The half-life of 234U, uranium-234, is 2.52 105 yr. If 97.7% of the uranium in the original sample is present, what length of time (to the nearest thousand years) has elapsed?
.977=e^(-.693T/thalf)
take the ln of each side, solve for T
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?
The half-life of 234U, uranium-234, is 2.52 105 yr. If 98.2% 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 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, we are given that the half-life of uranium-234 is 2.52 * 10^5 years. We are also told that 97.7% of the uranium in the original sample is still present.
To find the length of time that has elapsed, we need to determine how many half-lives have occurred. We can do this by dividing the natural logarithm of the remaining fraction (97.7%) by the natural logarithm of 0.5 (since half of the substance decays in each half-life).
Let's calculate it step by step:
1. Calculate the remaining fraction:
Remaining fraction = 97.7% = 97.7/100 = 0.977
2. Calculate the number of half-lives:
Number of half-lives = ln(remaining fraction) / ln(0.5)
Using a calculator:
Number of half-lives ≈ ln(0.977) / ln(0.5) ≈ -0.02305 / -0.69315
Number of half-lives ≈ 0.03325
3. Calculate the elapsed time:
Elapsed time = number of half-lives * half-life
Elapsed time ≈ 0.03325 * 2.52 * 10^5 years
Elapsed time ≈ 8,361 years
Therefore, to the nearest thousand years, the length of time that has elapsed is 8,000 years.