Final answer:
To find the time when there will be 7 grams of krypton-91 remaining, we calculate the decay using the half-life concept and an exponential decay formula. The exact time is roughly 13.516 seconds, which is between the first and second half-lives of the substance.
Step-by-step explanation:
The question involves calculating the time when there will be 7 grams of the radioactive element krypton-91 remaining, given that it has a half-life of 10 seconds and the initial amount is 16 grams. To solve this problem, we would use the concept of half-life, which is the time it takes for half of the radioactive nuclei to decay.
Since the initial quantity is 16 grams, after one half-life (10 seconds), there would be 8 grams left. After each subsequent half-life, the quantity remaining is halved again. Thus, we can set up a sequence like this:
- After 1 half-life (10 seconds) = 16 g / 2 = 8 g
- After 2 half-lives (20 seconds) = 8 g / 2 = 4 g
We can see that after two half-lives, or 20 seconds, only 4 grams are left, which is less than 7 grams. Therefore, the time at which 7 grams remains must be between 10 and 20 seconds. To find the exact time, we can use the formula for exponential decay:
N = N0(1/2)^(t/T)
where:
- N is the final amount of the substance (7 grams)
- N0 is the initial amount of the substance (16 grams)
- t is the time that has elapsed
- T is the half-life of the substance (10 seconds)
Plugging in the values we get:
7 = 16(1/2)^(t/10)
To solve for t, we take the natural logarithm (ln) of both sides:
ln(7) = ln(16(1/2)^(t/10))
ln(7) = ln(16) + ln((1/2)^(t/10))
ln(7) - ln(16) = (t/10) * ln(1/2)
Finally, we solve for t:
t = [ln(7) - ln(16)] / [ln(1/2)] * 10
Using a calculator, we find that t is approximately equal to 13.516 seconds, rounded to the nearest thousandths.