Gallium-67 is used medically in tumor-seeking agents. The half-life of gallium-67 is 78.2 hours. If you begin with 36.6 mg of this isotope, what mass remains after 190 hours have passed?
k = 0.693/t1/2
Substitute k into the following:
ln(36.6/X) = kt
The only unknown is X. Solve for that.
27g
To determine the mass of gallium-67 remaining after 190 hours, we need to calculate the number of half-lives that have passed and use the half-life formula. The half-life of gallium-67 is given as 78.2 hours.
First, let's determine the number of half-lives that have elapsed after 190 hours. We can do this by dividing 190 by the half-life:
Number of half-lives = 190 hours / 78.2 hours ≈ 2.43
Since we can't have partial half-lives, we round down to the nearest whole number. Therefore, 2 half-lives have passed.
Now, to calculate the amount remaining, we use the formula:
Remaining mass = Initial mass × (1/2)^(number of half-lives)
Plugging in the values we have:
Remaining mass = 36.6 mg × (1/2)^2
= 36.6 mg × (1/4)
= 9.15 mg
Therefore, after 190 hours, approximately 9.15 mg of gallium-67 remains.