13 half-lives have passed
<h3>Further explanation
</h3>
General formulas used in decay:
![\large{\boxed{\bold{N_t=N_0(\dfrac{1}{2})^{t/t\frac{1}{2} }}}](https://tex.z-dn.net/?f=%5Clarge%7B%5Cboxed%7B%5Cbold%7BN_t%3DN_0%28%5Cdfrac%7B1%7D%7B2%7D%29%5E%7Bt%2Ft%5Cfrac%7B1%7D%7B2%7D%20%7D%7D%7D)
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 :
![\tt No=\dfrac{100}{0.01}\times 18~mg=180,000](https://tex.z-dn.net/?f=%5Ctt%20No%3D%5Cdfrac%7B100%7D%7B0.01%7D%5Ctimes%2018~mg%3D180%2C000)
The duration of decay (T) :
![\tt \dfrac{Nt}{No}=\dfrac{1}{2}^{T/4}\rightarrow Nt=0.01\%No](https://tex.z-dn.net/?f=%5Ctt%20%5Cdfrac%7BNt%7D%7BNo%7D%3D%5Cdfrac%7B1%7D%7B2%7D%5E%7BT%2F4%7D%5Crightarrow%20Nt%3D0.01%5C%25No)
![\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](https://tex.z-dn.net/?f=%5Ctt%200.01%5C%25%3D%5Cdfrac%7B1%7D%7B2%7D%5E%7BT%2F4%7D%5C%5C%5C%5C10%5E%7B-4%7D%3D%5Cdfrac%7B1%7D%7B2%7D%5E%7BT%2F4%7D%5C%5C%5C%5C%28%5Cdfrac%7B1%7D%7B2%7D%29%5E%7B13%7D%3D%5Cdfrac%7B1%7D%7B2%7D%5E%7BT%2F4%7D%5C%5C%5C%5C13%3DT%2F4%5Crightarrow%20T%3D52~days)
Half-lives passed :
![\tt \dfrac{52}{4}=13~half-lives](https://tex.z-dn.net/?f=%5Ctt%20%5Cdfrac%7B52%7D%7B4%7D%3D13~half-lives)
<span>Ca^(2+). Calcium is an alkaline earth metal with an atomic number of 20. Its electronic configuration is (2, 8, 8, 2). From the electronic configuration it would be observed that calcium has two valence electrons. Calcium shares these two electrons with non metals like sulphur in an ionic bond to obtain an outer shell which is full octet. Let me use Sulphur. When calcium gives out its two valence electrons its electronic configuration becomes (2, 8, 8) thereby obtaining a stable out shell and the calcium ion becomes Ca^(2+). The sulphur also obtains a stable octet because it has 6 valence electron initially.</span>