The Sum Rule for Neutral Compounds

Introduction

The oxidation numbers of all atoms in a neutral compound sum to zero. Once the usual values for some elements are known, this rule lets us solve for an unknown value. It is an accounting equation, not a measurement of whole-atom charges in every bonded molecule.

Core explanation

Take water, H₂O. Hydrogen is usually +1 when bonded to a nonmetal, and oxygen is usually −2. There are two hydrogen atoms, so 2(+1) + (−2) = 0. This confirms the assigned values for a neutral molecule. The subscript two multiplies hydrogen's contribution; it does not make each hydrogen +2.

For carbon dioxide, CO₂, oxygen is usually −2. Let carbon's oxidation number be x. The total is x + 2(−2) = 0, so x = +4. Carbon monoxide, CO, similarly gives x + (−2) = 0, so carbon is +2. Comparing these results shows carbon is oxidised when CO becomes CO₂, provided the specified reaction has a matching reduction elsewhere.

Sulfur dioxide, SO₂, gives sulfur +4 because x + 2(−2) = 0. Sulfur trioxide, SO₃, gives sulfur +6 because x + 3(−2) = 0. The oxidation number can be calculated without drawing every electron pair, though a Lewis structure may help understand bonding. The formula and a trusted oxygen assignment are sufficient for this simple formal calculation.

In ammonia, NH₃, hydrogen is +1 in the usual nonmetal compound, so nitrogen satisfies x + 3(+1) = 0 and has −3. This example shows that an element's oxidation number can be negative. Moving from nitrogen at −3 in NH₃ to 0 in N₂ is an increase and therefore oxidation, even though zero may look “smaller” if the sign is ignored.

Check that the compound is actually neutral before setting the sum to zero. SO₄²⁻ is a polyatomic ion, so its numbers sum to −2, not zero. Treating it as neutral would give sulfur +8 instead of its usual +6 assignment. Charge written outside the formula belongs to the entire species and changes the right-hand side of the sum equation.

The usual rules have exceptions. Oxygen is −1 in a peroxide such as H₂O₂, not −2. Applying −2 to each oxygen in H₂O₂ while assigning hydrogen +1 would give a nonzero sum, which flags a problem with the chosen rule. The sum rule still holds; the assumption about oxygen must be adjusted. A correct solution lists assumptions and checks that their weighted sum matches the species charge.

An oxidation number may be an average when several atoms of the same element occupy different environments. A formula-based calculation can then give an average value without fully specifying each atom's local assignment. Introductory examples such as CO₂ and NH₃ avoid that complication, but it is another reason not to equate a formal number automatically with a physical ionic charge.

Step-by-step reasoning

1. Confirm that the formula represents a neutral compound. 2. Write the usual oxidation number for each known element. 3. Multiply each number by that element's subscript. 4. Set the total equal to zero and solve for the unknown. 5. Substitute the answer back to verify the sum and check exceptions.

Visual explanation

Write CO₂ as one C box and two O boxes. Place x in the carbon box and −2 in each oxygen box. Below them write x − 2 − 2 = 0, then x = +4. Beside it write CO with only one −2 box to show why carbon becomes +2 there.

Real-world analogy

For a balanced account, several positive and negative entries add to zero. If all but one entry are known, the last can be calculated. Formula subscripts tell how many entries of each kind exist. Oxidation-number accounting has the same arithmetic, while the chemical rules supply the known entries.

Real-world example

Carbon monoxide can be converted to carbon dioxide in pollution-control chemistry. The two neutral formulas make carbon's formal change easy to see: +2 in CO and +4 in CO₂. A complete process also includes an oxidant, often oxygen, whose oxidation number falls.

Why?

Why is the target sum zero for a neutral molecule? Oxidation numbers distribute electron ownership formally among atoms while preserving the whole species's charge. A neutral species has no net charge, so the signed atomic assignments must sum to zero.

Common misconception

“Because CO₂ has two oxygens, carbon must be +2.” Two oxygens at −2 contribute −4 total. Carbon must be +4 to bring a neutral molecule's sum to zero.

Worked example

Find nitrogen's oxidation number in N₂O₅, treating the two nitrogens by their average assignment. Five oxygens contribute 5(−2) = −10. Let each nitrogen average x; 2x − 10 = 0, so x = +5. Check: 2(+5) + 5(−2) = 0. This is a formal average for the formula, not a claim that two separate N⁵⁺ ions exist.

Quick check

1. If oxygen is −2 in neutral SO₃, what is sulfur's oxidation number? Answer: Sulfur is +6 because one sulfur and three oxygens must sum to zero: +6 + 3(−2) = 0.

Exam focus

Include subscripts in the equation and use the species's total charge as the target sum. For a neutral compound that target is zero. State when a usual assignment has an exception rather than forcing an impossible total.

Advanced insight

The sum rule works for all species, not only simple neutral molecules. Its right-hand side becomes the ion charge for a polyatomic ion. What requires chemical judgment is assigning known atoms, particularly where peroxides or unusual bonding make a common shortcut invalid.

Summary

Oxidation numbers multiplied by atom counts sum to zero for neutral compounds. This rule gives carbon +4 in CO₂, +2 in CO and nitrogen −3 in NH₃ under usual assignments. Check formula subscripts, charge and exceptions before interpreting redox change.

Practice questions

1. Calculate carbon's oxidation number in CO. Answer: +2, because x + (−2) = 0. 2. Calculate sulfur's number in neutral SO₂. Answer: +4, because x + 2(−2) = 0. 3. Calculate nitrogen's number in NH₃. Answer: −3, because x + 3(+1) = 0. 4. Why should SO₄²⁻ not be solved by setting the sum to zero? Answer: It is an ion with overall charge −2, so its oxidation-number sum must equal −2.