Why Molar Mass Equals Relative Mass in Grams

Introduction

The periodic table tells you that carbon has a relative atomic mass of 12.0, and chemists say that one mole of carbon has a mass of 12.0 g. Oxygen is 16.0, and one mole of oxygen atoms is 16.0 g. The numbers match every time. Is this a lucky coincidence? Not at all. The match was designed on purpose, and the Avogadro constant is the link that makes it work. Understanding this link turns the mole from a rule to memorise into an idea you can explain.

Core explanation

Two different scales. Relative atomic masses belong to the atomic world. They compare the mass of an atom with one-twelfth of the mass of a carbon-12 atom. That reference mass is called the unified atomic mass unit , u. Its value is tiny:

1 u ≈ 1.66 × 10⁻²⁴ g

Grams, on the other hand, belong to the laboratory world of balances and beakers. To connect the two worlds we need a conversion factor between u and g.

The Avogadro constant is that factor. Multiply the atomic mass unit by the Avogadro constant:

1.66 × 10⁻²⁴ g × 6.022 × 10²³ ≈ 1.00 g

So a mole of particles that each have a mass of 1 u has a total mass of 1 g. This is not an accident: historically the mole was defined as the number of atoms in exactly 12 g of carbon-12, which forces Nᴀ × 1 u to equal 1 g.

Scaling up to any atom. A carbon-12 atom has a mass of 12 u. One mole of them has a mass of 12 × (Nᴀ × 1 u) = 12 × 1 g = 12 g. A sodium atom has an average mass of 23.0 u, so one mole of sodium atoms has a mass of 23.0 g. Whatever the relative mass is, the same factor of Nᴀ converts "u per particle" into "grams per mole", and the number stays the same.

The same logic for compounds. A water molecule, H₂O, has a relative formula mass of 2 × 1.0 + 16.0 = 18.0, meaning one molecule has a mass of about 18.0 u. One mole of water molecules therefore has a mass of 18.0 g. The rule works for atoms, molecules and formula units alike.

What changes and what stays the same.

Quantity Carbon Water --- --- --- Relative mass (no units) 12.0 18.0 Mass of one particle 12.0 u ≈ 1.99 × 10⁻²³ g 18.0 u ≈ 2.99 × 10⁻²³ g Molar mass 12.0 g/mol 18.0 g/mol

The number is the same in all three rows; only the unit, and what it refers to, changes. Relative mass has no unit, the particle mass is in u, and the molar mass is in g/mol.

Formulae

1 u × Nᴀ ≈ 1 g/mol, where 1 u ≈ 1.66 × 10⁻²⁴ g and Nᴀ = 6.022 × 10²³ mol⁻¹. Therefore M (in g/mol) = Aᵣ or Mᵣ numerically.

Step-by-step reasoning

To explain why one mole of magnesium has a mass of 24.3 g:

1. Magnesium has Aᵣ = 24.3, so an average magnesium atom has a mass of 24.3 u. 2. One mole contains Nᴀ = 6.022 × 10²³ atoms. 3. Total mass = 24.3 u × 6.022 × 10²³ = 24.3 × (1 u × Nᴀ). 4. Since 1 u × Nᴀ ≈ 1 g, the total mass is 24.3 g.

Visual explanation

Picture a ladder with two rungs. On the bottom rung is a single atom labelled "24.3 u". An arrow marked "× 6.022 × 10²³" climbs to the top rung, labelled "24.3 g". The number 24.3 travels up the ladder unchanged; only the unit changes from u to g.

Real-world analogy

Imagine every sweet in a factory weighs "1 unit", and the factory packs sweets into bags of such a size that one bag of 1-unit sweets weighs exactly 1 kg. Then a bag of 5-unit sweets weighs 5 kg, and a bag of 12-unit sweets weighs 12 kg. The bag size plays the role of the Avogadro constant.

Real-world example

Pharmaceutical chemists routinely switch between the two scales. A mass spectrometer reports the mass of a drug molecule in u (often written Da, for dalton), while the production team weighs the same drug in grams. Because of the Avogadro link, a drug molecule of 180 u, such as aspirin, has a molar mass of 180 g/mol with no extra conversion.

Why?

Why did chemists choose such an awkward number as 6.022 × 10²³? They did not choose the number first. They chose the gram and the carbon-12 atom as reference points, and then measured how many atoms fit in 12 g. The number turned out to be 6.022 × 10²³, and it is whatever it needs to be to make the scales match.

Common misconception

"The molar mass of carbon is 12, so one carbon atom weighs 12 g." A single carbon atom has a mass of only about 2 × 10⁻²³ g. The value 12 g is the mass of 6.022 × 10²³ carbon atoms, not of one atom.

Worked example

Question: A molecule of glucose, C₆H₁₂O₆, has a mass of 180 u. Using 1 u = 1.66 × 10⁻²⁴ g and Nᴀ = 6.02 × 10²³ mol⁻¹, show that the molar mass of glucose is 180 g/mol.

Reasoning: Mass of one molecule = 180 × 1.66 × 10⁻²⁴ g = 2.99 × 10⁻²² g. Mass of one mole = 2.99 × 10⁻²² g × 6.02 × 10²³ mol⁻¹ = 180 g/mol (to 3 significant figures).

Answer: 180 g/mol, the same number as the relative formula mass.

Quick check

1. Nitrogen has Aᵣ = 14.0. What is the mass of one mole of nitrogen atoms, and why is the number the same? Answer: 14.0 g, because each atom has a mass of 14.0 u and multiplying by the Avogadro constant turns u per atom into g per mole.

Exam focus

You may be asked to "explain why the molar mass of an element in g/mol is numerically equal to its Aᵣ". Mention three things: Aᵣ is measured relative to one-twelfth of a carbon-12 atom, one mole is the number of atoms in 12 g of carbon-12, and so Nᴀ × 1 u = 1 g. Always write molar mass with its unit, g/mol; Aᵣ and Mᵣ have none.

Advanced insight

Since 2019 the mole has been defined by fixing Nᴀ at exactly 6.022 140 76 × 10²³ mol⁻¹, rather than by 12 g of carbon-12. As a result, Nᴀ × 1 u is no longer exactly 1 g/mol, but it differs by less than one part in a billion — far too small to matter in any school or industrial calculation.

Summary

Relative masses compare particles with 1 u, one-twelfth of a carbon-12 atom. Multiplying 1 u by the Avogadro constant gives 1 g, so a mole of particles of relative mass X has a mass of X g. That is why molar mass in g/mol has the same number as Aᵣ or Mᵣ, while a single particle has an extremely small mass.

Practice questions

1. State the approximate mass of 1 u in grams. Answer: About 1.66 × 10⁻²⁴ g. 2. Show that 1 u × Nᴀ is approximately 1 g. Answer: 1.66 × 10⁻²⁴ g × 6.022 × 10²³ = 0.9997 g, which is 1.00 g to 3 significant figures. 3. Carbon dioxide, CO₂, has Mᵣ = 44.0. State the mass of one molecule in u and the molar mass in g/mol. Answer: One molecule has a mass of 44.0 u; the molar mass is 44.0 g/mol. 4. Explain why it is wrong to say that one oxygen atom has a mass of 16 g. Answer: 16 g is the mass of one mole (6.022 × 10²³) of oxygen atoms; one atom has a mass of 16 u, about 2.66 × 10⁻²³ g.