# The atomic weight of a newly discovered element is 110.352 amu. It has two naturally occurring isotopes. One has a mass of 111.624 amu. The other has an isotopic mass of 109.75 amu. What is the percent abundance of the last isotope (109.75 amu)?

Nov 3, 2015

67.876%

#### Explanation:

The idea here is that each isotope will contribute to the average atomic mass of the element proportionally to their respective abundance.

Now, the key to this problem lies in how you can write the abundances of the two isotopes.

Let's assume that the decimal abundance, which is simply the percent abundance divided by $100$, of the isotope that has an atomic mass of $\text{109.75 u}$ is $x$.

Since you only have two isotopes, it follows that their decimal abundances must add up to give $1$. This means that the decimal abundance of the first isotope will be $\left(1 - x\right)$.

The average atomic mass of the element can be calculated using

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

In your case, you would have

$\text{110.352 u" = "111.624 u" xx (1-x) + "109.75 u} \times x$

This is equivalent to

$110.352 = 111.624 - 111.624 \cdot x + 109.75 \cdot x$

$1.874 \cdot x = 1.272 \implies x = \frac{1.272}{1.874} = 0.67876$

The percent abundances of the two isotopes will be

• "111.624 u " -> " 32.124%
• "109.75 u " -> " 67.876%