If a radioactive isotope has half life of 5 days,how long will it take for its radioactive strength to be a maximum of one thousandth,(1/1000) of its original value?
1/2 = e^-5k
ln .5 = -5 k
k = .1386
so
x = Xo e^-.1386 t
.001 = e^-.1386 t
-6.91 = -.1386 t
t = 49.8 days
check
about 10 five day periods
.5^10 = .000976
close enough to .001
To find out how long it will take for a radioactive isotope to reach one thousandth of its original value, we can use the concept of half-life.
The half-life is the time it takes for the radioactivity of a substance to decrease by half. In this case, the half-life of the isotope is 5 days.
To calculate the number of half-lives needed to reach one thousandth of the original value, we can use the formula:
Number of half-lives = (log(1/1000) / log(1/2))
Let's calculate it step by step:
1. Take the logarithm base 2 of 1/1000:
log(1/1000) / log(1/2) ≈ 9.966
2. Since fractional half-lives don't make sense, we need to round up to the next whole number:
Number of half-lives ≈ 10
So, it will take approximately 10 half-lives for the radioactive strength of the isotope to decrease to one thousandth (1/1000) of its original value.
Now, to find the total time it will take, we can multiply the half-life by the number of half-lives:
Total time = Half-life × Number of half-lives
= 5 days × 10
= 50 days
Therefore, it will take approximately 50 days for the radioactive strength of the isotope to reach one thousandth (1/1000) of its original value.