How do you calculate the number of moles? What is the formula of a mole?

The mole is a fundamental unit in chemistry that represents a specific number of particles (atoms, molecules, ions, or other entities). One mole contains exactly 6.022 × 10²³ particles, known as Avogadro's number. Understanding how to calculate the number of moles is essential for stoichiometry, chemical reactions, and quantitative analysis.

Complete Table of Number of Moles Formulas

Detailed Explanation of Each Formula

1. Basic Mole Formula (From Mass)

Formula: n = m/M

This is the most fundamental formula for calculating moles. The molar mass (M) can be found on the periodic table and represents the mass of one mole of a substance.

Example: Calculate moles of water (H₂O) in 36 grams.

• Molar mass of H₂O = 2(1) + 16 = 18 g/mol
• n = 36 g ÷ 18 g/mol = 2 moles

2. Mole Formula for Gases at STP

Formula: n = V/Vm

At Standard Temperature and Pressure (0°C or 273 K, and 1 atm pressure), one mole of any ideal gas occupies 22.4 liters. This is called the molar volume.

Example: How many moles are in 44.8 L of nitrogen gas at STP?

• n = 44.8 L ÷ 22.4 L/mol = 2 moles

3. Ideal Gas Law Formula

Formula: n = PV/RT

This universal gas equation works for gases at any temperature and pressure. Remember to convert temperature to Kelvin (K = °C + 273.15).

Example: Calculate moles of gas at 2 atm pressure, 5 L volume, and 27°C.

• T = 27 + 273 = 300 K
• n = (2 atm × 5 L) ÷ (0.0821 L·atm/mol·K × 300 K) = 0.406 moles

4. Mole Formula Using Avogadro's Number

Formula: n = N/NA

Avogadro's number (6.022 × 10²³) represents the number of particles in one mole. This formula connects microscopic particles to macroscopic amounts.

Example: How many moles are in 1.204 × 10²⁴ molecules of CO₂?

• n = (1.204 × 10²⁴) ÷ (6.022 × 10²³) = 2 moles

5. Mole Formula from Concentration

Formula: n = C × V

Molarity (C) expresses concentration as moles per liter. Volume must be in liters for this formula.

Example: How many moles are in 500 mL of 2 M NaCl solution?

• V = 500 mL = 0.5 L
• n = 2 mol/L × 0.5 L = 1 mole

6. Mole Fraction Formula

Formula: χA = nA/ntotal

Mole fraction is dimensionless and represents the ratio of moles of one component to total moles. The sum of all mole fractions equals 1.

Example: In a mixture of 2 moles of oxygen and 8 moles of nitrogen:

• χO₂ = 2 ÷ (2 + 8) = 0.2
• χN₂ = 8 ÷ (2 + 8) = 0.8

7. Mole Percentage Formula

Formula: Mole % = (ni/ntotal) × 100

This expresses composition as a percentage rather than a fraction.

Example: Using the mixture above:

• Mole % of O₂ = (2 ÷ 10) × 100 = 20%

8. Moles from Density

Formula: n = (ρ × V)/M

This formula combines density with volume to first calculate mass, then converts to moles.

Example: Calculate moles in 100 mL of ethanol with density 0.789 g/mL (M = 46 g/mol):

• n = (0.789 g/mL × 100 mL) ÷ 46 g/mol = 1.72 moles

Derived Relationships and Rearrangements

From the Basic Formula n = m/M:

• To find mass: m = n × M
• To find molar mass: M = m/n

From the Gas Volume Formula n = V/Vm:

• To find volume: V = n × Vm

From the Ideal Gas Law n = PV/RT:

• Complete form: PV = nRT
• To find pressure: P = nRT/V
• To find volume: V = nRT/P
• To find temperature: T = PV/nR

From the Particle Formula n = N/NA:

• To find number of particles: N = n × NA

From the Concentration Formula n = C × V:

• To find molarity: C = n/V
• To find volume: V = n/C

Important Constants to Remember

Quick Reference: When to Use Which Formula

You have mass? → Use n = m/M

You have gas volume at STP? → Use n = V/22.4

You have gas at non-STP conditions? → Use n = PV/RT

You have number of particles? → Use n = N/NA

You have solution concentration? → Use n = C × V

You have density? → Use n = (ρ × V)/M

You need composition of mixture? → Use mole fraction or mole percentage formulas

Common Mistakes to Avoid

1. Unit Inconsistency: Always ensure units match (liters with liters, grams with grams)
2. Temperature in Celsius: Always convert to Kelvin when using gas law formulas
3. Molar Mass Calculation: Double-check atomic masses from the periodic table
4. Volume Conversion: Remember 1 L = 1000 mL
5. STP Conditions: Molar volume of 22.4 L/mol only applies at STP