Isochron Plots

[Phasmid]

Hi,

 

I'm currently reading through Universal: A Guide to the Cosmos by Brian Cox and Jeff Forshaw and have a question related to one of the calculations detailed therein. The calculation specifically is to do with the use of isochron plots for dating rock samples using the decay of ⁸⁷Rb into ⁸⁷Sr. An example of this kind of isochron is attached.

 

Now the passage in the text says that if g = the fraction of ⁸⁷Rb that has decayed since the rock was formed, then the tangent of the tilt of the line is given by g/(1-g). The text then goes on to determine the value of g by determining the slope, which in this particular example is given by (0.7325-0.699)/0.5 = delta-y/delta-x = 0.067. Note that this does not correspond to the graph attached, the attachment is simply to show how the axes are labelled and what a typical isochron plot consists of.

 

Now that we have the slope of 0.067 using the deta-y and delta-x of the plot, the text goes on to state that this value of 0.067 implies a value for g of 0.063. Plugging 0.063 into the g/(1-g) provides a value of 0.067, so the numbers all work out.

 

The (0.7325-0.699)/0.5 = delta-y/delta-x calculation is used for determining the slope of the graph but so too is the g/(1-g) calculation as both give the same answer of 0.067. Now where I'm getting lost is the step used to determine the value for g of 0.063. Is there a way of manipulating g/(1-g) to give an answer in the form g = 0.063? If not then how is this value determined? It surely cannot be a case of trial and error.

 

I anticipate having missed something obvious and would be most appreciative of someone pointing it out to me. Of course if the information I've provided is too sparse I'm happy to provide any further details. If any of you own the book the specific section is at the bottom of page 30.

 

Kind regards,

 

Phasmid

[Sensei]

I don't have book. You also didn't attach anything to your post.

 

But Rubidium 87 has very large half-life 49.23 billion years.

 

If g is fraction of abundance,

g=0.063 (6.3%) of Rubidium 87 decayed to Strontium 87.

 

Sr-87 will have abundance in the sample rock 6.3%,

and Rb-87 will have abundance in the rock 100%-6.3% = 93.7%.

Measured by mass spectrometer, or using chemical reaction unique to these elements, to separate them, and being able measure their mass and abundance.

Rubidium compounds readily (explosively) react with water (RbOH) and are very good soluble in water (similar like Na,K,Cs),

while Strontium compounds are barely soluble in water (Sr(OH)₂) (like Mg,Ca).

 

 

The only decay mode of Rb-87 is by beta decay minus to Sr-87:

Rubidium-87 -> Strontium-87 + e- + Ve + 0.282271 MeV

 

You can make Spread sheet in Open Office, or Excel, with normal half-life equation, like I did here:

Rb 87 Decay.zip

 

And you will see data with 93.7% of Rb-87 and 6.3% of Sr-87 is between 4.5 bln y to 4.8 bln y.

Load file in OpenOffice and decrease step in A3 field from 0.15 to lower value, and Edit > Fill > Down, for better precision.

 

ps. Why it's in Mathematics, not Physics section.. ?

 

Edited April 17, 2017 by Sensei