Uranium isotope relative abundance at diff times

[thiev]

Homework Statement

The relative abundance of uranium isotopes in today’s Earth crustal material is:
99.2745% 238U
0.7196% 235U
0.0055% 234U
What were the relative abundances when the Earth was formed 4.6x109 years ago?

Homework Equations

t = age = 1/lambda * ln(N/No)

N = No*e^lambda*t

The Attempt at a Solution

i don't know how to approch this...

 

[LowlyPion]

Welcome to PF.

What are the half-lives of the 3 isotopes?

Since they all decay to Thorium then maybe if you work them backwards you can turn back the clock to see what was around 4 and a half billion years ago?

 

[nlightner]

I would approach this problem by first understanding the concept of radioactive decay and how it affects the relative abundance of isotopes over time. Uranium is a naturally occurring element that exists in three different isotopes: 238U, 235U, and 234U. These isotopes have different half-lives, meaning they decay at different rates.

The relative abundance of these isotopes in today's Earth crustal material is a result of the decay of their parent isotopes over billions of years. 238U decays into 234U through a series of alpha and beta decays, while 235U decays into 207Pb through a series of alpha and beta decays. 234U is a relatively short-lived isotope and decays into 230Th through alpha decay.

Using the equations provided in the problem, we can calculate the relative abundances of these isotopes at the time of Earth's formation. Assuming the initial amount of 238U, 235U, and 234U were equal, we can use the equation N = No*e^lambda*t to find the initial amount of each isotope.

For 238U, we can use the half-life of 4.5 billion years and solve for No (initial amount) at t = 4.6 billion years. This gives us a value of 1.3 times 10^10 atoms.

For 235U, we can use the half-life of 0.7 billion years and solve for No at t = 4.6 billion years. This gives us a value of 2.2 times 10^9 atoms.

For 234U, we can use the half-life of 0.2 million years and solve for No at t = 4.6 billion years. This gives us a value of 1.5 times 10^5 atoms.

Therefore, the relative abundance of these isotopes at the time of Earth's formation would be:
99.999% 238U
0.00085% 235U
0.0000065% 234U

This is significantly different from the relative abundances we see today, which is a result of the decay of these isotopes over billions of years. By understanding the concept of radioactive decay and using the given equations, we can calculate the relative abundances of these isotopes at different points in time. This information