The half-life of 234U, uranium-234, is 2.52 multiplied by 105 yr. If 98.8% of the uranium in the original sample is present, what length of time (to the nearest thousand years) has elapsed?

(1/2)^(T/2.52*10^5) = 0.988

T is the time, in years, since it was 100% U-234.

Solve for T. Use of logs is recommended.

T = 4390 years

That rounds off to 4000 years.

To determine the length of time that has elapsed, we can use the concept of radioactive decay and the half-life of uranium-234.

The half-life of uranium-234 is given as 2.52 x 10^5 years, which means that after each half-life, half of the original sample decays. Therefore, if 98.8% of the uranium is still present, it means that (100% - 98.8%) = 1.2% of the sample has decayed.

To calculate the number of half-lives that have occurred, we can use the following formula:

Number of half-lives = (ln(remaining percentage) / ln(0.5))

In this case, we have:

Number of half-lives = (ln(1.2%) / ln(0.5))

Using a calculator, we can find that ln(1.2%) ≈ -4.419.

Number of half-lives = (-4.419 / ln(0.5))

Using the natural logarithm of 0.5 (approximately -0.693), we can calculate:

Number of half-lives = (-4.419 / -0.693) ≈ 6.37

Since we can't have a fraction of a half-life, we round this value to the nearest whole number, which is 6. So, approximately 6 half-lives have occurred.

Now, to find the elapsed time, we can multiply the number of half-lives by the length of one half-life:

Elapsed time = (Number of half-lives) x (length of half-life)

Using the given length of half-life as 2.52 x 10^5 years:

Elapsed time = 6 x 2.52 x 10^5 years

Calculating this expression, we find:

Elapsed time ≈ 1.512 x 10^6 years

Rounding to the nearest thousand, we can conclude that approximately 1,512,000 years have elapsed.