Relative Molecular Mass Calculations Chemistry Tutorial

Key Concepts

⚛ Relative molecular mass is also known as:

· molecular mass

· relative molecular weight

· molecular weight

· formula mass

· formula weight

⚛ Relative molecular mass is usually given the symbol M_r. Other symbols commonly used are⁽¹⁾:

· MM (molecular mass)

· MW (molecular weight)

· FM (formula mass)

· FW (formula weight)

⚛ Relative molecular mass of a compound (M_r) is defined as the "ratio of the mass of a molecule to the unified atomic mass unit"⁽²⁾, or, the mass of a formula unit of the compound relative to the mass of a carbon-12 atom taken as exactly 12.

⚛ In practice, the relative molecular mass of a compound, M_r, is the sum of the relative atomic masses (atomic weights) of the atomic species as given in the chemical formula.

Chemical formula refers to either:

· formula of an ionic compound

· formula of a molecule (or molecular compound)

⚛ Relative molecular mass can be treated as a dimensionless quantity, that is, a quantity that has no units.⁽³⁾

⚛ There are 4 steps to calculating the relative molecular mass of a molecule or compound given its chemical formula:

Step 1: Use the chemical formula to determine how many atoms of each element are present in the compound.
Step 2: Write a mathematical expression to calculate the total mass of all the elements in the compound (the relative molecular mass of the compound)
Step 3: Use the Periodic Table to find the relative atomic mass (atomic weight) of each element
Step 4: Substitute the values for the relative atomic mass (atomic weight) of each element into the equation and solve

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Relative Molecular Mass Concepts

Consider the diagram below:

o				o
		o

The diagram above shows a box containing 3 balls:

one red ball shown as o
two black balls shown as o

If a red ball has a mass of 16 g and a black ball has a mass of 1 g, then the mass of all three balls in the box is 16 + 1 + 1 = 18 g

We could write a mathematical expression to find the total mass of the balls in the box:

mass(total) = mass(red ball) + 2 × mass(black ball)

mass(total) = 16 + 2 × 1 = 16 + 2 = 18 g

Chemists often think of atoms as really tiny balls, and refer to this as the particle theory of matter.

If each ball in the box represents an atom making up a water molecule, H₂O, then the diagram below shows a box containing a molecule of water:

o				o
	\		/
		o

In the diagram of a water molecule above:

the red ball o is an oxygen atom
each black ball o is hydrogen atom
each diagonal line ( \ and / ) represents a chemical bond between an oxygen atom and a hydrogen atom

If an oxygen atom has a mass of 16 and a hydrogen atom has a mass of 1, then the mass of all three atoms in the water molecule is 16 + 1 + 1 = 18

We could write a mathematical expression to find the total mass of all the atoms in a water molecule:

mass(water molecule) = mass(oxygen atom) + 2 × mass(hydrogen atom)

mass(water molecule) = 16 + 2 × 1 = 16 + 2 = 18

This is exactly what we are doing when we calculate the relative molecular mass of a compound:

Step 1: Use the chemical formula to determine how many atoms of each element are present in the compound.

For example: compund X_aY_b contains:

a atoms of element X

b atoms of element Y

Step 2: Write a mathematical expression to calculate the total mass of all the elements in the compound (called the relative molecular mass of the compound)

M_r(X_aY_b) = relative molecular mass of compound X_aY_b

M_r(X) = relative atomic mass of element X

M_r(Y) = relative atomic mass of element Y

a = number of atoms of element X

b = number of atoms of element Y

M_r(X_aY_b) = a × M_r(X) + b × M_r(Y)

Step 3: Use the Periodic Table to find the relative atomic mass (atomic weight) of each element

M_r(X) = relative atomic mass of element X = some number

M_r(Y) = relative atomic mass of element Y = another number

Step 4: Substitute the values for the relative atomic mass (atomic weight) of each element into the equation and then solve the equation to find the relative molecular mass of the compund

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Worked Examples of Relative Molecular Mass Calculations

Worked Example: Calculating the Relative Molecular Mass of a Diatomic Molecule

Carbon monoxide is a diatomic molecule, a molecule made up of two atoms: an atom of carbon (C) and an atom of oxygen (O).
Carbon monoxide has the molecular formula CO

Calculate the relative molecular mass (M_r) of the compound carbon monoxide, CO

What is the question asking you to do?

