How long does it take for 10.0 g of a radioactive isotope to decay to 1.25 g if its half-life is 17.0 d?
k = 0.693/t1/2
Then substitute k into the below.
ln(No/N) = kt
No = 10g
N = 1.25
k from above
Solve for k in days.
To determine the time it takes for a given amount of a radioactive isotope to decay, we can use the concept of half-life. The half-life is the time it takes for half of the initial amount of a substance to decay.
In this case, the half-life of the radioactive isotope is given as 17.0 days, which means that after 17.0 days, half of the initial amount will decay.
To find out how long it takes for 10.0 g of the isotope to decay to 1.25 g, we need to calculate the number of half-lives that have passed.
1. Start by calculating the number of half-lives it takes for 10.0 g to decay to 1.25 g:
Initial amount = 10.0 g
Final amount = 1.25 g
The ratio of the final amount to the initial amount is:
Final amount / Initial amount = 1.25 g / 10.0 g = 0.125
2. Next, use the formula for the number of half-lives:
Number of half-lives = log(base 2) (Final amount / Initial amount)
Calculating this value gives:
Number of half-lives = log(base 2) (0.125) ≈ -3.976
Note that we take the negative value as the ratio is less than 1.
3. Since each half-life is 17.0 days, multiply the number of half-lives by 17.0 to find the time it takes:
Time = Number of half-lives × Half-life duration
Time = -3.976 × 17.0 days ≈ -67.692 days
The negative value does not make sense in this context, so we can ignore it. Therefore, it takes approximately 67.692 days for 10.0 g of the radioactive isotope to decay to 1.25 g.
Keep in mind that in reality, the isotope will continue to decay beyond this point, but the question specifically asks for the time it takes to decay to 1.25 g.