# The formula for finding the percentage of isotopes in an element's atomic weight?

In my class my teacher showed us how to find it, with a method, but he didn't really state the formula. I conjured up a formula from what I observed and it works. Here it is:

$$\frac{i_1x + i_2y}{100} = A$$

$A$ = atomic mass
$i_1$ = First's isotope's atomic weight
$i_2$ = Second isotope's atomic weight
$x \text{ and } y$ = Percentages of isotopes, respectively
$x+y = 100$

Therefore: $$\frac{i_1x + i_2(100-x)}{100} = A$$

I'm just wondering, is there an official formula, or is this method it? Thank you in advance.

Ps. If I didn't word the title correctly, feel free to edit it to something that would make much more sense! thank you!

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Have you studied weighted averages before? It's exactly that type of mathematics. – Nicolau Saker Neto Mar 7 '14 at 22:08
In my class we just did three questions. My teacher would give us an element such as Rubidium, the isotope's atomic mass, and the percentage of one isotope, then we would have to find the atomic mass. Or he would give us enough information to find another variable in my equation. (When I see my equation I'm not saying I made it, it's probably out there somewhere, I just don't know where) – Samir Chahine Mar 7 '14 at 22:12

The average relative atomic mass of an element comprised of $n$ isotopes with relative atomic masses $A_i$and relative fractional abundances $p_i$ is given by:

$$A = p_1A_1 + p_2A_2 + ... + p_nA_n = \sum\limits_{i=1}^npiAi$$

Isotope Isotopic Mass × % Abundance

12C $12.000000 * 0.98892 = 11.867$

13C $13.003354 * 0.01108 = 00.144$

Since we have $p_1A_1$ and $p_2A_2$, we add those together to find A.

Chemical relative atomic mass of carbon: $$A = 00.144 + 11.867 = 12.0011$$

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I'm sorry to be a pain, but can you please use an example with $$Br - 79 \rightarrow{} 50.69%$$ $$Br - 81 \rightarrow{} 49.31$$ To find the atomic mass? – Samir Chahine Mar 7 '14 at 22:18
@SamirChahine I provided an example with carbon 12 and carbon 13. – anosdsd Mar 7 '14 at 22:25
Thank you so much, this helps a lot, I aprreciate it! I didn't quite understand how the whole concept worked but I now have a much more intricate understanding, thank you. – Samir Chahine Mar 7 '14 at 22:30
@SamirChahine My pleasure! – anosdsd Mar 7 '14 at 22:31

Once you know how it works, you never want to do that by hand again ;) For bromine:

isotopes = [79, 81]
abundances = [50.69, 49.31]

weighted = map(lambda x, y: x * y, isotopes, abundances)
# multiply isotope mass and abundance

weighted = sum(weighted)/100
# sum them up and divide by 100

print weighted
>>>> 79.9862

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Huh, I should of done that ahah! Python is really handy sometimes, thank you for the idea! – Samir Chahine Mar 8 '14 at 1:12