Calculate the relative molecular mass of carbon monoxide

M_r(CO) = ?

What information (data) has been given in the question?

Chemical formula for carbon monoxide: CO

What is the relationship between a chemical formula and its relative molecular mass?

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

A molecule of carbon monoxide (CO) contains 1 carbon atom (C) and 1 oxygen atom (O)

relative molecular mass (CO) = 1 × relative atomic mass (C) + 1 × relative atomic mass (O)

M_r(CO) = 1 × M_r(C) + 1 × M_r(O)

Use the Periodic Table to find the relative atomic mass (atomic weight) for each element present:

M_r(C) = relative atomic mass of C (carbon) = 12.01

M_r(O) = relative atomic mass of of O (oxygen) = 16.00

Calculate the relative molecular mass of carbon monoxide by substituting the values into the expression to find M_r(CO)

M_r(CO) = 1 × M_r(C) + 1 × M_r(O)

M_r(CO) = (1 × 12.01) + (1 × 16.00) = 28.01

Worked Example: Calculating the Relative Molecular Mass of a Triatomic Molecule

Carbon dioxide is a triatomic molecule, a molecule made up of 3 atoms.
Carbon dioxide has the molecular formula CO₂
Each molecule of carbon dioxide is made up of 1 atom of carbon (C) and 2 atoms of oxygen (O).

Calculate the relative molecular mass (M_r) of the compound carbon dioxide, CO₂.

What is the question asking you to do?

Calculate the relative molecular mass of carbon dioxide

M_r(CO₂) = ?

What information (data) has been given in the question?

Chemical formula for carbon dioxide: CO₂

What is the relationship between a chemical formula and its relative molecular mass?

relative molecular mass = sum of the relative atomic masses of each element in the compound

A molecule of carbon dioxide (CO₂) is composed of 1 atom of carbon (C) and 2 atoms of oxygen (O)

relative molecular mass (CO₂) = 1 × relative atomic mass (C) + 2 × relative atomic mass (O)

M_r(CO₂) = 1 × M_r(C) + 2 × M_r(O)

Use the Periodic Table to find the relative atomic mass (atomic weight) for each element present:

M_r(C) = relative atomic mass of C (carbon) = 12.01

M_r(O) = relative atomic mass of O (oxygen) = 16.00

Calculate the relative molecular mass of carbon dioxide by substituting the values into the expression to find M_r(CO₂)

M_r(CO₂) = 1 × M_r(C) + 2 × M_r(O)

M_r(CO₂) = (1 × 12.01) + (2 × 16.00) = 12.01 + 32.00 = 44.01

Worked Example: Calculating the Relative Molecular Mass of a Polyatomic Compound

Calcium hydroxide is a polyatomic compound, a compound made up of many "atoms".
Calcium hydroxide has the formula Ca(OH)₂
The compound is made up of calcium ions, Ca²⁺, and hydroxide ions, OH^-
For every 1 ion of calcium there are 2 hydroxide ions.
Each hydroxide ion, OH^-, is made up of 1 atom of oxygen (O) and 1 atom of hydrogen (H).

Calculate the relative molecular mass (M_r) of the compound calcium hydroxide, Ca(OH)₂

What is the question asking you to do?

Calculate the relative molecular mass of calcium hydroxide

M_r(Ca(OH)₂) = ?

What information (data) has been given in the question?

Chemical formula for calcium hydroxide: Ca(OH)₂

What is the relationship between a chemical formula and its relative molecular mass?

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

Calcium hydroxide contains one Ca atom (actually an ion) and two hydroxide ions.

