Balance Fe₂O₃ + C → Fe + CO₂

We have four elements to balance: Fe, O, and C. Let’s count atoms on each side with coefficients all = 1:

Left side: 2 Fe, 3 O, 1 C
Right side: 1 Fe, 2 O (in CO₂), 1 C

Fe is unequal (2 vs 1), O is unequal (3 vs 2), C will need adjustment too. We’ll use the algebraic method (or trial method) to balance all three simultaneously.

Let the balanced equation be:

Write atom balance equations for each element:

• Fe: $2a = c$
• O: $3a = 2d$
• C: $b = d$

We have three equations and four variables — we need to fix one. Start with $a = 2$ (a convenient value to avoid fractions, since oxygen gives $3a = 2d$, and we need $d$ to be a whole number):

From $3a = 2d$: $3(2) = 2d \Rightarrow d = 3$

From $b = d$: $b = 3$

From $2a = c$: $2(2) = c \Rightarrow c = 4$

So: $a = 2, b = 3, c = 4, d = 3$

The balanced equation is:

All elements balance. The equation is correctly balanced.

This is an important metallurgical reaction — it represents the reduction of iron ore (haematite, Fe₂O₃) by carbon (coke) in a blast furnace to produce iron metal.

• Fe₂O₃ is reduced (gains electrons, loses oxygen) → Fe
• C is oxidised (loses electrons, gains oxygen) → CO₂
• This is a redox reaction (also a displacement reaction)

The reaction is used industrially to extract iron from haematite ore. The actual blast furnace process uses CO (carbon monoxide) as the primary reducing agent, but carbon directly reduces iron oxide at high temperatures too.