What Will the Concentration of Uranium 235 Be in One Billion Years?

[JDiorio]

Homework Statement

At present, the concentration of uranium 235 in naturally occurring uranium deposits is approximately 0.77%. What will the concentration be one billion years from now?

Homework Equations

N(t) = No e^ -(ln(2)/half-life)t

The Attempt at a Solution

i attempted this problem and got an answer of .69%.. What i did was put XNo= .77No e^ -(ln(2)/half-life)t .. that was the No can cancel out.. and then I just solved for X.. I feel like I am close but just making a slight mistake.

 

[Redbelly98]

You need to know the half-life of uranium to solve this problem.

 

[JDiorio]

well i looked it up and found that it was equal to 7.04E8.. sorry that i did not include this information.. it wasn't included in the original question but i looked it up in the textbook.. but with that information, am i going about solving the problem correctly?

 

[Grogs]

Your equation is right, although that .77 should either be .77% or .0077. I get something around half the value you reported when I plug the half-life and 1 Billion years into my calculator.

One thing you're not accounting for is the decay of the U-238 though. Does the problem say to ignore that? About 15% of it would decay away in a billion years and you would have to account for that to get the new concentration of U-235.

 

[Redbelly98]

U-235 is not part of the decay chain of U-238, so the U-238 decay does not matter here.

I agree with Grogs, the % U-235 is less than the .69% calculated by JDioro. JDioro, what do you calculate for the quantity:

(ln(2)/half-life)t​

It's probably a simple arithmetic error somewhere.

 

[Grogs]

U-235 is not part of the decay chain of U-238, so the U-238 decay does not matter here.

It's not in the decay chain, but it is in the denominator of the equation used to find the fraction of U-235 in natural uranium:

frac(U-235) = N(235) / [N(235) + N(238)] (Ignoring the really small amount of U-234)

That's why most of the times I saw this type of question in basic physics courses the question stated to ignore U-238 decay. Otherwise you have to solve for both and find the new fraction.

If the question is asking what's the concentration in the rock (it seems a little ambiguous the way it's worded - "uranium deposit" sounds like we're just talking just uranium enrichment) then the U-238 just converts to Th-234 and so on so you don't have to account for it. If you're just doing basic radioactive decay I suspect that's what the instructor is looking for, but it's good to state that you're making that assumption.

 

[JDiorio]

my calculations were as follows:

(ln(2)/half-life)t = .0985

and then i just calculate e^ -.0985 and now i got .906.. and then when i multiply by the .77 I get .697

 

[Redbelly98]

If the half-life is 7.04e8 years, and t=1e9 years, then that expression should be 0.985, not 0.0985.

It's not in the decay chain, but it is in the denominator of the equation used to find the fraction of U-235 in natural uranium:

Yes, good point. But I agree, we're probably supposed to ignore that, and that should have been stated explicitly.

 

[JDiorio]

UGH!... I am using the E button on my calculator and didn't account for an extra.. i was doing 1E8 instead of 10E8.. thanks for the help.. really appreciate it..