Radioactive decay is a first order process.
Calculate the first-order rate constant
t_½ = ln2/k
k = ln2/t_½ = ln2/"2.62 h" = 0.2645 color(red)(cancel(color(black)("h"^"-1"))) × (1 color(red)(cancel(color(black)("h"))))/"60 min" = 4.409 × 10^"-3"color(white)(l) "min"^"-1"
Calculate the original mass
The integrated rate law for a first order reaction is
color(blue)(bar(ul(|color(white)(a/a)ln("A"_0/"A"_t) = ktcolor(white)(a/a)|)))" "
where
"A"_0 and "A"_t are the amounts at time t = 0 and at time t and
k is the rate constant
In your problem,
"A"_t = "47.1 g"
k = 4.409 × 10^"-3"color(white)(l) "min"^"-1"
t = "55.2 min"
ln("A"_0/"A"_t) = 4.409 × 10^"-3" color(red)(cancel(color(black)("min"^"-1"))) × 55.2 color(red)(cancel(color(black)("min"))) = 0.2434
"A"_0/"A"_t = e^0.2434 = 1.276
"A"_0 = 1.276"A"_t = "1.276 × 47.1 g = 60.1 g"
The original mass was 60.1 g.
