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

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

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."##

##color(blue)("Mass of boys = 600 kg")## ##color(blue)("Mass of girls = 1100 kg")## ##color(blue)("Total mass of students = 1700 kg""##

##color(blue)("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 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.

##"Mass of the 800 atoms = 800 × 42.0 u = 33 600 u"## ##"Mass of the 200 atoms = 200 × 43.0 u = 8600 u"## ##"Mass of 1000 atoms = 42 200 u"##

##"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,

##"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.