Determination of Relative Molecular Masses

Determination of Relative Molecular Masses: Relative mass is an essential concept in Chemistry used to simplify the calculation of the mass of an atom or molecule. The mass of a proton is about \(1.69 \times {10^{ – 27}}\) kilograms (kg). That of a neutron is very slightly greater, about \(1.69 \times {10^{ – 27}}\,{\rm{kg}}\), and that of an electron is \(9.11 \times {10^{ – 31}}\,{\rm{kg}}\). To deal with such complex calculations, scientists have defined the relative atomic mass of a carbon atom as \(12\). Thus, Carbon\(- 12\) is the standard while measuring the atomic masses.

What is the Relative Mass?

Relative Mass is defined as the mass of an atom or molecule relative to \(1/12{\text{th}}\) the mass of a carbon \(- 12\) atom.

Carbon\(- 12\) was chosen as a basis because-
(a) It was already being used as a standard reference in mass spectrometers.
(b) Many elements can combine with carbon \(-12\).
(c) It exists in solid form at room temperature and thus can be handled easily.
(d) It is the most abundant carbon isotope, occurring about \(98.89{\text{ }}\% \). Hence, the mass of exactly \(12\) units assigned to one carbon\(-12\) atom is an accurate value.

Periodic Table and Isotopes

The relative atomic masses of different elements on the periodic table is the average of the masses contributed from different isotopes.

For example, chlorine has two isotopes with masses \(35\) and \(37\). About three-quarters of the chlorine is found in the form of chlorine\(-35\), and the remaining quarter is chlorine\(-37\).

The formula for calculating the relative masses of isotopes is:

Relative Atomic Mass
\(= \frac{{\left( {{\text{Isotope}}\,1\,{\text{Mass}}\, \times \,{\text{Isotope}}\,1\,{\text{abundance}}\, + \,{\text{Isotope}}\,2\,{\text{Mass}}\, \times \,{\text{Isotope 2}}\,{\text{abundance + }}….} \right)}}{{100}}\)

So for chlorine, this is:
Relative Atomic Mass \(= \frac{{\left( {35 \times 75 + 37 \times 25} \right)}}{{100}}\)

\(= \frac{{2,625 + 925}}{{100}} = 35.5\)

Hence, for chlorine, the relative atomic mass is \(35.5\).

Relative Molecular Mass

Relative Molecular Mass or RMM of a compound is defined as the mass of a formula unit of the compound relative to the mass of a carbon\(-12\) atom.

Practically, RMM is the sum of the relative atomic masses or atomic weights of the atomic species as given in the chemical formula. It is a dimensionless quantity, hence has no units.

Illustration

The diagram alongside shows a box containing three balls. Out of which-

One red ball is shown as:

Two black balls are shown as

If the mass of a red ball is \(16{\text{ g}}\) and the mass of a black ball is \(1\,{\text{g}}\), then the mass of all three balls in the box is \(16{\text{ }} + {\text{ }}1{\text{ }} + {\text{ }}1{\text{ }} = {\text{ }}18{\text{ g}}\)

Hence, the total mass of the balls in the box is given by the expression-

\({\text{mass}}\,\left( {{\text{total}}} \right)\, = \,1 \times \,{\text{mass}}\,\left( {{\text{red}}\,{\text{ball}}} \right) + \,2\, \times \,{\text{mass}}\,\left( {{\text{black}}\,{\text{ball}}} \right)\)

\({\text{mass}}\,\left( {{\text{total}}} \right) = 16 + 2 \times 1=16+ 2=18\,{\text{g}}\)

If the black ball represents the hydrogen atom and the red ball represents the oxygen atom, then the box will contain a molecule of water. This can be diagrammatically represented as-

If the mass of an oxygen atom is \(16\) and that of the hydrogen atom is \(1\), then the mass of all three atoms comprising the water molecule is \(16{\text{ }} + {\text{ }}1{\text{ }} + {\text{ }}1{\text{ }} = {\text{ }}18\)

Hence, the relative molecular mass of a compound can be calculated by multiplying the relative atomic mass of each atom by the number of those atoms present in the molecule and then adding the results together. This looks like this:

Relative molecular mass \(= \,\left( {{\text{number}}\,{\text{of}}\,{\text{atoms}}\,{\text{of}}\,{\text{element}}\,1\, \times \,{\text{relative}}\,{\text{mass}}\,{\text{of}}\,{\text{element}}\,1} \right)\)
\(+ \left( {{\text{number}}\,{\text{of}}\,{\text{atoms}}\,{\text{of}}\,{\text{element}}\,2\, \times \,{\text{relative}}\,{\text{mass}}\,{\text{of}}\,{\text{element}}\,2} \right)\)
\(+ ………\)

