To calculate the initial mass we must use the half life equation
[tex]N_t=N_0\cdot e^{-\lambda\cdot t}[/tex]
Where
λ the dacay constant
Nt is the ending amount
N0 is the initial amount
t is the elapsed time
First, we must calculate λ
[tex]\lambda=\frac{\ln (2)}{t_{\frac{1}{2}}}[/tex]
Where
t1/2 is the half life
So λ will be
[tex]\begin{gathered} \lambda=\frac{\ln(2)}{4} \\ \lambda=0.1733 \end{gathered}[/tex]
Then, we must solve the initial equation for N0
[tex]N_0=\frac{N_t}{e^{-\lambda\cdot t}}[/tex][tex]\begin{gathered} N_0=\frac{3mg}{e^{-0.1733\cdot20}} \\ N_0=\frac{3mg}{0.0312}=96.1538 \end{gathered}[/tex]
So, the initial mass of the sample was 96.1538 mg.
2.
To calculate the mass after 4 weeks me must convert weeks to days and then we will use the same formula
[tex]4\text{weeks}=28\text{days}[/tex][tex]\begin{gathered} N_{28}=96.1538\cdot e^{-0.1733\cdot28} \\ N_{28}=96.1538\cdot7.8096\cdot10^{-3} \\ N_{28}=0.7509mg \end{gathered}[/tex]
So, the mass after 4 weeks will be 0.7509 mg
