The half-life of 234U, uranium-234, is 2.52 105 yr. If 97.4% of the uranium in the original sample is present, what length of time (to the nearest thousand years) has elapsed?
2.52*10^5 = 252 thousand
do it all in thousands of years
.5 = e^-k(252)
ln .5 = -252 k
solve for k
then
.974 = e^-kt
ln .974 = - k t solve for t
To determine the length of time that has elapsed, we can use the formula for radioactive decay:
N(t) = N₀ * (1/2)^(t / t₁/₂)
Where:
N(t) = Final amount of uranium remaining after time t
N₀ = Initial amount of uranium in the sample
t = Time that has elapsed
t₁/₂ = Half-life of uranium-234 (2.52 * 10^5 yr)
We are given that 97.4% of the uranium is present in the original sample. This means that 97.4% of N₀ is equal to N(t), so we can rewrite the formula as:
0.974N₀ = N₀ * (1/2)^(t / t₁/₂)
Dividing both sides by N₀ gives:
0.974 = (1/2)^(t / t₁/₂)
To solve for t, we need to take the logarithm of both sides with base 1/2:
log₁/₂(0.974) = t / t₁/₂
Now we can substitute the values into the formula and solve for t:
log₁/₂(0.974) = t / (2.52 * 10^5)
Using logarithmic properties, we can rewrite the equation as:
t = log₁/₂(0.974) * (2.52 * 10^5)
Using a calculator, we find:
t ≈ 7007
Therefore, the length of time that has elapsed is approximately 7007 years (rounded to the nearest thousand years).