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Why an equation balances

Atoms are rearranged and never created, so an equation is balanced with coefficients and never by changing a subscript.

Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.

1. What you will learn

By the end of this lesson you will be able to say why a chemical equation has to balance — atoms are rearranged, never created or destroyed — and to check any equation by counting each element on both sides. You will be able to name the one kind of number balancing is allowed to change, explain why changing a subscript makes the equation true of a different reaction, and say why a balanced equation is written in the smallest whole numbers that work.

2. What you already have

You can read a formula as a count of atoms, brackets included, and you have met the difference between a subscript and a number written in front. This lesson is about why that difference matters more than any other distinction in the subject.

3. Words for this lesson

4. Weigh it before, weigh it after

Before this lesson tells you what a reaction does to the atoms in a flask, here is the evidence that settles it, and it is the oldest evidence in chemistry: a balance.

Each reaction below is run twice — once in an open beaker on the pan, and once in a stoppered flask that nothing can get into or out of. The balance is read before and after.

Two accounts are on offer, and the table decides between them.

• Account A: a reaction can make matter and can destroy it. Sometimes you end up with more and sometimes with less, and the first column shows both.
• Account B: nothing is made and nothing is destroyed. When the balance moves, something crossed the boundary of the beaker.

Work through the rows before reading on. Which account explains both columns at once? Under Account B, what must have left the open beaker of marble chips, and what must have joined the burning magnesium — and where would it have come from? And why is the third row, where nothing changes in either column, worth putting in a table meant to show a difference?

Then say what the stoppered column is evidence for, in one sentence, and hold on to it: the rest of the lesson is about what follows from that sentence when you write the reaction down.

5. Atoms are rearranged, never created

A chemical reaction breaks bonds and makes new ones. It does not make atoms and it does not destroy them. Every carbon atom that goes into a flask is still a carbon atom when the reaction stops, whatever it is now attached to.

That one fact is what a balanced equation records, and it is why balancing is not a formatting rule. An equation with four hydrogens on the left and two on the right is a claim that two hydrogen atoms stopped existing, and that claim is false.

So balancing is a search for numbers that make the counts agree — and there is only one kind of number you are allowed to search over.

• A subscript says what the substance is. $\mathrm{H_2O}$ and $\mathrm{H_2O_2}$ differ by one subscript and are water and a bleaching agent. Changing a subscript does not adjust the equation; it replaces one of the substances in it.
• A coefficient says how much of it there is. $2\mathrm{H_2O}$ is still water, two units of it. You may choose this number freely, and it is the only one you may choose.

And then a third rule, which is about tidiness rather than truth: if a set of coefficients balances, so does twice that set, and three times, forever. The convention is the smallest whole numbers that work.

6. The method, step by step, and how to check it

List the elements. Every element that appears anywhere in the equation.

Count each element on the left. Subscript times coefficient, added over every formula on that side.

Count each element on the right. The same way.

Compare. Every element must agree. If one does not, change a coefficient — never a subscript — and count again.

Reduce to lowest terms. If all the coefficients share a factor, divide it out.

Check the work. Did every coefficient multiply every atom in its formula, not just the first? Did an element that appears in two formulas on a side get both counts added? Are the formulas exactly the ones you were given, with no subscript changed? And in an experiment, did anything cross the boundary of the container — a gas escaping, air joining — that would make the balance reading move?

7. Why each step is allowed

Requiring every element to agree is allowed because atoms are neither created nor destroyed in a chemical reaction. The stoppered flask shows it: whatever happens inside, the balance reading does not move.

Changing only coefficients is allowed because a coefficient counts how many units react, and the amounts are exactly what balancing has to find. A subscript is part of the identity of a substance, fixed before the equation was written.

Reducing to lowest terms is a convention, not a law, but a useful one. Every multiple of a balanced set is also balanced, so without the convention two correct answers could look different.

Explaining a change in an open beaker by matter crossing the boundary is allowed because the stoppered column rules out the alternative. If reactions could create or destroy matter, the stoppered flask would change too, and it never does.

8. What the counting looks like

Take $\mathrm{2H_2 + O_2 \rightarrow 2H_2O}$.

Two things in that table are worth saying out loud. First, the coefficient multiplies every atom in the formula it stands in front of: the 2 in $2\mathrm{H_2O}$ doubles the oxygen as well as the hydrogens. Second, nothing in the table is a mass. A balanced equation is a statement about counting, and the fact that mass also comes out equal is a consequence of the counting rather than the thing being checked.

