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%

Atomic Mass and Relative Atomic Mass

Atom is the smallest particle which takes part in a chemical reaction.

A molecule is the smallest unit of any substance which can exist independently, and shows all the properties of that substance.

Each molecule is a group of chemically bonded atoms.

Each atom or a molecule has its own characteristic mass.

So, the atomic and molecular masses may be expressed in any unit of mass, i.e., in gram, kilograms, etc.

But the actual mass of a single atom is very small.

An atom of hydrogen (the lightest atom) is
0.000 000 000 000 000 000 000 000 001 673 kg (1.673 × 10^–27 kg).

It is not possible to measure such small masses even with the help of the most sensitive balance. Such small numbers are not very convenient to write also.

IUPAC accepted in 1961 the atomic mass unit scale to be used for expressing atomic and molecular masses. In this scale of atomic masses, the ¹²C isotope has been assigned an atomic mass of 12.000000 atomic mass units.

The atomic mass unit is abbreviated as amu, and denoted
by the symbol u, or m_u.

Atomic mass unit

Atomic masses are very small. To express atomic masses, a unit called atomic mass unit (abbreviation, amu; symbol, u) is commonly used. The atomic mass unit is defined as follows:

The mass equal to 1/12th of the mass of a ¹²C atom
is called one atomic mass unit.

1 atomic mass unit = Mass of a ¹²C atom / 12

Absolute mass of a ¹²C atom is 1.9924 × 10^–23 g.

Therefore,

1 atomic mass unit = 1.9924 × 10^–23 g / 12 = 1.66 × 10^–24 g = 1.66 × 10^–27 kg

Atomic mass (A)

The average mass of an atom of an element is known as its atomic mass.

Atomic mass of an element is designated as A.

Since, atomic mass is actual mass, hence it has the unit of mass, viz., gram, kilogram, atomic mass unit, etc.

The internationally accepted unit for describing atomic mass is atomic mass unit.

The average mass of an atom of an element in atomic mass unit is called its atomic mass.

Relative atomic mass (A_r)

The relative atomic mass (A_r) of an element is defined as the average mass of an atom of the element compared with an atom of ¹²C taken as 12 u. Thus,

Relative atomic mass of an element (A_r) = Average mass of 1 atom of the

element  /  [(Mass of one ¹²C atom) / 12]

The relative atomic mass is denoted by _Ar.

The relative atomic mass is a pure number, and hence it has no unit.

The relative atomic mass of an element indicates the number of times one atom of that element is heavier than 1/12th of a ¹²C atom.

For example,

the average relative mass of chlorine is 35.45.

This means that an atom of chlorine on average is (35.45 / 12) times heavier than one atom of ¹²C.

The names, symbols and relative atomic masses of some common elements are presented in Table (base ¹²C = 12.000 u)

Element		Relative atomic mass (Ar)
Name	Symbol	Exact	Common value*
Hydrogen	H	1.008	1.0
Carbon	C	12.010	12.0
Oxygen	O	15.999	16.0
Nitrogen	N	14.007	14.0
Chlorine	Cl	35.45	35.5
Sodium	Na	22.99	23.0
Silver	Ag	107.87	108.0
Copper	Cu	63.54	63.5

Note: Magnitude of the relative atomic mass and the atomic mass of an element are the same, but the two differ in units.

• The relative atomic mass has no unit.

• The atomic mass is expressed in the atomic mass units.

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