# How would you calculate the percent relative abundance of Cu-63 with the mass 62.9296 g and Cu-65 with the mass 64.9278 g, when the average mass of Cu is 63.546?

Dec 13, 2015

69.152% -> ""^63"Cu"
30.848% -> ""^65"Cu"

#### Explanation:

As you know, the average atomic mass of an element is determined by taking the weighted average of the atomic masses of its naturally occurring isotopes.

Simply put, an element's naturally occurring isotopes will contribute to the average atomic mass of the element proportionally to their abundance.

color(blue)("avg. atomic mass" = sum_i ("isotope"_i xx "abundance"_x))

When it comes to the actual calculation, it's easier to use decimal abundances, which are simply percent abundances divided by $100$.

So, you know that copper has two naturally occurring isotopes, copper-63 and copper-65. This means that their respective decimal abundance must add up to give $1$.

If you take $x$ to be the decimal abundance of copper-63, you can say that the decimal abundance of copper-65 will be equal to $1 - x$.

Therefore, you can say that

overbrace(x * 62.9296 color(red)(cancel(color(black)("u"))))^(color(blue)("copper-63")) + overbrace((1-x) * 64.9278 color(red)(cancel(color(black)("u"))))^(color(red)("copper-65")) = 63.546 color(red)(cancel(color(black)("u")))

Solve this equation for $x$ to get

$62.9296 \cdot x - 64.9278 \cdot x = 63.546 - 64.9278$

$1.9982 \cdot x = 1.3818 \implies x = \frac{1.3818}{1.9982} = 0.69152$

This means that the percent abundances of the two isotopes will be

• 69.152% -> ""^63"Cu"
• 30.848% -> ""^65"Cu"