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?
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4000
To find the length of time that has elapsed, we can use the formula for exponential decay:
N(t) = N₀ * (1/2)^(t / T)
Where:
- N(t) is the amount of the substance remaining at time t,
- N₀ is the initial amount of the substance,
- t is the elapsed time, and
- T is the half-life of the substance.
In this case, we know that 98.8% of the original uranium is present, so N(t) = 0.988 * N₀. We are given that the half-life of uranium-234 is 2.52 * 10^5 years, so T = 2.52 * 10^5 years.
Substituting these values into the equation, we have:
0.988 * N₀ = N₀ * (1/2)^(t / (2.52 * 10^5))
To solve for t, we can divide both sides of the equation by N₀ and take the logarithm of both sides:
0.988 = (1/2)^(t / (2.52 * 10^5))
log(0.988) = log((1/2)^(t / (2.52 * 10^5)))
Using logarithm properties, we can rewrite this equation as:
log(0.988) = (t / (2.52 * 10^5)) * log(1/2)
Now, we can solve for t by isolating it:
t = (2.52 * 10^5) * (log(0.988) / log(1/2))
Using a calculator, we can evaluate the right-hand side of the equation to find t.
Calculating this, we get:
t ≈ 1.07 * 10^5 years
Therefore, to the nearest thousand years, approximately 107,000 years have elapsed.