Steps to follow while determining the relative molecular masses:

1. The number of atoms of each element in the compound is determined by using the compound’s chemical formula. For example- compound \({\text{X}}_{\text{a}}{\text{Y}}_{\text{b}}\) contains:

a. atoms of element \({\text{X}}\)
b. atoms of element \({\text{Y}}\)

2. A mathematical expression is framed to calculate the relative molecular mass of the compound, i.e. the total mass of all the elements present in the compound. Let \({\text{M}}_{\text{r}}\left( {\text{X}}_{\text{a}}{\text{Y}}_{\text{b}} \right)\) represent the relative molecular mass of the compound \({\text{X}}_{\text{a}}{\text{Y}}_{\text{b}}\)

\({{\text{M}}_{\text{r}}}\left( {\text{X}} \right)\)= relative atomic mass of element \({\text{X}}\)
\({{\text{M}}_{\text{r}}}\left( {\text{Y}} \right)\)= relative atomic mass of element \({\text{Y}}\)
\({\text{a}}\; = \)number of atoms of element \({\text{X}}\)
\({\text{b}}\; = \)number of atoms of element \({\text{Y}}\)

\({\text{M}}_{\text{r}}\left( {\text{X}}_{\text{a}} {\text{Y}}_{\text{b}} \right)\, = \,{\text{a}}\, \times \,{\text{M}}_{\text{r}}\left( {\text{X}} \right)\, + \,{\text{b}}\, \times \,{\text{M}}_{\text{r}}\,\left( {\text{Y}} \right)\)

3. Each element’s relative atomic mass or atomic weight is determined from the Periodic Table.

4. The values obtained in step \(3\) is substituted in the equation obtained in step \(2\) and is solved to find the relative molecular mass of the compound.

Calculating the Relative Molecular Mass of-

A Diatomic Molecule

Carbon monoxide is a diatomic molecule. It is made up of two atoms: an atom of carbon \(\left( {\text{C}} \right)\)and an atom of oxygen \(\left( {\text{O}} \right)\). Hence, it has the molecular formula \({\text{CO}}\).

Relative molecular mass \(=\) Sum of the relative atomic mass of each element in the compound

A molecule of carbon monoxide \(\left( {{\text{CO}}} \right)\) contains \(1\) carbon atom \(\left( {\text{C}} \right)\) and \(1\) oxygen atom \(\left( {\text{O}} \right)\).Hence,

Relative molecular mass or RMM of CO \(= 1\, \times \,{\text{relative}}\,{\text{atomic}}\,{\text{mass}}\left( {\text{C}} \right)\, + \,1\, \times \,{\text{relative}}\,{\text{atomic}}\,{\text{mass}}\left( {\text{O}} \right)\)

The relative atomic mass or atomic weight for each element is determined from the Periodic Table:

Relative atomic mass of \({\text{C }}\left( {{\text{carbon}}} \right){\text{ }} = \)\(12.01\)

The relative atomic mass of \({\text{O }}\left( {{\text{oxygen}}} \right){\text{ }} = \)\(16.00\)

Substituting the relative atomic masses of carbon and oxygen in the Relative molecular mass formula we get-

Relative molecular mass or RMM of CO \(= \left( {1\, \times \,12.01} \right)\, + \,\left( {1\, \times \,16.00} \right)\, = \,28.01\)

A Triatomic Molecule

Carbon dioxide is a triatomic molecule. It is made up of \(3\) atoms: an atom of carbon \(\left( {\text{C}} \right)\) and two atoms of oxygen \(\left( {\text{O}} \right)\). Hence, it has a molecular formula \({\text{C}}{{\text{O}}_2}\).

Relative molecular mass \(=\) sum of the relative atomic masses of each element in the compound

A molecule of carbon dioxide \(\left( {{\text{C}}{{\text{O}}_{\text{2}}}} \right)\) is composed of \(1\) atom of carbon \(\left( {\text{C}} \right)\) and \(2\) atoms of oxygen \(\left( {\text{O}} \right)\). Hence,

Relative molecular mass or RMM of \({\text{C}}{{\text{O}}_2} \ = 1\, \times {\text{relative}}\,{\text{atomic}}\,{\text{mass}}\,\left( {\text{C}} \right)\, + \,2\, \times \,{\text{relative}}\,{\text{atomic}}\,{\text{mass}}\,\left( {\text{O}} \right) \)

The relative atomic mass or atomic weight for each element is determined from the Periodic Table:

