If a radioisotope has a physical half-life 𝑡𝐻 and a biological half-life 𝑡𝐵, what fraction of the substance decays within the body? Calculate that fraction for I-131 with radioactive half-life 8 days and biological half-life 4 days.
The Correct Answer and Explanation is:
The fraction of a substance that decays within the body, considering both its physical and biological half-lives, can be calculated using the combined effective half-life formula:1teff=1tH+1tB\frac{1}{t_{\text{eff}}} = \frac{1}{t_H} + \frac{1}{t_B}teff1=tH1+tB1
Where:
- tHt_HtH is the physical half-life (radioactive decay time),
- tBt_BtB is the biological half-life (the time for the body to remove half of the substance),
- tefft_{\text{eff}}teff is the effective half-life, which is the overall time for the substance to decrease by half in the body, accounting for both decay and removal.
Step 1: Calculate the effective half-life
Given:
- tH=8 dayst_H = 8 \, \text{days}tH=8days
- tB=4 dayst_B = 4 \, \text{days}tB=4days
We can plug these values into the formula to find the effective half-life tefft_{\text{eff}}teff:1teff=18+14\frac{1}{t_{\text{eff}}} = \frac{1}{8} + \frac{1}{4}teff1=81+41
First, find a common denominator:18+14=18+28=38\frac{1}{8} + \frac{1}{4} = \frac{1}{8} + \frac{2}{8} = \frac{3}{8}81+41=81+82=83
Thus,teff=83 days≈2.67 dayst_{\text{eff}} = \frac{8}{3} \, \text{days} \approx 2.67 \, \text{days}teff=38days≈2.67days
Step 2: Calculate the fraction of decay within the body
The fraction of the substance decaying within the body is related to the physical half-life and the effective half-life. To find the fraction decaying within the body, use the ratio of the physical half-life to the effective half-life:Fraction decayed within the body=tHteff\text{Fraction decayed within the body} = \frac{t_H}{t_{\text{eff}}}Fraction decayed within the body=tefftH
Substituting the values:Fraction decayed within the body=82.67≈3.00\text{Fraction decayed within the body} = \frac{8}{2.67} \approx 3.00Fraction decayed within the body=2.678≈3.00
This result indicates that the substance decays approximately 3 times faster in the body than in isolation, which aligns with the fact that the biological process helps to eliminate the substance more rapidly.
Conclusion:
For I-131 with a radioactive half-life of 8 days and a biological half-life of 4 days, the fraction decaying within the body is approximately 3.
