The half-life of 234U, uranium-234, is 2.52 105 yr. If 98.3% of the uranium in the original sample is present, what length of time (to the nearest thousand years) has elapsed?
To determine the length of time that has elapsed, we need to use the formula for exponential decay:
Remaining Amount = Initial Amount * (0.5)^(time / half-life)
Given that 98.3% of the uranium is present, this means the remaining amount is 0.983 times the initial amount.
0.983 = 1 * (0.5)^(time / half-life)
Taking the natural logarithm of both sides:
ln(0.983) = ln(0.5) * (time / half-life)
To solve for time, we rearrange the equation:
(time / half-life) = ln(0.983) / ln(0.5)
Now, we can substitute the given values:
(time / 2.52 * 10^5) = ln(0.983) / ln(0.5)
Using a calculator, we find:
(time / 2.52 * 10^5) ≈ -0.01798
To solve for time, we multiply both sides by 2.52 * 10^5:
time ≈ -0.01798 * 2.52 * 10^5
time ≈ -4547.296
Since time cannot be negative, we discard the negative value:
time ≈ 4547.296
Therefore, approximately 4547 years have elapsed.
To determine 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 the amount of the substance to decay by half.
In this case, the half-life of uranium-234 is given as 2.52 x 10^5 years.
We are also given that 98.3% of the uranium in the original sample is still present. This means that only 1.7% has decayed.
To find the elapsed time, we need to determine the number of half-lives that have occurred. Since each half-life corresponds to a decay of 50% (or 0.5), we can use the equation:
(0.5)^n = 0.017
Here, n represents the number of half-lives. Taking the logarithm of both sides, we have:
n * log(0.5) = log(0.017)
Solving for n, we find:
n = log(0.017) / log(0.5)
Using a calculator, we get:
n ≈ 4.167
So, approximately 4 half-lives have occurred.
Since the half-life of uranium-234 is 2.52 x 10^5 years, we can calculate the elapsed time by multiplying the half-life by the number of half-lives:
elapsed time ≈ 4 * 2.52 x 10^5 years
Calculating this, we find:
elapsed time ≈ 1.008 x 10^6 years
Rounding this to the nearest thousand years, we get:
elapsed time ≈ 1,008,000 years
Therefore, approximately 1,008,000 years have elapsed.