Tricky radiation half-life question

[lando45]

Hey,

I have been set this question as an assignment and I spent about an hour researching it yesterday and came up with what I thought was the right answer, but it has turned out to be wrong.

"A theory of astrophysics proposes that all the elements heavier than iron are formed in supernova explosions ending the lives of massive stars. If we assume that at the time of the explosion the amounts of 235U and 238U were equal, how long ago did the star(s) explode that released the elements that formed our Earth? The present 235U / 238U ratio is 0.00725. The half-lives of 235U and 238U are 0.704 x 10^9 years and 4.47 x 10^9 years."

Here is what I did.

Reciprocal of 0.00725 = 137.93
Used log to find that 2^7.1078 = 137.93
Multiplied 7.1078 by 0.704 x 10^9 to get 5,003,891,200 years.

But this answer turned out to be wrong. Can anyone guide me in the right direction?

 

[andrevdh]

The exponential decay process is described by
$$n=n_oe^{-\lambda t}$$
where $n_o$ is the initial amount of nuclei present (the same for both types) and $\lambda$ is the decay constant of the particular isotope. Its relationship with the halflife $T_{\frac{1}{2}}$ is
$$\lambda T_{\frac{1}{2}} = \ln(2)$$

 

[HallsofIvy]

Hey,

I have been set this question as an assignment and I spent about an hour researching it yesterday and came up with what I thought was the right answer, but it has turned out to be wrong.

"A theory of astrophysics proposes that all the elements heavier than iron are formed in supernova explosions ending the lives of massive stars. If we assume that at the time of the explosion the amounts of 235U and 238U were equal, how long ago did the star(s) explode that released the elements that formed our Earth? The present 235U / 238U ratio is 0.00725. The half-lives of 235U and 238U are 0.704 x 10^9 years and 4.47 x 10^9 years."

Here is what I did.

Reciprocal of 0.00725 = 137.93
Used log to find that 2^7.1078 = 137.93
Multiplied 7.1078 by 0.704 x 10^9 to get 5,003,891,200 years.

But this answer turned out to be wrong. Can anyone guide me in the right direction?

Just telling us what arithmetic you did doesn't make it very clear WHY you did it. What equations did you have? What reason do you have for thing taking the reciprocal of 0.00725, etc. will give the correct answer?

We can take the "equal amounts" of U235 and U238 created to be 1. Since U235 has a half life of 0.2704 x 10⁹ year, the amount after T years will be
$$U235= \left(\frac{1}{2}\right)^{\frac{T}{0.2704x10^9}}$$
Since U238 has a half life of 4.47 x 10⁹, the amount left after T years will be
$$U238= \left(\frac{1}{2}\right)^{\frac{T}{4.47x10^9}}$$
The ratio of those is
$$\frac{U235}{U238}= \left(\frac{1}{2}\right)^{\frac{T}{0.2704x10^9}- \frac{T}{4.47x10^9}}= 0.00725$$

Solve that for T.

 

[mark96]

So could someone please tell me what the hell the answer is.

Thanks

 

[Checkfate]

Nevermind... got names mixed up. sigh. lol