Answer : The correct option is, (2) 0.5 grams
Solution : Given,
As we know that the radioactive decays follow first order kinetics.
First we have to calculate the half life of Americium-241.
Formula used : [tex]t_{1/2}=\frac{0.693}{k}[/tex]
Putting value of half-life in this formula, we get the rate constant.
[tex]432years=\frac{0.693}{k}[/tex]
[tex]k=1.6\times 10^{-3}year^{-1}[/tex]
The expression for rate law for first order kinetics is given by :
[tex]k=\frac{2.303}{t}\log\frac{a}{a-x}[/tex]
where,
k = rate constant = [tex]1.6\times 10^{-3}year^{-1}[/tex]
t = time taken for decay process = 432 years
a = initial amount of the Americium-241 = 1 g
a - x = amount left after decay process = ?
Putting values in above equation, we get
[tex]1.6\times 10^{-3}=\frac{2.303}{432}\log\frac{1}{a-x}[/tex]
[tex]a-x=0.502g=0.5g[/tex]
Therefore, the amount remain in 432 years will be, 0.5 grams
