Writing Formulae by Crossing Valencies

Introduction

Now that you know the valencies of common elements, you can put them to work. Instead of memorising that aluminium oxide is Al₂O₃ or that calcium chloride is CaCl₂, you can build these formulae yourself in a few seconds. The trick is called crossing valencies , or the swap-and-drop method. It turns formula writing into a reliable routine that works for almost every binary compound you will meet.

Core explanation

The idea behind it. In a compound, the total combining power of each element must be equal. If aluminium has valency 3 and oxygen has valency 2, you need numbers of atoms that make 3 × (Al count) = 2 × (O count). The smallest solution is two aluminium atoms and three oxygen atoms: 2 × 3 = 6 and 3 × 2 = 6. Crossing valencies finds this answer automatically.

The swap-and-drop method.

1. Write the symbols, metal first: Al O. 2. Write each valency above its symbol: Al (3), O (2). 3. Swap the valencies: aluminium takes oxygen's 2, oxygen takes aluminium's 3. 4. Drop them down as subscripts: Al₂O₃. 5. Leave out any subscript of 1, and simplify if both numbers share a factor.

Examples.

Compound Valencies After swapping Final formula --- --- --- --- Sodium chloride Na 1, Cl 1 Na₁Cl₁ NaCl Calcium chloride Ca 2, Cl 1 Ca₁Cl₂ CaCl₂ Aluminium oxide Al 3, O 2 Al₂O₃ Al₂O₃ Magnesium nitride Mg 2, N 3 Mg₃N₂ Mg₃N₂ Iron(III) chloride Fe 3, Cl 1 Fe₁Cl₃ FeCl₃

Simplifying. Sometimes the crossed numbers share a common factor. Magnesium (2) and oxygen (2) cross to give Mg₂O₂, but ionic formulae show the lowest whole-number ratio, so this simplifies to MgO. Carbon (4) and oxygen (2) cross to C₂O₄, which simplifies to CO₂. Always check for a common factor before writing your final answer.

Using Roman numerals. For metals with variable valency, the Roman numeral gives you the valency to cross. Copper(I) oxide: Cu (1), O (2) → Cu₂O. Lead(IV) oxide: Pb (4), O (2) → Pb₂O₄ → PbO₂.

Checking your answer. Multiply each subscript by its valency. The two products must be equal. For Mg₃N₂: 3 × 2 = 6 and 2 × 3 = 6. If they do not match, something has gone wrong.

Step-by-step reasoning

To write the formula of potassium sulfide:

1. Symbols: K and S. 2. Valencies: potassium 1 (Group 1), sulfur 2 (Group 6). 3. Swap and drop: K₂S₁. 4. Remove the 1: K₂S. 5. Check: 2 × 1 = 2 and 1 × 2 = 2. Balanced.

Visual explanation

Write the two symbols side by side with their valencies above them. Now draw two arrows forming an X: one from the valency above the metal diagonally down to the bottom right of the non-metal, and one from the non-metal's valency down to the bottom right of the metal. The X shape is why the method is also called criss-cross.

Real-world analogy

Crossing valencies is like matching paired items sold in different pack sizes. If bread rolls come in packs of 3 and burgers in packs of 2, buying 2 packs of rolls and 3 packs of burgers gives six of each, with nothing left over. Each pack size tells you how many of the other you need.

Real-world example

Aluminium oxide, Al₂O₃, forms a thin, tough layer on the surface of aluminium window frames and drinks cans, protecting the metal beneath from further corrosion. Crossing aluminium's valency of 3 with oxygen's 2 gives its formula straight away.

Why?

Why does swapping work? The product of valency and number of atoms must be the same for both elements. Giving each element the other's valency as its count makes both products equal to valency A × valency B, so balance is guaranteed. Simplifying then removes any unnecessary common factor.

Common misconception

"The subscripts in a formula are the valencies of the elements they sit beside." After crossing, each subscript is actually the valency of the other element. In Al₂O₃ the 2 is oxygen's valency, not aluminium's.

Worked example

Question: Write the formula of calcium nitride.

Reasoning: Calcium is in Group 2, valency 2. Nitrogen is in Group 5, valency 3. Swap and drop: Ca₃N₂. There is no common factor. Check: 3 × 2 = 6 and 2 × 3 = 6.

Answer: Ca₃N₂.

Quick check

1. Use crossing valencies to write the formula of lithium oxide. Answer: Li₂O (lithium 1, oxygen 2, so Li₂O₁ becomes Li₂O).

Exam focus

Show your working: write the valencies, cross them, simplify, then check. Common mark losses come from forgetting to simplify (writing Ca₂O₂ instead of CaO), using the wrong valency for a transition metal, and writing subscripts as full-size numbers or superscripts.

Advanced insight

Crossing valencies works well for simple ionic and molecular compounds but is a shortcut, not a law. Some real compounds do not follow the usual valencies, such as iron's mixed oxide Fe₃O₄, which contains both iron(II) and iron(III), and molecular compounds like N₂O₄, whose formula shows the actual molecule rather than a simplified ratio.

Summary

To write the formula of a binary compound, write the symbols with their valencies, swap the valencies, drop them down as subscripts, leave out 1s and simplify any common factor. Check by confirming that subscript × valency is equal for both elements. For metals with variable valency, use the Roman numeral as the valency.

Practice questions

1. Write the formula of aluminium chloride. Answer: AlCl₃ (aluminium 3, chlorine 1). 2. Write the formula of magnesium oxide and explain the simplifying step. Answer: Crossing gives Mg₂O₂; both subscripts share a factor of 2, so the formula simplifies to MgO. 3. Write the formula of iron(II) oxide. Answer: FeO (iron 2, oxygen 2 gives Fe₂O₂, which simplifies to FeO). 4. Write the formula of sodium nitride and check it. Answer: Na₃N; check: 3 × 1 = 3 and 1 × 3 = 3. 5. A student writes the formula of barium chloride as Ba₂Cl. What went wrong, and what is the correct formula? Answer: The valencies were not swapped; barium has valency 2 and chlorine 1, so the correct formula is BaCl₂.