Atomic structure

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How can you calculate the relative abundance of two isotopes when given the atomic masses of both isotopes and also the average atomic mass for the element?

Igloo says
 
The conventional method is to use algebra.
 
Let’s take an example. Copper consists mainly of two isotopes, ⁶³Cu and ⁶⁵Cu, and its (average) atomic mass is 63.55 (to 2 d.p.)
 
Let’s assume next that the percentage abundance of ⁶³Cu is x
This means that the percentage abundance of ⁶⁵Cu will be 100-x
Given 100 random copper atoms, x will each have a mass of 63 [total mass = 63x]
And 100-x will have a mass of 65 [total mass = (100-x) x 65 = 6500-65x]
So the total mass of 100 atoms = 63x + 6500–65x = 6500–2x
This means that the average mass = (6500–2x) / 100
 
But we are told in the question that average mass = 63.55
Therefore (6500-2x) / 100 = 63.55
So 6500–2x = 6355
Hence 2x = 6500-6355 = 145
And x = 72.5
 
So, in a typical sample of copper 72.5% of the atoms are ⁶³Cu and 27.5% are ⁶⁵Cu.
 
Another (shorter) method is to use a number scale and intuition, but perhaps you are safer with algebra!

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updated: 09 November 2004