It would indeed be problematic if scientists had to assume the original amounts of trace elements in rocks in order to get at its age.
But they do not. Let's look at how this is possible.
In my early post (
LINK) I described the primary equation of radioactive decay.
If N is the parent atom and D is the daughter atom then:-
N (t) = N(0)*exp(-Kt) where K is the decay constant and N(0) is the original number of parent atoms at the beginning of calculation.
If the number of original daughter atoms (say lead) at time t=0 was D(0)
then the total daughter atoms formed at time t by this decay process is
D(t) = D(0) + N(t)*[exp(Kt) - 1].
Also Half Life T = 0.693/K
Consider the case where Rubidium 87 (Rb_87) decays into Strontium 87 (Sr_87) and a beta particle with a half life of 48.9 billion years. Then:-
Sr_87 (today) = Sr_87 (initial) + Rb_87 (today)* [exp(Kt)-1]
Strontium has a stable naturally occuring isotope Strontium_86 and its usually convenient to measure ratios than actual amounts in a rock. So defining:-
Y(t) = Sr_87/Sr_86 ratio today
Y(0) = Sr_87/Sr_86 ratio originally
X(t) = Rb_87/Sr_86 ratio today
we get
Y(t) = Y(0) + X(t)*[exp(Kt) - 1]
Now the only thing we do not know is Y(0) as both Y(t) and X(t) are concentration ratios today and can be directly measured.
Defining original time
The question naturally occurs is what exactly does time zero signify.
For geologists, its the time when the rock (like granite or basalt) first solidified from liquid volcanic lava. Consider a large slab of basalt rock.
Let time t=0 be the time when the rock solidified out of molten magma. We are trying to determine how much time has elapsed since the rock solidified.
Now such rocks are mixtures of various types of crystals and minerals, all of which will have different initial composition of Strontium to Rubidium. Let is sample
100 such various crystals from the same slab of rock and measure its current ratios. So:-
Y1(t) = the measured strontium strontium ratio of sample 1
Y2(t) = the measured strontium strontium ratio of sample 2
etc.
Similarly we X1(t), X2(t),.....
Since the slab had solidified from liquid magma at the same time the expression
m = [exp(Kt)-1] is a constant for that given slab since decay rate is constant and the time since solidification is the same.
Y(0) is a constant for all rock crystals as crystals do not distinguish between isotopes of the same element.
So the equation for a given slab of rock that has solidified at the same time is
Y1(t) = Y(0) + X1(t)*m
Y2(t) = Y(0) + X2(t)*m
......
For all hundred samples.
These are equations for points in a single straight line with slope m and intercept Y(0).
So if we plot Y vs X, then the experimentally determined values of Y1, Y2,... and X1, X2,.... will follow a single straight line in that plot. This diagram is called the isochron diagram, and the line, the isochron.
Then
we can evaluate from the fitted line the slope "m" of the line.
Since m = [exp(Kt)-1]
Hence age of the rock is
t = [1+log(m)]/K where K is the decay constant.
Thus one can determine the age of a rock without needing to know anything about the initial composition.
Edit:- For those people who like to think in pictures,
the graphical derivation of the above relations can be seen in the link below, as well as refutation of common creationist objections to...mathematics?!
Radioactive Dating