The half-life of uranium-235 is 713 million years. Suppose a fossil originally had 26 grams of uranium-235. A geologist had the fossil tested and found that it now has only 3.25 grams of uranium-235.
Approximately how old is the fossil?
N=N₀exp(-ln2•t/T)
ln{N/N₀) = -ln2•t/T
t= - ln{N/N₀)•T/ln2=
= - ln(3.25/26) •713•10⁶/ln2 =
= 2.14•10⁹ years
To determine the age of the fossil, we can use the formula for exponential decay:
N = N0 * (1/2)^(t / T)
Where:
N is the final amount of uranium-235 (3.25 grams),
N0 is the initial amount of uranium-235 (26 grams),
t is the time that has passed,
T is the half-life of uranium-235 (713 million years).
Substituting the known values into the equation, we get:
3.25 = 26 * (1/2)^(t / 713 million)
Now we need to solve for t. Taking the natural logarithm (ln) of both sides of the equation, we can isolate t:
ln(3.25) = ln(26) + (t / 713 million) * ln(1/2)
Using a calculator, we find that ln(3.25) is approximately 1.1786549963416462 and ln(26) is approximately 3.258096538021482. The equation becomes:
1.1786549963416462 = 3.258096538021482 + (t / 713 million) * ln(1/2)
Simplifying:
1.1786549963416462 - 3.258096538021482 = (t / 713 million) * ln(1/2)
-2.079441541679836 = (t / 713 million) * ln(1/2)
Now we can solve for t by multiplying both sides of the equation by (713 million / ln(1/2)):
t = (-2.079441541679836) * (713 million / ln(1/2))
Using a calculator, we can find that ln(1/2) is approximately -0.6931471805599453, and substitute it into the equation:
t = (-2.079441541679836) * (713 million / -0.6931471805599453)
Calculating this expression, we find:
t ≈ 2.139443280187887 * 10^8
Therefore, the fossil is approximately 213.9 million years old.
To determine the age of the fossil, we can use the concept of radioactive decay and the half-life of uranium-235.
The half-life of uranium-235 is given to be 713 million years. This means that in 713 million years, half of the original amount of uranium-235 would decay.
In this case, the fossil originally had 26 grams of uranium-235 and now has 3.25 grams. We can calculate how many half-lives have elapsed by dividing the current amount of uranium-235 by the original amount:
3.25 grams / 26 grams = 0.125
So, 0.125 half-lives have occurred. To find the age of the fossil, we need to multiply the half-life (713 million years) by the number of half-lives that have elapsed:
0.125 * 713 million years = 89.125 million years
Therefore, the approximate age of the fossil is 89.125 million years.