Relative atomic mass of \({\text{C}}\,\,\left( {{\text{carbon}}} \right){\text{ }} = \)\(12.01\)

The relative atomic mass of \({\text{O }}\left( {{\text{oxygen}}} \right){\text{ }} = \)\(16.00\)

Substituting the relative atomic masses of carbon and oxygen in the Relative molecular mass formula we get-

Relative molecular mass or \({\text{C}}{{\text{O}}_2} = \left( {1 \times 12.01} \right)\, + \,\left( {2 \times 16.00} \right)\, = \,44.01\)

A Polyatomic Compound

Calcium hydroxide is a polyatomic compound, with a chemical formula \({\text{Ca}}{\left( {{\text{OH}}} \right)_2}\). For every \(1\) calcium ion, there are \(2 \) hydroxide ions. Each of the hydroxide ions, \({\text{O}}{{\text{H}}^ – }\) is made up of \(1\) atom of oxygen \(\left( {\text{O}} \right)\) and \(1\) atom of hydrogen \(\left( {\text{H}} \right)\).

Relative molecular mass of calcium hydroxide \(= 1 \times \,{\text{relative}}\,{\text{atomic}}\,{\text{mass}}\,\left( {{\text{Ca}}} \right) + \,2\, \times \,{\text{relative}}\,{\text{atomic}}\,{\text{mass}}\,\left( {\text{O}} \right) \)

The relative atomic mass or atomic weight for each element is determined from the Periodic Table:

Relative atomic mass calcium \(= {\text{ }}40.08\)

Relative atomic mass hydrogen \(= {\text{ }}1.008\)

The relative atomic mass of oxygen \( = {\text{ }}16.00\)

Substituting the relative atomic masses of calcium, hydrogen and oxygen in the Relative molecular mass formula we get-

Relative molecular mass of calcium Hydroxide \( = 1\, \times \,{\text{relative}}\;{\text{atomic}}\;{\text{mass}}\,\left( {{\text{Ca}}} \right) + \,2\, \times \,{\text{relative}}\;{\text{atomic}}\;{\text{mass}}\left( {\text{O}} \right) +\)
\(2\, \times {\text{relative}}\;{\text{atomic}}\;{\text{mass}}\,\left( {\text{H}} \right)\)
\(= 1\, \times \,40.08)\, + \,\left( {2\, \times \,16.00} \right)\, + \,\left( {2\, \times \,1.008} \right) = \,40.08\, + \,32.00\, + \,2.016\, = \,74.10\)

Similarly in \({{\text{H}}_2}{\text{S}}{{\text{O}}_4}\) ,there are \(2\) hydrogen \(\left( {\text{H}} \right)\)atoms, \(1\) sulphur atom, and \(3\) oxygen atoms.

The relative atomic mass or atomic weight for each element is determined from the Periodic Table:

Relative atomic mass of Sulphur \(= 32\)

Relative atomic mass of hydrogen \(= 1.008\)

Relative atomic mass of oxygen \(= 16.00\)

The relative molecular mass of \({\rm{H}_2}{\text{S}}{{\text{O}}_4}\)
\( = \,\left( {{\text{number}}\;{\text{of}}\;{\text{atoms}}\;{\text{of}}\;{\text{H}}\, \times \,{\text{relative}}\;{\text{mass}}\;{\text{of}}\;{\text{H}}} \right)\, +\)
\(\left( {\text{number}}\;{\text{of}}\,{\text{atoms}}\,{\text{of}}\;{\text{S}}\, \times \,{\text{relative}}\;{\text{mass}}\;{\text{of}}\;{\text{S}} \right)\, +\)
\(({\text{number}}\;{\text{of}}\;{\text{atoms}}\;{\text{of}}\;{\text{O}}\, \times \,{\text{relative}}\;{\text{mass}}\;{\text{of}}\;{\text{O}}\)
\(= \,\left( {2\, \times \,1.008} \right)\, + \,\left( {1\, \times \,32} \right)\, + \,\left( {4\, \times \,16} \right) = 2.016 + 32 + 64 = 98.016\).

Solved Examples

Q.1: A chemist finished a detailed analysis of an unknown compound and found that it is made up of two phosphorus atoms and some oxygen atoms. It has a relative molecular mass of \(141.94\). Determine the molecular formula for this molecule.

Solution:

Step 1: Using the Periodic Table, write the symbols for all the elements present in the molecule.

From the Periodic Table, the symbol for phosphorus is \({\text{P}}\), and oxygen is \({\text{O}}\).

Step 2: Based on the information given, write the partial chemical formula of the compound

Let the number of oxygen atoms be \({\text{b}}\).

