[tex]\lambda =\frac{0.693}{t_{\frac{1}{2}}}=\frac{0.693}{105}=6.6\times 10^{-3}{\text {billion years^{-1}}[/tex]
[tex]N=N_o\times e^{-\lambda t}[/tex]
N = amount left after time t = ?
[tex]N_0[/tex] = initial amount = 240 g
[tex]\lambda[/tex] = rate constant =[tex]6.6\times 10^{-3}billion years^{-1}[/tex]
t= time = 525 billion years
Putting in the values, we get
[tex]N=240\times e^{- 6.6\times 10^{-3}billion years^{-1}\times 525 {\text {billion years}}[/tex]
[tex]N=7.50g[/tex]
