Balancing Chemical Equations with Worked Solutions

Counting atoms alone, this reaction has infinitely many solutions: you can add the decomposition 2H₂O₂ = 2H₂O + O₂ in any proportion and the atoms still match. The textbook answer 2KMnO₄ + 5H₂O₂ + 3H₂SO₄ = K₂SO₄ + 2MnSO₄ + 5O₂ + 8H₂O comes from the electron balance: all the oxygen with oxidation number −1 ends up as O₂. The tool shows that answer and says openly that atoms alone allow more.

Write the oxidation number above every element and look for the ones that change. The element that gains electrons (its oxidation number goes down) is reduced, and the substance containing it is the oxidizing agent. The element that loses electrons (its oxidation number goes up) is oxidized, and its substance is the reducing agent. In KMnO₄ + HCl manganese goes from +7 to +2 and gains 5 electrons, so KMnO₄ is the oxidizing agent; chlorine goes from −1 to 0, so HCl is the reducing agent.

Chlorine, oxygen, hydrogen, nitrogen and the other diatomic elements exist as the molecules Cl₂, O₂, H₂ and N₂. So the line takes two atoms at once: 2Cl⁻¹ − 2e⁻ → Cl₂⁰. The multiplier in front of the line then becomes the coefficient of the molecule directly, with no halving later.

When amounts of two reactants are given, one of them is usually in excess and must not be used for the calculation: not all of it reacts. Divide the moles of each reactant by its coefficient in the balanced equation; the smaller result marks the limiting reactant, and the product is calculated from it. For 13 g of zinc (0.2 mol) and 7.3 g of HCl (0.2 mol): zinc gives 0.2 / 1 = 0.2, HCl gives 0.2 / 2 = 0.1, so HCl is limiting and 0.1 mol of hydrogen forms.

22.4 L/mol is the volume of one mole of gas at 0 °C and 1 atm (101.325 kPa), the STP most textbooks use. IUPAC redefined standard pressure as 100 kPa, which gives 22.7 L/mol at 0 °C, and at 25 °C and 100 kPa a mole takes about 24.8 L. Choose the conditions in the settings; the Vm used is always written in the solution.

The rule “oxidation numbers add up to zero” gives +8/3 for iron. A single atom cannot have a fractional oxidation number: Fe₃O₄ is the mixed oxide FeO·Fe₂O₃, with one iron atom at +2 and two at +3, and +8/3 is their average. The tool labels such a value as an average and does not pass it off as a real one.