Each hydroxide ion is composed of one hydrogen atom and one oxygen atom

So calcium hydroxide contains 1 calcium atom (Ca), 2 oxygen atoms (O) and 2 hydrogen atoms (O)

relative molecular mass of calcium hydroxide = 1 × relative atomic mass (Ca) + 2 × relative atomic mass (O) + 2 × relative atomic mass (H)

M_r(Ca(OH)₂) = 1 × M_r(Ca) + 2 × M_r(O) + 2 × M_r(H)

Use the Periodic Table to find the relative atomic mass (atomic weight) for each element present:

M_r(Ca) = relative atomic mass calcium = 40.08

M_r(H) = relative atomic mass hydrogen = 1.008

M_r(O) = relative atomic mass of oxygen = 16.00

Calculate the relative molecular mass of calcium hydroxide by substituting the values into the expression to find M_r(Ca(OH)₂)

M_r(Ca(OH)₂) = 1 × M_r(Ca) + 2 × M_r(O) + 2 × M_r(H)

M_r(Ca(OH)₂) = (1 × 40.08) + (2 × 16.00) + (2 × 1.008) = 40.08 + 32.00 + 2.016 = 74.10

Alternatively, relative molecular mass = relative atomic mass of calcium + (2 × relative molecular mass of hydroxide ions)
M_r(Ca(OH)₂) = 40.08 + [2 × (16.00 + 1.008)] = 40.08 + [2 × 17.008]= 40.08 + 34.016 = 74.10

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Problem Solving : Relative Molecular Mass Calculations

The Problem: Chris the Chemist has just finished a detailed analysis of a molecule and has found that it is made up of two elements, phosphorus and oxygen.
The molecule contains two phosphorus atoms and has a relative molecular mass of 141.94.

Determine the molecular formula for this molecule.

Solving the Problem using the StoPGoPS model for problem solving:

