Question

How many milligrams of a 6.5×10-2kg sample of the x parent isotope will remain at the end of
(a)? In a half life is 3 and time elapsed is 13.5 years

Answer 1

To calculate how much of a parent isotope remains after a given time period, we can use the half-life formula. The formula for the remaining mass of a substance after a certain time \( t \) is:

\[
N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}
\]

Where:
- \( N \) = remaining amount of the isotope
- \( N_0 \) = initial amount of the isotope
- \( t \) = elapsed time
- \( T_{1/2} \) = half-life of the isotope

Given:
- Initial mass, \( N_0 = 6.5 \times 10^{-2} \) kg
- Half-life, \( T_{1/2} = 3 \) years
- Time elapsed, \( t = 13.5 \) years

First, we need to determine how many half-lives have passed in the given time:

\[
\text{Number of half-lives} = \frac{t}{T_{1/2}} = \frac{13.5 \text{ years}}{3 \text{ years}} = 4.5
\]

Now we can plug this value into the formula:

\[
N = N_0 \left( \frac{1}{2} \right)^{4.5}
\]

Calculating \( \left( \frac{1}{2} \right)^{4.5} \):

\[
\left( \frac{1}{2} \right)^{4.5} = \frac{1}{2^{4.5}} = \frac{1}{\sqrt{2^9}} = \frac{1}{\sqrt{512}} \approx \frac{1}{22.6274} \approx 0.0442
\]

Now plug in the initial amount:

\[
N = (6.5 \times 10^{-2} \text{ kg}) \times 0.0442 \approx 2.874 \times 10^{-3} \text{ kg}
\]

To convert kilograms to milligrams (1 kg = 1,000,000 mg):

\[
N \approx 2.874 \times 10^{-3} \text{ kg} \times 1,000,000 \text{ mg/kg} \approx 2874 \text{ mg}
\]

Therefore, at the end of 13.5 years, approximately **2874 mg** of the parent isotope will remain.