S1.4.2 Relative atomic and formula masses

Relative Atomic Mass ( $A_{r}$ ) and Relative Formula Mass ( $M_{r}$ )

Why Do We Need Relative Masses in Chemistry?

You are trying to weigh a single atom on a balance. It's impossible—atoms are so small that their masses are expressed in atomic mass units (amu), which are defined relative to the mass of a carbon-12 atom. Why carbon-12?

It serves as a universal reference: one atom of carbon-12 is assigned a mass of exactly 12 amu.

Relative Atomic Mass ( $A_{r}$ ): A Comparison to Carbon-12

Definition

Relative atomic mass

The relative atomic mass of an element is a weighted average of the masses of its isotopes, based on their natural abundances, compared to 1/12th the mass of a carbon-12 atom. Since it is a ratio, $A_{r}$ has no units.

Key Points:

Reference to Carbon-12: The mass of an atom is expressed relative to 1/12th the mass of a carbon-12 atom.
Weighted Average: For elements with multiple isotopes, $A_{r}$ accounts for both the mass and natural abundance of each isotope.
Data Booklet Values: $A_{r}$ values are provided in the IB Chemistry Data Booklet and are typically rounded to two decimal places.

Example

Calculating $A_{r}$ for Chlorine

Chlorine has two isotopes:

$^{35} Cl$ , with a mass of 34.97 and an abundance of 75.78%.
$^{37} Cl$ , with a mass of 36.97 and an abundance of 24.22%.To calculate $A_{r}$ :
$A_{r} = (34.97 \times \frac{75.78}{100}) + (36.97 \times \frac{24.22}{100})$ $A_{r} = 26.51 + 8.95 = 35.46$

Thus, the relative atomic mass of chlorine is approximately 35.46.

Note

In exams, you’ll typically use the $A_{r}$ values provided in the data booklet rather than calculating them yourself.

Relative Formula Mass ( $M_{r}$ ): Adding Up Atomic Masses

Definition

Relative formula mass

The relative formula mass ( $M_{r}$ ) is the sum of the relative atomic masses ( $A_{r}$ ) of all the atoms in a chemical formula.

It applies to both molecular compounds (like water) and ionic compounds (like sodium chloride).

Key Points:

Molecular Compounds: For molecules, $M_{r}$ is the sum of the $A_{r}$ values of all atoms in the molecule.
Ionic Compounds: For ionic compounds, $M_{r}$ is calculated using the smallest formula unit (e.g., NaCl for sodium chloride).
No Units: Like $A_{r}$ , $M_{r}$ is dimensionless.

Determining $M_{r}$ : Step-by-Step Guide

1. Identify the Chemical Formula

Write down the chemical formula of the compound to determine the number and type of atoms.

2. Look Up $A_{r}$ Values

Use the periodic table or data booklet to find the $A_{r}$ values for each element.

3. Multiply and Add

Multiply the $A_{r}$ of each element by the number of atoms of that element in the formula, then sum the results.

Example

Calculating $M_{r}$ for Water ( $H_{2} O$ )

Chemical formula: $H_{2} O$ (2 hydrogen atoms, 1 oxygen atom).
- $A_{r}$ values:
  Hydrogen ( $H$ ): 1.01
- Oxygen ( $O$ ): 16.00
Calculation:
$M_{r} = (2 \times 1.01) + (1 \times 16.00) = 2.02 + 16.00 = 18.02$

Thus, the relative formula mass of water is 18.02.

Analogy

Think of $M_{r}$ as adding up the weights of all the ingredients in a recipe to find the total weight of the dish.

Application to Ionic and Hydrated Compounds

For ionic compounds, $M_{r}$ is calculated using the smallest formula unit.
Hydrated compounds include water molecules in their structure, and the $M_{r}$ must account for these.

Example

Calculating $M_{r}$ for Copper(II) Sulfate Pentahydrate ( ${CuSO}_{4} \cdot 5 H_{2} O$ )

- Formula: ${CuSO}_{4} \cdot 5 H_{2} O$
  1 copper ( $C u$ ): $A_{r} = 63.55$
- 1 sulfur ( $S$ ): $A_{r} = 32.07$
- 4 oxygen ( $O$ ): $A_{r} = 16.00$
- 5 water molecules ( $5 \times H_{2} O$ ): $M_{r} = 5 \times 18.02$
Calculation: $M_{r} = 63.55 + 32.07 + (4 \times 16.00) + (5 \times 18.02)$ $M_{r} = 63.55 + 32.07 + 64.00 + 90.10 = 249.72$

Thus, the relative formula mass of copper(II) sulfate pentahydrate is 249.72.

Common Mistake

Don’t forget to include the water of crystallization when calculating $M_{r}$ for hydrates. This is a common error in exams!

Reflection and Broader Implications

Self review

Calculate the $M_{r}$ for:
- a) Ammonia ( ${NH}_{3}$ )
- b) Sulfuric acid ( $H_{2} {SO}_{4}$ )
- c) Sodium sulfate decahydrate ( ${Na}_{2} {SO}_{4} \cdot 10 H_{2} O$ )
A compound has the formula ${MgCl}_{2}$ . What is its $M_{r}$ ?
Determine the $M_{r}$ of glucose ( $C_{6} H_{12} O_{6}$ ).

Theory of Knowledge

To what extent does defining relative masses as dimensionless numbers simplify scientific communication?
Could this approach lead to any misunderstandings?

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S1.4.2 Relative atomic and formula masses

Relative Atomic Mass (Ar) and Relative Formula Mass (Mr)

Why Do We Need Relative Masses in Chemistry?

Relative Atomic Mass (Ar): A Comparison to Carbon-12

Key Points:

Calculating Ar for Chlorine

Relative Formula Mass (Mr): Adding Up Atomic Masses

Key Points:

Determining Mr: Step-by-Step Guide

1. Identify the Chemical Formula

2. Look Up Ar Values

3. Multiply and Add

Calculating Mr for Water (H2O)

Application to Ionic and Hydrated Compounds

Reflection and Broader Implications

You've read 2/2 free chapters this week.

Introduction to Relative Masses

Relative Atomic Mass ( $A_{r}$ ) and Relative Formula Mass ( $M_{r}$ )

Relative Atomic Mass ( $A_{r}$ ): A Comparison to Carbon-12

Calculating $A_{r}$ for Chlorine

Relative Formula Mass ( $M_{r}$ ): Adding Up Atomic Masses

Determining $M_{r}$ : Step-by-Step Guide

2. Look Up $A_{r}$ Values

Calculating $M_{r}$ for Water ( $H_{2} O$ )