The equation $4\mathrm{H_2} + 2\mathrm{O_2} \rightarrow 4\mathrm{H_2O}$ passes the same table. It is a true statement and it is not the answer, because every coefficient can be halved.

9. Reading the open beaker

The open-beaker column is not a failure of conservation; it is conservation with an unwatched door. When marble chips react with acid, carbon dioxide gas bubbles out and leaves the beaker: the 2.2 g that the balance lost is the carbon dioxide, now in the room. When magnesium burns, it takes oxygen from the air into the solid: the 1.60 g it gained is that oxygen. When copper sulfate crystals are heated, water is driven off as steam.

In every case the arithmetic tells you how much crossed the boundary, and the chemistry tells you what it was. That is how a chemist uses conservation in practice: a change of mass in an open container is a measurement of the gas that came or went.

10. In the world: a cement plant's carbon dioxide

Making cement is one of the largest sources of carbon dioxide in the United States, and conservation of mass is how plants in Texas, Missouri and California account for it. The heart of the process is heating limestone, calcium carbonate, in a kiln: $\mathrm{CaCO_3 \rightarrow CaO + CO_2}$. The equation balances — one calcium, one carbon and three oxygens on each side — and the masses follow: 100 tons of calcium carbonate give 56 tons of calcium oxide and 44 tons of carbon dioxide.

The kiln is an open beaker on an enormous scale. The calcium oxide comes out as clinker, which is weighed; the carbon dioxide goes up the stack. Conservation says the difference between the limestone fed in and the clinker coming out is the gas released, so a plant that knows its tonnage of limestone knows its carbon dioxide emissions without measuring the stack at all.

That is exactly how the Environmental Protection Agency's reporting rules let cement plants calculate this part of their emissions: from the mass of carbonate fed to the kiln, multiplied by the 44-to-100 ratio the balanced equation fixes. A plant processing a million tons of limestone a year releases about 440,000 tons of carbon dioxide from the chemistry alone, before any fuel is burned.

11. In the world: why a burned log seems to vanish

A log in a Vermont wood stove leaves only a little ash, but nothing has been destroyed. The carbon and hydrogen left up the chimney as carbon dioxide and water vapor, joined by oxygen from the air; weighed all together, the products outweigh the log.

12. The move that always works and is always wrong

A subscript says what the substance is; a coefficient in front says how much of it there is. Changing a subscript to make an equation balance does balance it, and the equation is then about a different substance — which is why it is the one move balancing never allows.

It is worth seeing it happen. A student trying to balance $\mathrm{H_2 + O_2 \rightarrow H_2O}$ notices two oxygens on the left and one on the right, and writes $\mathrm{H_2 + O_2 \rightarrow H_2O_2}$. Both sides now have two hydrogens and two oxygens. The equation balances perfectly, and it says that burning hydrogen makes hydrogen peroxide, which it does not.

The reason this mistake survives is that nothing about the page objects. There is no red ink, no leftover atom, no contradiction to trip over. The only defense is the rule itself: if a change is to a subscript, it is not balancing.

The second common error is quieter. A learner balances an equation correctly and leaves it as $\mathrm{4H_2 + 2O_2 \rightarrow 4H_2O}$. Nothing is false about it. It is simply not the smallest set, and the convention exists so that two people who balance the same equation write the same thing down.

A third is to read an open beaker's change of mass as matter created or destroyed. It is matter that crossed the boundary, and the stoppered flask is the proof.

13. Balancing by counting, one element at a time

1. Count each element.

   $\mathrm{Mg + O_2 \rightarrow MgO}: \text{O } 2 \text{ against } 1$

   Find the element that disagrees.

2. Double the magnesium oxide.

   $\mathrm{Mg + O_2 \rightarrow 2MgO}$

   Oxygen is now two and two.

3. Recount the magnesium.

   $1 \text{ against } 2$

   Fixing one element usually breaks another.

4. Double the magnesium.

   $\mathrm{2Mg + O_2 \rightarrow 2MgO}$

   Magnesium two and two.

5. Check for a common factor.

   $2, 1, 2: \text{ none}$

   Balanced, and in lowest terms.

14. Why the counting has to come before the mass

1. Read the balanced equation.

   $\mathrm{2Mg + O_2 \rightarrow 2MgO}$

   Two magnesium atoms take one oxygen molecule.

2. Turn the counts into grams.

   $48 + 32 = 80 \text{ g}$

   Magnesium and oxygen make the oxide.

3. Check the masses balance.

   $80 = 80$

   As conservation requires.

