Nuclear u-235 and u-238 isotope half life question

[debwaldy]

Homework Statement

The present day value of the uranium isotopic ratio U-235/U-238 is 0.00723. The half life of U-238 is 4.51 x 10^9 and the half life of U-235 is 7.13 x 10^8 years. What was the U-235/U-238 isotopic ratio 2 billion years ago when the Oklo natural reactor was active?

Homework Equations

T1/2 = ln (2) / λ
τ = 1/λ
N(t) = No e^(-kt)

The Attempt at a Solution

Calculated the decay constant lambda for each isotope:

U-235: λ = 9.72 x 10^-10 per year

U-238: λ = 1.54 x 10^-10 years,

I'm not sure how to proceed or what I need to do next

New to this radioactive decay stuff so any tips or suggestions would be much appreciated.

Thanks,
Debbie

 

[daveb]

You have 2 equations, one for the activity of U-235 and one for the activity of U-238, each as a function of time. What happens when you divide one by the other?

 

[debwaldy]

As in the equation N(t) equations? will the No be different in each case?

 

[debwaldy]

If i do this and substitute in t = 2 billion years i get a ratio of 0.1947, although I don't think that this is correct?

 

[asteriatic]

Hello Debbie,

Thank you for your question. To solve this problem, we will use the equations you have listed in your attempt at a solution. We will also make use of the fact that the ratio of U-235 to U-238 isotopes is constant throughout the decay process. This means that the ratio of the initial amount of U-235 to U-238 is equal to the ratio of the present day amount of U-235 to U-238. We will use this concept to solve for the initial U-235/U-238 isotopic ratio 2 billion years ago when the Oklo natural reactor was active.

First, we need to find the initial amount of U-235 and U-238 isotopes. We can do this by rearranging the equation N(t) = No e^(-kt) to solve for No. This gives us No = N(t) / e^(-kt). We can then substitute in our known values for N(t) (present day value) and t (2 billion years) to find the initial amount of each isotope.

Next, we can use the equation T1/2 = ln (2) / λ to solve for the decay constant k. We will use the half-life values given in the homework statement to find the decay constants for U-235 and U-238.

Finally, we can use the ratio of the initial amounts of U-235 and U-238 to find the initial U-235/U-238 isotopic ratio. This will give us the answer to the homework question.

I hope this helps. Please let me know if you have any further questions or need clarification on any of the steps. Keep up the good work in your studies of radioactive decay!

Best,