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?
.5 = e^2.52*10^5 k
ln .5 = 2.52 * 10^5 k
k =-2.75*10^-6
ln .977 = -2.75*10^-6 t
t = 8460 years
= 8,000 years
.977=1e^(.693t/thalf)
take lne of each side.
ln.977=.692t/th
t= thalf*ln.977/.693
To find the length of time that has elapsed, we can use the concept of exponential decay and the formula for calculating half-life.
The half-life of uranium-234 is given as 2.52 x 10^5 years, which means that after this amount of time, the mass of uranium-234 is reduced by half.
Since 97.7% of the original uranium-234 is still present, we can say that the remaining amount is 97.7% = 0.977 times the original amount. This is equal to 0.977 * 100% = 97.7%.
Let's assume the original amount of uranium-234 was 100 grams. After one half-life, 50 grams would remain. So, we need to find out how many half-lives it takes for the remaining amount to reach 97.7 grams (0.977 * 100 grams), which is 97.7% of the original amount.
Let's set up an equation to solve for the number of half-lives (n):
(1/2)^n = remaining amount / original amount
(1/2)^n = 97.7 grams / 100 grams
(1/2)^n = 0.977
To solve this equation for n, we can take the logarithm of both sides of the equation:
log((1/2)^n) = log(0.977)
Using the logarithmic property log(a^b) = b * log(a):
n * log(1/2) = log(0.977)
Now we can solve this equation to find n:
n = log(0.977) / log(1/2)
Using a calculator, we find that n ≈ 1.371.
Since n represents the number of half-lives, we can approximate the elapsed time by multiplying n by the half-life (2.52 x 10^5 years):
Elapsed time ≈ n * half-life
Elapsed time ≈ 1.371 * 2.52 x 10^5 years
Calculating this, we get:
Elapsed time ≈ 3.23 x 10^5 years
So, to the nearest thousand years, the length of time that has elapsed is approximately 323,000 years.