Counting Particles by Weighing Them: The Mole, Avogadro’s Number, and Molar Mass

Counting Particles by Weighing Them

Introduction to Counting Particles

In chemistry, it is often necessary to determine the number of atoms or molecules present in a given mass of a substance. Because even a small mass contains an enormous number of particles, chemists use a special counting unit called the mole to group particles for practical calculations.

• Mole: A standard scientific unit for measuring large quantities of very small entities such as atoms, molecules, or other specified particles.

• Even a few grams of a substance contain a huge number of particles.

The Mole and Avogadro’s Number

Definition and Significance

The mole is defined as the amount of substance that contains as many elementary entities (atoms, molecules, ions, etc.) as there are atoms in exactly 12.00 grams of carbon-12. This number is known as Avogadro’s Number.

• Avogadro’s Number: $6.022 \times 10^{23}$ particles per mole.

• This number allows chemists to relate the mass of a substance to the number of particles it contains.

• Conversion factors:

  • $1\ \text{mol atoms} = 6.022 \times 10^{23}\ \text{atoms}$

  • $6.022 \times 10^{23}\ \text{atoms} = 1\ \text{mol atoms}$

• Example: There are $6.022 \times 10^{23}$ carbon atoms in 12.00 grams of carbon.

Relationship Between Atomic Mass and Molar Mass

• 1 atom of carbon (C) weighs 12.00 atomic mass units (amu).

• 1 mole of carbon atoms weighs 12.00 grams.

• Key Principle: Whatever the atomic or molecular mass is in amu, the mass of one mole of those particles is the same number in grams.

Molar Mass

Definition and Use

Molar mass is the mass of one mole of a substance, expressed in grams per mole (g/mol). It serves as a conversion factor between the mass of a substance and the number of particles it contains.

• For elements: The molar mass in grams per mole is numerically equal to the atomic mass in amu.

• For molecules: The molar mass is equal to the molecular weight (sum of atomic masses of all atoms in the molecule).

• Example: The molecular weight of water (H2O) is 18.0 amu, so the molar mass is 18.0 g/mol.

Examples of Molar Masses

Substance	Molar Mass (g/mol)	Number of Particles in 1 mol
Aluminum (Al)	26.98	$6.022 \times 10^{23}$ atoms
Carbon (C)	12.01	$6.022 \times 10^{23}$ atoms
Helium (He)	4.003	$6.022 \times 10^{23}$ atoms
Water (H2O)	18.0	$6.022 \times 10^{23}$ molecules
Gold (Au)	197	$6.022 \times 10^{23}$ atoms

Using Molar Mass in Calculations

• To find the number of atoms or molecules in a given mass:

  1. Convert mass to moles using molar mass.

  2. Convert moles to number of particles using Avogadro’s number.

• To find the mass of a given number of atoms or molecules:

  1. Convert number of particles to moles using Avogadro’s number.

  2. Convert moles to mass using molar mass.

Sample Calculation: Number of Atoms in a Given Mass

Question: How many magnesium atoms are there in 150.0 g of magnesium?

• Step 1: Convert grams to moles: $\frac{150.0\ \text{g Mg}}{24.305\ \text{g/mol}} = 6.175\ \text{mol Mg}$

• Step 2: Convert moles to atoms: $6.175\ \text{mol Mg} \times 6.022 \times 10^{23}\ \text{atoms/mol} = 3.717 \times 10^{24}\ \text{Mg atoms}$

Sample Calculation: Mass of a Given Number of Atoms

Question: What is the mass of 1000 atoms of Mg?

• Step 1: Convert atoms to moles: $\frac{1000\ \text{atoms}}{6.022 \times 10^{23}\ \text{atoms/mol}} = 1.661 \times 10^{-21}\ \text{mol}$

• Step 2: Convert moles to grams: $1.661 \times 10^{-21}\ \text{mol} \times 24.305\ \text{g/mol} = 4.04 \times 10^{-20}\ \text{g}$

Sample Calculation: Mass of an Element in a Compound

Question: How many grams of hydrogen are there in 100.0 g of water?

• Step 1: Convert grams of water to moles: $\frac{100.0\ \text{g H}_2\text{O}}{18.02\ \text{g/mol}} = 5.55\ \text{mol H}_2\text{O}$

• Step 2: Convert moles of water to moles of hydrogen atoms (2 H per H2O): $5.55\ \text{mol H}_2\text{O} \times 2 = 11.1\ \text{mol H}$

• Step 3: Convert moles of hydrogen to grams: $11.1\ \text{mol H} \times 1.008\ \text{g/mol} = 11.2\ \text{g H}$

Key Definitions and Relationships

• Atomic Mass Unit (amu): A unit of mass used to express atomic and molecular weights, defined as one twelfth the mass of a carbon-12 atom.

• Molecular Weight: The sum of the atomic masses of all atoms in a molecule, expressed in amu.

• Molar Mass: The mass of one mole of a substance, numerically equal to the molecular or atomic mass in grams per mole.

Summary Table: Key Quantities

Quantity	Symbol/Unit	Definition
Mole	mol	Amount of substance containing $6.022 \times 10^{23}$ entities
Avogadro’s Number	$N_A$	$6.022 \times 10^{23}$ entities/mol
Molar Mass	g/mol	Mass of 1 mole of substance
Atomic Mass Unit	amu	1/12 the mass of a carbon-12 atom

Additional info: These concepts are foundational for stoichiometry, chemical reactions, and quantitative analysis in general chemistry.