Comparing Amounts: Equal Moles, Different Masses

Introduction

The previous comparison fixed mass and found different numbers of particles. Reverse the condition: fix the amount in moles. Then the specified-entity counts are equal because one mole always contains the same number of specified entities. Their masses can still differ greatly because different entities have different masses.

Core explanation

For a stated amount n, the entity count is N = nNₐ. If two samples each contain 1.00 mol of their specified entities, both contain the same number of those entities, about 6.02 × 10²³ using the rounded classroom value. Their masses are m₁ = nM₁ and m₂ = nM₂. At equal n, the mass ratio m₁/m₂ equals M₁/M₂. The Avogadro constant establishes equal count; molar mass determines different weight.

One mole of H₂O molecules has mass about 18 g. One mole of CO₂ molecules has mass about 44 g. Each sample contains Nₐ molecules, but one CO₂ molecule is heavier than one H₂O molecule in the same relative-mass model. The CO₂ sample is therefore 44/18 ≈ 2.44 times heavier. This comparison is about molecule numbers and masses; it does not say the two samples contain equal numbers of all atoms, although both H₂O and CO₂ happen to have three atoms per molecule.

For one mole of Mg atoms and one mole of Cu atoms, the atom counts are equal. Using rounded M(Mg) ≈ 24.3 g mol⁻¹ and M(Cu) ≈ 63.5 g mol⁻¹, the copper sample is heavier. A mole is a count, not a fixed number of grams. One mole of anything does not “weigh one gram” by definition.

Half-mole examples work the same way. A 0.500 mol sample of O₂ molecules and a 0.500 mol sample of N₂ molecules each contain 0.500Nₐ molecules. With O = 16 and N = 14, their masses are 0.500 × 32 = 16 g and 0.500 × 28 = 14 g, respectively. Both molecules are diatomic, so each sample also contains 1.00 mol atoms, though those atoms are of different elements.

For ionic substances, one mole of NaCl formula units and one mole of CaCl₂ formula units each contain Nₐ formula units, but their formula masses differ. The ion counts do not follow the same equality: NaCl represents two ions per unit and CaCl₂ represents three. One mole of NaCl units represents two moles of ions total in the simple accounting, while one mole of CaCl₂ units represents three moles of ions total. Always name the counted unit before saying “same number of particles.”

Hydrates extend the same idea. One mole of CuSO₄ formula units and one mole of CuSO₄·5H₂O hydrate units have equal counts of their respective units, but the hydrate is heavier because each unit includes five waters. With the rounded masses used earlier, 160 g versus 250 g represents one mole of each. It is incorrect to use anhydrous molar mass for the hydrated material simply because both have one CuSO₄ part.

Step-by-step reasoning

1. Confirm the two mole amounts are equal and specify their counted entities. 2. Use N = nNₐ to explain the equal count of those entities. 3. Calculate each correct molar mass and multiply each by the shared n. 4. Compare masses through the M ratio, then check constituent atom or ion counts separately if asked.

Visual explanation

Draw two boxes with the same number of symbols, one box of H₂O icons and one of CO₂ icons. Place “1 mol molecules = Nₐ molecules” above both. Under the boxes, place “18 g” and “44 g” to show equal count with unequal mass.

Real-world analogy

One dozen paperclips and one dozen metal hammers both contain twelve items. Their total weights differ because a hammer weighs more than a paperclip. The mole is a much larger fixed item count, and molar mass measures how heavy that count is for the named entity.

Real-world example

A teacher sets out 0.100 mol samples of sodium chloride and sugar, using glucose C₆H₁₂O₆ as the sugar. With M(NaCl) ≈ 58.5 and M(glucose) ≈ 180 g mol⁻¹, the masses are 5.85 g and 18.0 g. Both samples contain the same number of their specified formula units or molecules, but the glucose sample is heavier.

Why?

Why do equal moles ensure equal counts even when masses differ? The unit mole is defined by a fixed number of specified entities, independent of chemical identity. The mass per entity changes from substance to substance, so the mass of that fixed count changes with molar mass.

Common misconception

“A mole of NaCl and a mole of CaCl₂ contain equal numbers of ions because they contain equal numbers of formula units.” Their formula-unit counts are equal, but CaCl₂ has three ions per unit versus two for NaCl. The word “particles” must be made specific.

Worked example

Compare 0.250 mol O₂ molecules and 0.250 mol CO₂ molecules. Both contain 0.250 × 6.02 × 10²³ = 1.505 × 10²³ molecules in the rounded model. With M(O₂) = 32 and M(CO₂) = 44 g mol⁻¹, their masses are 8.0 g and 11 g. The CO₂ sample is heavier by 3.0 g and by a factor 44/32 = 1.375, despite equal molecule numbers.

Quick check

1. Do 0.50 mol H₂O molecules and 0.50 mol CO₂ molecules contain equal molecule counts? Answer: Yes. Each contains 0.50Nₐ molecules, though their masses differ.

Exam focus

Say explicitly whether the equal count refers to atoms, molecules or formula units. Use m = nM for mass and identify the larger M before calculating. Do not infer equal atom or ion totals from equal formula-unit amounts without checking formulas.

Advanced insight

An equal-mole comparison can be made at any scale. One millimole of each substance contains the same 0.001Nₐ specified entities; one kilomole of each contains the same 1000Nₐ. The mass ratio remains M₁/M₂ at every equal amount because n cancels from m₁/m₂.

Summary

Equal mole amounts contain equal numbers of their specified entities, while masses follow molar masses. H₂O and CO₂ samples with equal molecule amounts therefore have equal molecule counts but unequal masses. Formula details still determine how many constituent atoms or ions those equal units represent.

Practice questions

1. Find the masses of 1.00 mol H₂O and 1.00 mol CO₂ with M = 18 and 44 g mol⁻¹. Answer: 18 g H₂O and 44 g CO₂; both contain Nₐ molecules. 2. Compare the masses of 0.50 mol N₂ and 0.50 mol O₂ with M = 28 and 32 g mol⁻¹. Answer: 14 g N₂ and 16 g O₂; the molecule counts are equal. 3. Do 1 mol NaCl formula units and 1 mol CaCl₂ formula units represent equal total ion counts? Answer: No. NaCl represents 2 mol individual ions; CaCl₂ represents 3 mol individual ions. 4. Why does a hydrated salt weigh more than its anhydrous form for an equal mole amount of units? Answer: Each hydrate unit includes the specified water molecules, increasing its molar mass.