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****Atomic Weights and Isotopic Abundance**

The atomic weight of an element is the relative atomic mass of that element.
It is actually a weighted mass of the elements isotopes (if any) and
their relative abundance.
You know that the sum of the percentages of the isotopes is equal to 1
(100%), so the relative abundance of the isotopes can be found using
simple algebra.

__Example #1__:
Silver (Atomic weight 107.868) has two naturally-occurring isotopes with
isotopic weights of 106.90509 and 108.90470. What is the percentage
abundance of the lighter isotope?

To avoid mistakes, use "x" as the multiplier for the isotope
percentage you wish to find. In this case, you want to find the
percentage of the lighter isotope, so the "x" is associated with
106.90509. Since the sum of the isotopic abundance percentages is equal to
1 (100%), the formula is:

108.90470 (1 - x) + 106.90509 (x) = 107.868

Multiplying, re-arranging and condensing the above formula results in:

108.90470 - 108.90470x + 106.90509x = 107.868
- 108.90470x + 106.90509x = - 108.90470 + 107.868

- 1.9996x = - 1.0367

x = 0.5185

Therefore, the answer is 51.85 %

__Example #2__:
An imaginary element (Atomic weight 93.7140) has three naturally-occurring
isotopes with isotopic weights of 92.9469, 93.2923 and 94.9030. The abundance
of the lightest isotope is 42.38 %. What is the percentage abundance of the
heaviest isotope?

In this case, we know the abundance of one of the isotopes. We know the
percentages of the lighter isotope (42.38 %) and the percentage of the
heavier isotope (x), so the percentage of the middle isotope is equal to
1 (100%) minus the other two percentages (1 - 0.4238 - x).

92.9469 (0.4238) + 93.2923 [(1-0.4238)-x] + 94.9030x = 93.7140
39.3909 + 53.7550 - 93.2923x + 94.9030x = 93.7140

93.14 + 1.6107x = 93.7140

1.6107x = 0.4217

x = 26.18

Therefore, the answer is 26.18%

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