Answer:
The half-life of A is 17.1 days.
Explanation:
Hi there!
The half-life of B is 1.73 days.
Let´s write the elapsed time (3 days) in terms of half-lives of B:
1.37 days = 1 half-life B
3 days = (3 days · 1 half-life B / 1.37 days) = 2.19 half-lives B.
After 3 days, the amount of A in terms of B is the following:
A = 4.04 B
The amount of B after 3 days can be expressed in terms of the initial amount of B (B0) and the number of half-lives (n):
B after n half-lives = B0 / 2ⁿ
Then after 2.19 half-lives:
B = B0 /2^(2.19)
In the same way, the amount of A can also be expressed in terms of the initial amount and the number of half-lives:
A = A0 / 2ⁿ
Replacing A and B in the equation:
A = 4.04 B
A0 / 2ⁿ = 4.04 · B0 / 2^(2.19)
Since A0 = B0
A0 / 2ⁿ = 4.04 · A0 / 2^(2.19)
Dividing by A0:
1/2ⁿ = 4.04 / 2^(2.19)
Multipliying by 2ⁿ and dividing by 4.04 / 2^(2.19):
2^(2.19) / 4.04 = 2ⁿ
Apply ln to both sides of the equation:
ln( 2^(2.19) / 4.04) = n ln(2)
n = ln( 2^(2.19) / 4.04) / ln(2)
n = 0.1756
Then, if 3 days is 0.1756 half-lives of A, 1 half-life of A will be:
1 half-life ·(3 days / 0.1756 half-lives) = 17.1 days
The half-life of A is 17.1 days.
