Learning Objectives
1 objectiveBy the end of this note, you should be able to:
- 1.1.A — Calculate quantities of a substance or its relative number of particles using dimensional analysis and the mole concept.
The Mole Concept and Avogadro’s Number
Counting individual atoms or molecules during laboratory work is impossible, so chemists use the mole [a unit representing a fixed number of particles] to bridge macroscopic measurements and the particle scale.
Avogadro’s number (Nᴀ = 6.022 × 10²³ mol⁻¹) defines how many constituent particles exist in one mole of any substance. “Constituent particles” means whatever entity the formula specifies — atoms for an element like Fe, molecules for a covalent compound like H₂O, or formula units for an ionic compound like NaCl. Therefore:
- 1 mol of carbon (C) contains 6.022 × 10²³ atoms of carbon.
- 1 mol of water (H₂O) contains 6.022 × 10²³ molecules of water.
- 1 mol of sodium chloride (NaCl) contains 6.022 × 10²³ formula units of NaCl.
To convert between moles and number of particles, use dimensional analysis:
$$\text{Number of particles}=n\times {N}_{A}$$
where n = number of moles (mol) and Nᴀ = 6.022 × 10²³ mol⁻¹.
MisconceptionStudents sometimes apply Avogadro’s number to grams directly (e.g., multiplying mass by 6.022 × 10²³). Avogadro’s number converts moles to particles, not grams to particles. You must convert mass → moles first, then moles → particles.
Connecting Atomic Mass Units to Molar Mass
The atomic mass unit (amu) provides a quantitative bridge between the mass of a single particle and the molar mass of a substance in grams per mole.
One amu is defined as exactly one-twelfth the mass of a carbon-12 atom. On this scale, a single atom of carbon-12 has a mass of 12.00 amu, a single oxygen atom averages 16.00 amu, and a single water molecule has a mass of 18.02 amu.
The average mass of one particle (atom, molecule, or formula unit) expressed in amu is numerically equal to the molar mass of that substance expressed in grams per mole (g mol⁻¹). For example, because one water molecule has a mass of 18.02 amu, one mole of water has a mass of 18.02 g.
The relevant equation is:
$$n=\frac{m}{M}$$
Where: n = number of moles (mol), m = mass of the sample (g), M = molar mass (g mol⁻¹).
Worked Example:
A sample of glucose (C₆H₁₂O₆) has a mass of 45.0 g. Calculate the number of moles of glucose.
First, determine the molar mass of glucose from atomic masses:
$$M=6(12.01)+12(1.008)+6(16.00)$$
$$M=72.06+12.10+96.00$$
$$M=180.16{\text{ g mol}}^{-1}$$
Equation used
$$n=\frac{m}{M}$$
Given
$$m=45.0\text{ g}$$
$$M=180.16{\text{ g mol}}^{-1}$$
Working
$$n=\frac{45.0}{180.16}$$
$$n=0.250\text{ mol}$$
Practice Problem:
A sample of calcium carbonate (CaCO₃) has a mass of 12.5 g. Calculate the number of moles of CaCO₃. (Molar mass of CaCO₃ = 100.09 g mol⁻¹.)
Answer
Equation used
$$n=\frac{m}{M}$$
Given
$$m=12.5\text{ g}$$
$$M=100.09{\text{ g mol}}^{-1}$$
Working
$$n=\frac{12.5}{100.09}$$
$$n=0.125\text{ mol}$$
Examiner InsightAP free-response questions frequently require multi-step conversions: mass → moles → particles (or the reverse). Show every conversion step explicitly and include units at each stage — partial credit depends on visible dimensional analysis.
Exam TipWrite units beside every number in your calculation so graders can follow your logic even if your final answer is wrong.
MisconceptionStudents sometimes confuse the mass of one mole (in grams) with the mass of one particle (in amu). The numerical values are the same, but the units are different. On the exam, stating “one molecule of H₂O weighs 18.02 g” is incorrect — it weighs 18.02 amu. Only one mole of H₂O weighs 18.02 g.
Exam TipAlways check whether the question asks about one particle or one mole before assigning units.
QUICK RECAP
Key Points
- The mole connects macroscopic mass to the number of particles.
- Nᴀ = 6.022 × 10²³ mol⁻¹ defines one mole of any substance.
- Constituent particles can be atoms, molecules, or formula units.
- n = m / M converts mass in grams to moles.
- Number of particles = n × Nᴀ converts moles to particles.
- 1 amu = average mass of one particle; numerically equals molar mass in g mol⁻¹.
- Both amu and the mole are defined relative to carbon-12.
- The periodic table gives atomic mass in amu and molar mass in g mol⁻¹.
- Always convert mass → moles before using Avogadro’s number.
- Include units at every step of dimensional analysis on the exam.
CAN I…? PROGRESS CHECK
Self-Assessment
- Calculate the number of moles from a given mass and molar mass using n = m / M.
- Convert between moles and number of particles using Avogadro’s number.
- Explain why the average mass of one particle in amu equals the molar mass in g mol⁻¹.
- Perform multi-step conversions from mass → moles → number of particles (and the reverse).
- Identify the correct constituent particle (atom, molecule, or formula unit) for a given substance.
- Use dimensional analysis with proper units at every step of a calculation.