STOP!	State the question.	What is the question asking you to do? Determine the molecular formula for the molecule
PAUSE!	Pause to Plan.	What information (data) have you been given? elements in molecule are phosphorus and oxygen 2 atoms of phosphorus M_r(molecule) = relative molecular mass of molecule = 141.94 What is your plan for solving this problem? Step 1: Write the symbols for all the elements present in the molecule Use the Periodic Table to find the symbols for: phosphorus : oxygen : Step 2: Write a partial chemical formula based on the data provided Molecule contains only phosphorus and oxygen atoms 2 phosphorus atoms: phosphorus₂ let b equal the number of oxygen atoms: oxygen_b Partial chemical formula is phosphorus₂oxygen_b substitute in chemical symbols for phosphorus and oxygen Step 3: Calculate total mass of phosphorus present in molecule Determine the number of atoms of phosphorus in the molecule = 2 Use Periodic Table to find the relative atomic mass of phosphorus M_r(phosphorus) Calculate the total mass of phosphorus atoms present in the molecule: total mass (phosphorus) = number of phosphorus atoms × relative atomic mass (phosphorus) total mass (phosphorus) = 2 × M_r(phosphorus) Step 4: Calculate the total mass of oxygen atoms present in the molecule mass(molecule) = total mass of phosphorus atoms + total mass of oxygen atoms M_r(phosphorus₂oxygen_b) = 2 × M_r(phosphorus) + [b × M_r(oxygen)] so, total mass of oxygen atoms present = [b × M_r(oxygen)] = M_r(phosporus₂oxygen_b) - 2 × M_r(phosphorus) so, total mass of oxygen atoms present = [b × M_r(oxygen)] = 141.94 - 2 × M_r(phosphorus) Step 5: Calculate the number of oxygen atoms present Use the Periodic Table to find the relative atomic mass of oxygen M_r(oxygen) Substitute this value into the equation for the total of oxygen mass in the molecule: total mass of oxygen atoms present = b × M_r(oxygen) therefore, number of oxygen atoms = b = total mass of oxygen ÷ M_r(oxygen) Step 6: Write the chemical formula for the molecule Substitute the chemical symbols for phosphorus and oxygen, and, the value for b into the chemical formula for the molecule phosphorus₂oxygen_b
GO!	Go with the Plan.	Step 1: Write the symbols for all the elements present in the molecule Use the Periodic Table to find the symbols for: phosphorus : P oxygen : O Step 2: Write a partial chemical formula based on the data provided Molecule contains only phosphorus and oxygen atoms 2 phosphorus atoms: phosphorus₂ let b equal the number of oxygen atoms: oxygen_b Partial chemical formula is phosphorus₂oxygen_b substitute in chemical symbols for phosphorus and oxygen P₂O_b Step 3: Calculate total mass of phosphorus present in molecule Determine the number of atoms of phosphorus in the molecule = 2 Use Periodic Table to find the relative atomic mass of phosphorus M_r(phosphorus) = 30.97 Calculate the total mass of phosphorus atoms present in the molecule: total mass (phosphorus) = number of phosphorus atoms × relative atomic mass (phosphorus) total mass (phosphorus) = 2 × M_r(phosphorus) total mass (phosphorus) = 2 × 30.97 = 61.94 Step 4: Calculate the total mass of oxygen atoms present in the molecule mass(molecule) = total mass of phosphorus atoms + total mass of oxygen atoms M_r(phosphorus₂oxygen_b) = 2 × M_r(phosphorus) + [b × M_r(oxygen)] so, total mass of oxygen atoms present = [b × M_r(oxygen)] = M_r(P₂O_b) - 2 × 30.97 so, total mass of oxygen atoms present = [b × M_r(oxygen)] = 141.94 - 2 × 30.97 total mass of oxygen atoms present = 141.94 - 61.94 = 80.00 Step 5: Calculate the number of oxygen atoms present Use the Periodic Table to find the relative atomic mass of oxygen M_r(oxygen) = 16.00 Substitute this value into the equation for the total of oxygen mass in the molecule: total mass of oxygen atoms present = b × 16.00 = 80.00 therefore, number of oxygen atoms = b = total mass of oxygen ÷ M_r(oxygen) = 80.00 ÷ 16.00 = 5 Step 6: Write the chemical formula for the molecule Substitute the chemical symbols for phosphorus and oxygen, and, the value for b into the chemical formula for the molecule P₂O_b P₂O₅
PAUSE!	Ponder Plausability.	Have you answered the question that was asked? Yes, we have determined the molecular formula for this compound. Is your solution to the question reasonable? Work backwards by calculating the relative molecular mass for the molecular formula we have found: M_r(P₂O₅) = 2 × M_r(P) + 5 × M_r(O) = 2 × 30.97 + 5 × 16.00 = 61.94 + 80.00 = 141.94 Since the relative molecular mass calculated using our chemical formula is the same as the one given in the question, we are satisfied that the answer is correct.
STOP!	State the solution.	What is the molecular formula of the compound? The molecular formula for this compound is P₂O₅

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Sample Question: Calculating Relative Molecular Mass

Determine the relative molecular mass of sodium hydrogencarbonate.

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Footnotes:

(1) recommended IUPAC symbol for relative molecular mass (relative molar mass) is M_r.
Debate still rages about whether the term atomic mass, atomic weight, relative atomic mass, relative atomic weight, or standard atomic weight should be used. Don't worry about it, at High School you can treat all these terms as meaning the same thing.
M is the IUPAC symbol recommended for molar mass

(2) IUPAC Gold Book 2019

(3) The IUPAC periodic table uses standard atomic weights and provides the lower and upper bounds of the standard atomic weight for some elements.
Prior to 1961 the unit amu (atomic mass unit) was used. Since then the unit u (unified atomic mass unit) or Da (dalton) have been widely used but neither of these is an SI unit (as defined by the CIPM, International Committee for Weights and Measures). The issue will be further complicated if the dalton is redefined as being 0.001/N_A kg since it would no longer be a unit of atomic mass relative to carbon-12. For High School students it is probably best to avoid the problems and use relative atomic mass (atomic weight) as a quantity with no units.
Molar mass, M, does have units. The SI units for molar mass are kg mol^-1

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