# 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?

##### 1 Answer

#### 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

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

If you take

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

#62.9296 * x - 64.9278 * x = 63.546 - 64.9278#

#1.9982 * x = 1.3818 implies x = 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"#