Assume that when Earth formed, uranium-238 and uranium-235 were equally abundant. Their current percent natural abundances are \(99.28 \%\) uranium- 238 and \(0.72 \%\) uranium- \(235 .\) Given half-lives of \(4.5 \times 10^{9}\) years for uranium-238 and \(7.1 \times 10^{8}\) years for uranium-235, determine the age of Earth corresponding to this assumption.

Short Answer

Expert verified
The approximated age of Earth in this scenario is \(4.48 \times 10^{9}\) years.

Step by step solution

01

Determine the number of half-lives that has passed

First, find the ratio of the remaining amounts of uranium-238 and uranium-235. This is calculated by dividing the current amounts, which are \(99.28 \%\) for uranium-238 and \(0.72 \%\) for uranium-235.
02

Calculate the difference in half-lives

The ratio calculated in the previous step is equal to \(2^n\), where n is the difference in the number of half-lives that has passed for both uranium isotopes. To find out the difference, solve the equation \(99.28/0.72 = 2^n\) for n using logarithms or approximation.
03

Calculate the age of Earth

Now that the difference in the number of half-lives has been determined, the age of earth can be approximated. Multiply the number of additional half-lives that uranium-238 underwent compared to uranium-235, which is the value for n, by the half-life of uranium-235, which is \(7.1 \times 10^{8}\) years. The result will give an estimation for the age of the Earth.

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Most popular questions from this chapter

Complete the following nuclear equations. (a) \(\frac{23}{11} \mathrm{Na}+? \longrightarrow_{11}^{24} \mathrm{Na}+_{1}^{1} \mathrm{H}\) (b) \(_{27}^{59} \mathrm{Co}+_{0}^{1} \mathrm{n} \longrightarrow_{25}^{56} \mathrm{Mn}+?\) (c) \(?+_{1}^{2} \mathrm{H} \longrightarrow_{94}^{240} \mathrm{Pu}+_{-1}^{0} \beta\) (d) \(^{246} \mathrm{Cm}+? \longrightarrow_{102}^{254} \mathrm{No}+5_{0}^{1} \mathrm{n}\) (e) \(^{238} \mathrm{U}+? \longrightarrow_{99}^{246} \mathrm{Es}+6 \frac{1}{0} \mathrm{n}\)

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