# Why are atomic masses of most of the elements fractional?

Feb 13, 2014

The atomic masses of most elements are fractional because they exist as a mixture of isotopes of different masses.

#### Explanation:

Most elements occur as a mixture of isotopes of different masses. The fractional atomic masses arise because of this mixture.

Avg. mass = total mass of all atoms/number of atoms.

Before we calculate the average mass of atoms, let's use an analogy.

color(blue)("Assume that a class contains 10 boys (mass 60 kg) and 20 girls (mass 55 kg)."
color(blue)("What is the average mass of the students."

$\textcolor{b l u e}{\text{Mass of boys = 600 kg}}$
$\textcolor{b l u e}{\text{Mass of girls = 1100 kg}}$
color(blue)("Total mass of students = 1700 kg""

$\textcolor{b l u e}{\text{Avg. mass of 1 student"= "1700 kg"/"30 students" = "56.7 kg/student}}$

Now let's use this technique to calculate the average mass of a mixture of isotupes.

Example

If 80.0% of an element has an atomic mass of 42.0 u and 20.0% has an atomic mass of 43.0 u, what is the average mass of the mixture?

Solution

Let's assume we have 1000 atoms. Then 800 will have mass 42.0 u, and 200 will have mass 43.0 u.

$\text{Mass of the 800 atoms = 800 × 42.0 u = 33 600 u}$
$\text{Mass of the 200 atoms = 200 × 43.0 u = 8600 u}$
$\text{Mass of 1000 atoms = 42 200 u}$

$\text{Average mass of an atom" = "42 200 u"/"1000 atoms" ="42.2 u/atom}$

Alternate Solution

The average mass is the mass of each isotope multiplied by its percentage. Thus,

$\text{Average mass = 80.0 % × 42.0 u + 20.0 % × 43.0 u = 33.6 u + 8.6 u = 42.2 u}$

The second method is easier, but it doesn't explain intuitively why the answer is the average atomic mass.