How Do Geologists Determine the Age of Rock Samples Using Isotopic Ratios?

[molly16]

Homework Statement

Geologists are able to calculate ages of rock samples through measurements of certain isotopic rations combined with known radioactive decay rates (all such decays being considered to adhere to 1st order kinetic rate laws). Two such processes leading to stable isotopes are:

87Rb --> 87Sr
238Ur --> 206Pb

Half life of 87Rb is 4.90x10^10 years while half life of 238Ur is 4.51x10^9 years.The isotopic ratios are:

87Sr/87Rb = 0.051
206Pb/238Ur = 0.71

Assuming the time the rock was formed it contained no 87Sr or 206Pb Calculate the age of the rock indicated by the isotopic ratios.

Homework Equations

dA/dt = -k[A]

ln(A/Ao) = -kt

t1/2 = ln(2)/k

The Attempt at a Solution

I tried solving for the rate constant using t1/2 = ln(2)/k
and rearranging it to k = ln(2)/t1/2 = 1.41 x 10^-11

Then to solve for the time for the decay of 87Rb I plugged the value into the equation ln(A/Ao) = -kt

ln (A/Ao) = ln (87Sr/87Rb) = ln (0.051) = -(1.41x10^-11)t

then I solved for t which is t = 2.1x10^11 years for the decay of the 87Rb
but the answer is supposed to be t = 3.517x10^9 years

Can someone please explain this question to me and tell me what I'm doing wrong?
Thanks!

 

[Simon Bridge]

I tried solving for the rate constant using t1/2 = ln(2)/k

... converting half-lives to mean lives, good.

Can someone please explain this question to me and tell me what I'm doing wrong?

If you have isotope X' and Y' and they decay to X and Y respecively, but at different rates, then, from the decay equations of X' and Y' you can get equations for how the amounts of X and Y change over time ... and thus how the ratio of X:Y changes with time.

You are given the ratio X:Y, and initial quantities of X and Y, and you have to find the time that has elapsed to turn the initial quantities into the final ratio.

I suspect you have been concentrating on the wrong isotope - on X':Y' instead of X:Y ... start by deriving the equation for the quantity of each decay product as a function of time.

 

[molly16]

... converting half-lives to mean lives, good.

I suspect you have been concentrating on the wrong isotope - on X':Y' instead of X:Y ... start by deriving the equation for the quantity of each decay product as a function of time.

so then
Ao = A (1-e^-kt)
Ao/A = 1-e^-kt
ln (Ao/A) = +kt

but I still get the same answer??

 

[Simon Bridge]

Please show all your working with your reasoning at each stage.

 

[molly16]

Please show all your working with your reasoning at each stage.

Ahh never mind I figured it out. Thanks.

 

[Simon Bridge]

Well done :)

For others googling here later:
I made a slight mistake earlier - you are given X:X' rather than X:Y.
I think your repeated mistake was that you were putting the ration as A/Ao or Ao/A... which was similar to my misreading.