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
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.
- Chlorine has two isotopes:
- $^{35}\text{Cl}$, with a mass of 34.97 and an abundance of 75.78%.
- $^{37}\text{Cl}$, with a mass of 36.97 and an abundance of 24.22%.
- To calculate $A_r$:
$$A_r = \left(34.97 \times \frac{75.78}{100}\right) + \left(36.97 \times \frac{24.22}{100}\right)$$ $$A_r = 26.51 + 8.95 = 35.46$$ - Thus, the relative atomic mass of chlorine is approximately 35.46.
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
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.
Calculating $M_r$ for Water ($H_2O$)
- Chemical formula: $H_2O$ (2 hydrogen atoms, 1 oxygen atom).
- $A_r$ values:
Hydrogen ($H$): 1.01 - Oxygen ($O$): 16.00
- $A_r$ values:
- 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.
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.
Calculating $M_r$ for Copper(II) Sulfate Pentahydrate ($\text{CuSO}_4 \cdot 5\text{H}_2\text{O}$)
- Formula: $\text{CuSO}_4 \cdot 5\text{H}_2\text{O}$
- 1 copper ($Cu$): $A_r = 63.55$
- 1 sulfur ($S$): $A_r = 32.07$
- 4 oxygen ($O$): $A_r = 16.00$
- 5 water molecules ($5 \times H_2O$): $M_r = 5 \times 18.02$
- 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.
- Don’t forget to include the water of crystallization when calculating $M_r$ for hydrates.
- This is a common error in exams!
- Calculate the $M_r$ for:
- Ammonia ($\text{NH}_3$)
- Sulfuric acid ($\text{H}_2\text{SO}_4$)
- Sodium sulfate decahydrate ($\text{Na}_2\text{SO}_4 \cdot 10\text{H}_2\text{O}$)
- A compound has the formula $\text{MgCl}_2$. What is its $M_r$?
- Determine the $M_r$ of glucose ($\text{C}_6\text{H}_{12}\text{O}_6$).