4. Try going backwards from mass.

   $48 : 32$

   Not an obvious ratio of atoms.

5. Note what the equation is.

   $2, \ 1, \ 2$

   Counts, not masses.

6. Draw the rule.

   $\text{count first; weigh afterwards}$

   Mass is a consequence, not a route.

15. Reading the open beaker of marble chips

1. Read the two balance readings.

   $50.0 \to 47.8 \text{ g}$

   The open beaker lost mass.

2. Find the mass that left.

   $50.0 - 47.8 = 2.2 \text{ g}$

   Before less after.

3. Name what left.

   $\text{carbon dioxide gas}$

   It bubbled out of the acid.

4. Check with the stoppered flask.

   $50.0 \to 50.0 \text{ g}$

   With the gas trapped, nothing changes.

5. Rule out creation or destruction.

   $\text{the closed reading never moves}$

   Matter only crossed the boundary.

6. Write the balanced equation.

   $\mathrm{CaCO_3 + 2HCl \rightarrow CaCl_2 + H_2O + CO_2}$

   Every atom accounted for.

7. Link the two.

   $\text{the } 2.2 \text{ g is the } \mathrm{CO_2}$

   Conservation turns a mass loss into a measurement.

16. Your turn: balance $\mathrm{N_2 + H_2 \rightarrow NH_3}$.

1. Balance the nitrogen.

   $\mathrm{N_2 + H_2 \rightarrow 2NH_3}$

   Start with the element in one formula on each side.

2. Balance the hydrogen.

   $6 \text{ on the right} \Rightarrow 3\mathrm{H_2}$

   Three units of hydrogen are needed.

3. Your turn: work this step out. Its working is at the end of the packet.

   Write and check the equation.

17. Guided practice

A student balancing the equation for sulfuric acid neutralized by sodium hydroxide cannot make one of the elements agree, so they change a subscript inside one of the formulas. Both sides now have the same number of every atom. What is wrong with what they have done?

• One of the formulas is now a different substance from the one that reacted.
• Nothing is wrong. The two sides have the same atoms, so the equation is balanced.
• The coefficients are no longer in their lowest terms.

18. Guided practice

Complete the worked solution: $200$ g of calcium carbonate is heated in an open dish until only $112$ g of calcium oxide is left, because carbon dioxide gas escaped. Find the mass of carbon dioxide that left, and that mass as a percentage of the calcium carbonate.

1. Find the gas that left.

   $(\text{calcium carbonate}) - (\text{calcium oxide}) =$ _g g

   Nothing was destroyed; it crossed the edge of the dish.

2. Find the gas as a percentage.

   $(\text{gas}) \div (\text{calcium carbonate}) \times \text{a hundred} =$ _p

   A share of the starting mass.

3. Say what a stoppered flask would show.

   $\text{no change in mass}$

   The gas would stay inside and still be weighed.

19. Guided practice

The left-hand side of the equation for burning methane is already written. Conservation decides the two numbers on the right — you do not get to choose them. Fill them in.

$1\mathrm{CH_4} + 2\mathrm{O_2} \rightarrow$ _a $\mathrm{CO_2} \ + \ $ _b $\mathrm{H_2O}$

20. Practice

Here is sodium nitrate heated in a fertilizer works. One substance breaks into two, so every atom that leaves the first formula has to arrive in one of the other two. Put a whole number in front of each formula so that it does.

This task has no paper form; do it on a device.

21. Practice

In a welding class at a Cleveland community college, students burn $1.92$ g of magnesium ribbon in an open crucible and weigh $3.20$ g of white magnesium oxide afterwards. How many grams of oxygen from the air joined the metal?

The answer: _a.

22. Somewhere new

This one has two substances going in and two coming out, so no single formula fixes an element on its own. It is the same rule: every atom that goes in comes out. Balance the equation for zinc reacting with hydrochloric acid.

This task has no paper form; do it on a device.

23. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

24. Test question

Burning ethene in a flare at a chemical works is written $\mathrm{C_2H_4 + 3O_2 \rightarrow 2CO_2 + 2H_2O}$. Audit it: count the atoms of each element on each side. If the equation is a true statement, the two columns must agree row by row.

25. What you can do now

You can audit an equation element by element, and you know that only the coefficients may change. Tell someone why balancing hydrogen and oxygen by writing hydrogen peroxide is not allowed, even though both sides then have the same atoms. Next: the order to balance in, so that you are not chasing the same element around the equation.

Working for the steps left to you

16. Your turn: balance $\mathrm{N_2 + H_2 \rightarrow NH_3}$., step 3