Fractional Atomic Masses

Atomic masses of most elements are not whole number, i.e., these are fractional.

It is because most elements in nature occur as a mixture of two or more isotopes in certain fixed proportions.

This is illustrated below.

Calculating the atomic mass of an element from the mass numbers of its isotopes

The atomic mass of an element is the weighted arithmetic mean of the atomic masses of its isotopes present in the sample of the element.

Let us consider a sample of an element X containing its two isotopes X1 and X2. Then

This method is illustrated by taking the case of chlorine.

The two isotopes of chlorine, ³⁵₁₇Cl and ³⁷₁₇Cl occur in the ratio 3 : 1. Then,

Atomic mass of chlorine = (35 u × 3) + (37 u × 1) / 3 + 1

 = (105 u + 37u) / 4 = 142 u / 4      =      35.5 u

Mass numbers are whole numbers,

whereas atomic masses are generally fractional.

EXAMPLE 1  

If bromine occurs in the form of say two isotope ⁷⁹₃₅Br (49.7%) and ⁸¹₃₅Br (50.3%), then calculate the atomic mass of bromine atom.

Solution:

Atomic mass of bromine = (79 u × 49.7) + (81 u × 50.3) / (49.7 + 50.3) = 80.0 u

So, the atomic mass of bromine is 80.0 u.

EXAMPLE 2

The atomic mass of a sample of an element X is 16.2 u. What are the percentages of isotope ¹⁶₈X and ¹⁸₈X in the sample?

Solution:

Atomic mass of X = 16.2 u

Let,

Percentage of the isotope ¹⁶₈X be a

Then,

Percentage of the isotope ¹⁸₈X = (100 – a)

Atomic mass of the sample = (% ¹⁶₈X × 16 u) + ((100 – %¹⁶₈X) × 18 u) / 100

      16.2 u = (a × 16 u) + (100 – a) × 18 u / 100

                             1620 u = 16 a u + 1800 u – 18 a u

This gives,

        (18 a u – 16 a u) = 1800 u – 1620 u

                                2 a u = 180 u

                                     a = 180 / 2 = 90

 Thus,      Percentage of ¹⁶₈X = 90%

 and        Percentage of ¹⁸₈X = (100 – 90)% = 10%