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
Since we have a "half-life" situation I can use a base of 1/2
.977 = (1/2)^(t/2.52x10^5)
t/2.52x10^5 = ln.977/ln.5
t = 2.52x10^5(ln.97/ln.5) = 8459.22 years or 9000 years to the nearest thousand years
for an alternate solution see MathMate's
http://www.jiskha.com/display.cgi?id=1310960571
To determine the amount of time that has elapsed, you 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.
Since 97.7% of the uranium is still present, it means that only 2.3% (100% - 97.7%) has decayed. Therefore, the remaining amount is 0.023 times the original amount.
To find the number of half-lives that have passed, you can use the formula Nt/N0 = (1/2)^(t/T), where Nt is the remaining amount, N0 is the original amount, t is the time elapsed, and T is the half-life.
Plugging in the values, we have:
0.023 = (1/2)^(t/2.52 x 10^5)
To solve for t, we can take the logarithm of both sides:
log(0.023) = (t/2.52 x 10^5) * log(1/2)
Rearranging the equation to solve for t, we get:
t = (log(0.023) / log(1/2)) * 2.52 x 10^5
Now, let's calculate the value:
t ≈ (log(0.023) / log(1/2)) * 2.52 x 10^5 ≈ 1.6338 * 2.52 x 10^5
Therefore, the length of time, rounded to the nearest thousand years, that has elapsed is approximately 411,996 years.