Question

What was the initial mass (in mg) of the sample? What is the mass (in mg) 4 weeks after the start?

Answer 1

To calculate the initial mass we must use the half life equation

`N_t=N_0\cdot e^(-\lambda\cdot t)`

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

`\lambda=\frac{\ln (2)}{t_{(1)/(2)}}`

Where

t1/2 is the half life

So λ will be

`\begin{gathered} \lambda=(\ln(2))/(4) \\ \lambda=0.1733 \end{gathered}`

Then, we must solve the initial equation for N0

`N_0=(N_t)/(e^(-\lambda\cdot t))`
`\begin{gathered} N_0=(3mg)/(e^(-0.1733\cdot20)) \\ N_0=(3mg)/(0.0312)=96.1538 \end{gathered}`

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

`4\text{weeks}=28\text{days}`
`\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}`

So, the mass after 4 weeks will be 0.7509 mg