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I am very interested in dating rocks and am about to read the book of Gunter Faure, Principles of Isotope Geology. I came across the linear equation below, which I understand well.

(87Sr/86Sr)t = (87Sr/86Sr)i + (87Rb/86Sr)t * lambda * t

where the index t after the parenthesis refers to time dependency and i to the initial value, lambda is the decay constant.

However, I don't understand the interpretation that is given in the book, that is, the claim that (87Rb/86Sr)t * lambda is the slope. How can this be since a slope in linear equation must be independent of the variable t, which is not the case. In fact rubidium 87 decays to stable strontium 87 so (87Rb/86Sr)t is decreasing over time since strontium 86 is stable as well. Can someone explain this oddity?

Interesting question. I'm just thinking out liud here.
Isn't (87Rb/86Sr)t the ratio at a certain time 't'. I interpret it as the t not being a variable but rather a indication of the time dependancy of the ratio.

The (87Rb/86Sr)t is thus a given if you measure it, but you don't know t.

The 'constant' in the formula is then multiplied with Lambda to get the slope of the function as in y=ax+b a is the slope.

Yes, (87Rb/86Sr)t is the ratio at a certain time t. If for instance a rock is very old, you can take an initial value (87Sr/86Sr)i as it is found in meteorites. The other ratios you measure and then you can solve the equation for t and determine how old a rock is. But in Faure's book there is an illustration (see below).



The straight lines there have presumably constant slopes (87Rb/86Sr)t * lambda. But these ratios are certainly not constant over a time of 800 million years like in the illustration, unless I am missing something. But what?

The solution is that the ratio 87Rb/86Sr can be interpreted both as time dependent and as the initial ratio when the rock was formed. Because the number of daughter atoms can both be:

D = D0 + N(t) (Exp[lambda * t] - 1)

or:

D = D0 + N0 (1 - Exp[-lambda * t])

where N(t) is the number of parent atoms at time t, D0 the initial number of daughter atoms and correspondingly N0 the initial number of parent atoms.

So approximating Exp[±lambda * t] = 1 ± lambda * t for small lambda * t, we get what we are looking for. Unfortunately, only the first equation is explicitly discussed in the book...

Thanks for sharing. I wouldn‘t have come up with that...