Question

The ​half-life of a radioactive element is 130​ days, but your sample will not be useful to you after​ 80% of the radioactive nuclei originally present have disintegrated. About how many days can you use the​ sample? Round to the nearest day.

Answer 1

We can use the sample about 42 days.

Explanation:

Decay Equation:

`(dN)/(dt)\propto -N`

`\Rightarrow (dN)/(dt) =-\lambda N`

`\Rightarrow (dN)/(N) =-\lambda dt`

Integrating both sides

`\int (dN)/(N) =\int\lambda dt`

`\Rightarrow ln|N|=-\lambda t+c`

When t=0, N=
`N_0` = initial amount

`\Rightarrow ln|N_0|=-\lambda .0+c`

`\Rightarrow c= ln|N_0|`

`\therefore ln|N|=-\lambda t+ln|N_0|`

`\Rightarrow ln|N|-ln|N_0|=-\lambda t`

`\Rightarrow ln|(N)/(N_0)|=-\lambda t`.......(1)

`(N)/(N_0)=e^(-\lambda t)`.........(2)

Logarithm:

• 
  `ln|\frac mn|= ln|m|-ln|n|`
• 
  `ln|ab|=ln|a|+ln|b|`

• 
  `ln|e^a|=a`

• 
  `ln|a|=b \Rightarrow a=e^b`
• 
  `ln|1|=0`

130 days is the half-life of the given radioactive element.

For half life,

`N=\frac12 N_0`,
`t=t_\frac12=130` days.

we plug all values in equation (1)

`ln|(\frac12N_0)/(N_0)|=-\lambda * 130`

`\rightarrow ln|(\frac12)/(1)|=-\lambda * 130`

`\rightarrow ln|1|-ln|2|-ln|1|=-\lambda * 130`

`\rightarrow -ln|2|=-\lambda * 130`

`\rightarrow \lambda= (-ln|2|)/(-130)`

`\rightarrow \lambda= (ln|2|)/(130)`

We need to find the time when the sample remains 80% of its original.

`N=(80)/(100)N_0`

`\therefore ln|{\frac{\frac {80}{100}N_0}{N_0}|=-(ln2)/(130)t`

`\Rightarrow ln|{{\frac {80}{100}|=-(ln2)/(130)t`

`\Rightarrow ln|{{ {80}|-ln|{100}|=-(ln2)/(130)t`

`\Rightarrow t=(ln|80|-ln|100|)/(-(ln|2|)/(130))`

`\Rightarrow t=\frac)* 130{-2}`

`\Rightarrow t\approx 42`

We can use the sample about 42 days.