The half-life of 234U, uranium-234, is 2.52 105 yr. If 98.4% of the uranium in the original sample is present, what length of time (to the nearest thousand years) has elapsed?
x = A e^-kt
if x/A = 0.5, t = 2.52*10^5
.5 = e^-k(2.52*10^5)
ln .5 = -.693 = -2.52*10^5 k
k = 2.75*10^-6
0.984 = e^-2.75*-10^-6 t
t = 5863 years
To find the length of time that has elapsed, we can use the concept of half-life.
First, let's define the half-life. The half-life is the time it takes for half of the radioactive nuclei in a sample to decay.
In this case, the half-life of uranium-234 is 2.52 * 10^5 years.
We know that 98.4% of the uranium is still present in the sample. This means that 1 - 0.984 = 0.016 (or 1.6%) of the uranium has decayed.
Since each half-life corresponds to a decay of 50% of the sample, we can set up the following equation:
(0.5)^(n) = 0.016
Here, 'n' represents the number of half-lives that have occurred.
To find 'n,' we can take the logarithm of both sides of the equation:
log(0.5)^(n) = log(0.016)
n * log(0.5) = log(0.016)
n = log(0.016) / log(0.5)
Using a calculator, we can find that n ≈ 5.276.
Since the number of half-lives is not a whole number, we can round it to the nearest whole number, which is 5.
Now, to find the length of time that has elapsed, we can multiply the number of half-lives by the half-life of uranium-234:
Time elapsed = 5 * 2.52 * 10^5 years
Calculating this gives us approximately 1.26 * 10^6 years.
Rounding this to the nearest thousand years, the length of time elapsed is approximately 1,260,000 years.