The partial chemical formula of the compound is \({\text{phosphoru}}{{\text{s}}_2}\) \({\text{oxyge}}{{\text{n}}_{\text{b}}}\)

Substituting the chemical symbols we get- \({\text{P}}_2 {\text{O}}_{\text{b}}\)

Step 3: Calculating the relative atomic mass of the phosphorus atoms present in the molecule

From the Periodic Table, the relative atomic mass of phosphorus is \(30.97\)

The total mass of phosphorus \(=\) number of phosphorus atoms \(\times\) relative atomic mass of phosphorus \(= \;2\; \times \;30.97\; = \;61.94\)

Step 4: Calculating the relative atomic mass of oxygen atoms present in the molecule

The relative molecular mass of the molecule \(=\) total mass of phosphorus atoms
\( + \) total mass of oxygen atoms

\(141.94{\text{ }} = \;61.94{\text{ }} + {\text{ }}\left[ {{\text{b}} \times {\text{ relative atomic mass of oxygen}}} \right]\)

\({\text{b}} \times \)relative atomic mass of oxygen \(= \;141.94\; – \;61.94\; = \;80.00\).

Step 5: Calculating the number of oxygen atoms present

From the Periodic Table, the relative atomic mass of oxygen is \(16.00\)

From step 4, we have-

b \(\times \) relative atomic mass of oxygen\(= {\text{ }}80.00\)

Substituting the relative atomic mass of oxygen in the above equation, we get-

\({\text{b}}\; \times {\text{ }}16.00{\text{ }} = {\text{ }}80.00\)

\({\text{b}} = 80.00\; \div \;16.00\; = 5\)

Hence, the number of oxygen atoms in the unknown compound is \(5\).

Therefore, the chemical formula for the molecule \({\text{P}}_2{\text{O}}_{\text{b}}\) is \({{\text{P}}_2}{{\text{O}}_5}\).

Is the solution reasonable?

By backward calculation, the relative molecular mass of \({{\text{P}}_2}{{\text{O}}_5}\) is \(2 \times 30.97 + 5 \times 16.00 = 61.94 + 80.00 = 141.94\). The value is the same as the one given in the question; hence the answer is correct.

Summary

The concept of relative molecular mass can only be used for substances made up of molecules. However, for ionic compounds, instead of relative molecular mass, the term ‘relative formula mass’ or is similar to the relative molecular mass. It is calculated by adding up the relative atomic masses of all the atoms shown in the chemical formula of the ionic compound. This article explains the concept of Relative Molecular Mass and its determination. It also describes that no matter how complicated the formula is, we can quickly determine the Relative Molecular Mass of the compound by following some simple steps.

FAQs

Q.1. What is a relative molecular mass?
Ans. Relative molecular mass or RMM of a compound is defined as the mass of a formula unit of the compound relative to the mass of a carbon\(- 12\) atom. It is the sum of the relative atomic masses or atomic weights of the atomic species as given in the chemical formula. It is a dimensionless quantity, hence has no units.

Q.2. How is relative molecular mass determined?
Ans. The relative molecular mass of a compound can be calculated by multiplying the relative atomic mass of each atom by the number of those atoms present in the molecule and then adding the results together. This looks like this:
Relative Molecular Mass \(=\left( {{\text{number}}\,{\text{of}}\,{\text{atoms}} \,{\text{of}}\, {\text{element}}\,1 \, \times \,{\text{relative}} \,{\text{mass}} \,{\text{of}} \,{\text{element}} 1} \right)\)
\({\text{ + }}\left( {{\text{number}} \,{\text{of}} \,{\text{atoms}}\, {\text{of}}\, {\text{element}}\, 2 \, \times \, {\text{relative}} \,{\text{mass}}\, {\text{of}}\, {\text{element}}\, 2} \right) + ……….\)

Q.3. Are molecular weight and molar mass the same?
Ans. Molar mass gives the mass of a mole of a particular substance. At the same time, the molecular weight gives the mass of a molecule of a particular substance.

Q.4. Which one is the best method for molecular mass determination of polymers?
Ans. GPC or gel-permeation chromatography is the most effective technique to determine the molecular weight of polymers.

Q.5. Is relative molecular mass and relative formula mass the same?
Ans. The concept of relative molecular mass can only be used for substances that are made up of molecules. However, for ionic compounds, instead of relative molecular mass, the term ‘relative formula mass’ or \({{\text{F}}_{\text{r}}}\). It is similar to the relative molecular mass and is calculated by adding up the relative atomic masses of all the atoms shown in the chemical formula of the ionic compound.

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