the half-life of 234U, uranium-234, is 2.52 x 10^5yr. If 98.5% of thee uranium in the original sample is present, what length of time (to the nearest thousand years) has elasped
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The half-life of 234U, uranium-234, is 2.52 105 yr. If 98.5% of the uranium in the original sample is present, what length of time (to the nearest thousand years) has elapsed?
To find the length of time that has elapsed, we can use the concept of half-life. The half-life of a radioactive substance is the time it takes for half of the substance to decay. In this case, the half-life of uranium-234 is given as 2.52 x 10^5 years.
We are told that 98.5% of the uranium in the original sample is present. This means that only 1.5% of the uranium has decayed. Since each half-life corresponds to a decay of 50%, we can calculate the number of half-lives that have occurred.
Let's assume we start with 100 grams of uranium-234. If 98.5% is still present, we have 98.5 grams remaining, and 1.5 grams have decayed.
Now, let's divide the mass of the decayed uranium by the original mass to find the fraction that has decayed:
1.5 grams / 100 grams = 0.015
This fraction corresponds to the number of half-lives, as each half-life results in a 50% decay. We can use the following formula to find the number of half-lives (n):
(1/2)^n = 0.015
Taking the logarithm (base 0.5) of both sides, we get:
n = log(0.015) / log(0.5)
Using a calculator, we find that n ≈ 6.46. Since we cannot have a fraction of a half-life, we'll round this down to 6.
Now, we can multiply the number of half-lives (6) by the length of a half-life (2.52 x 10^5 years) to find the total elapsed time:
6 x 2.52 x 10^5 years = 1.512 x 10^6 years
Therefore, approximately 1,512,000 years (to the nearest thousand years) have elapsed.