13 half-lives have passed
General formulas used in decay:
[tex]\large{\boxed{\bold{N_t=N_0(\dfrac{1}{2})^{t/t\frac{1}{2} }}}[/tex]
T = duration of decay
t 1/2 = half-life
N₀ = the number of initial radioactive atoms
Nt = the number of radioactive atoms left after decaying during T time
t1/2 = 4 days
Nt=18 mg (0.01% of the original isotope)
18 mg (Nt) = 0.01% No
No = the original isotope :
[tex]\tt No=\dfrac{100}{0.01}\times 18~mg=180,000[/tex]
The duration of decay (T) :
[tex]\tt \dfrac{Nt}{No}=\dfrac{1}{2}^{T/4}\rightarrow Nt=0.01\%No[/tex]
[tex]\tt 0.01\%=\dfrac{1}{2}^{T/4}\\\\10^{-4}=\dfrac{1}{2}^{T/4}\\\\(\dfrac{1}{2})^{13}=\dfrac{1}{2}^{T/4}\\\\13=T/4\rightarrow T=52~days[/tex]
Half-lives passed :
[tex]\tt \dfrac{52}{4}=13~half-lives[/tex]
