Balancing Chemical Equations: The $\text{Fe}_2\text{O}_3$ and $\text{CO}$ Reaction

Balancing a chemical equation ensures that the law of conservation of mass is obeyed. This means the number of atoms of each element must be the same on both the reactant side and the product side of the equation. We are given the unbalanced equation:

$\text{aFe}_2\text{O}_3\text{(s)} + \text{bCO(g)} \rightarrow \text{cFe (s)} + \text{dCO}_2\text{(g)}$

Here, a, b, c, and d are the stoichiometric coefficients we need to find to balance the equation.

Steps to Balance the Chemical Equation

Let's count the number of atoms of each element on both sides of the equation:

• Reactant Side:
• Iron (Fe): 2 atoms in $\text{Fe}_2\text{O}_3$, multiplied by coefficient a. Total Fe atoms: $2a$.
• Oxygen (O): 3 atoms in $\text{Fe}_2\text{O}_3$ (multiplied by a) plus 1 atom in CO (multiplied by b). Total O atoms: $3a + b$.
• Carbon (C): 1 atom in CO, multiplied by coefficient b. Total C atoms: $b$.

• Product Side:
• Iron (Fe): 1 atom in Fe, multiplied by coefficient c. Total Fe atoms: $c$.
• Oxygen (O): 2 atoms in $\text{CO}_2$, multiplied by coefficient d. Total O atoms: $2d$.
• Carbon (C): 1 atom in $\text{CO}_2$, multiplied by coefficient d. Total C atoms: $d$.

Setting Up Equations

To balance the equation, the number of atoms of each element must be equal on both sides:

• For Iron (Fe): $2a = c$
• For Oxygen (O): $3a + b = 2d$
• For Carbon (C): $b = d$

Solving for the Coefficients (a, b, c, d)

We have a system of equations. Let's use the simplest relationship first, which is $b = d$.

Now substitute $d$ with $b$ in the oxygen equation:

$3a + b = 2(b)$

$3a + b = 2b$

Subtract $b$ from both sides:

$3a = 2b - b$

$3a = b$

So, we have the relationships: $b = 3a$ and $c = 2a$, and $d = b$. Since $b=3a$, we also have $d=3a$.

To find the smallest whole number coefficients, we can choose a simple value for one variable. Let's assume $a=1$.

• If $a=1$:
• $b = 3a = 3 \times 1 = 3$
• $c = 2a = 2 \times 1 = 2$
• $d = b = 3$

So, the coefficients are $a=1$, $b=3$, $c=2$, and $d=3$.

Verifying the Balanced Equation

Let's plug these values back into the equation:

$1\text{Fe}_2\text{O}_3\text{(s)} + 3\text{CO(g)} \rightarrow 2\text{Fe (s)} + 3\text{CO}_2\text{(g)}$

Let's check the atom counts:

All elements are balanced. The coefficients a, b, c, and d are 1, 3, 2, and 3 respectively.

Coefficients of the Balanced Equation

The values for a, b, c, and d are 1, 3, 2, and 3.

Revision Table: Key Concepts in Balancing Equations

Additional Information: Chemical Reactions

The reaction between iron(III) oxide ($\text{Fe}_2\text{O}_3$) and carbon monoxide ($\text{CO}$) is a common example of a redox (reduction-oxidation) reaction, specifically a reduction reaction used in the production of iron from its ore in a blast furnace. $\text{Fe}_2\text{O}_3$ is reduced to Fe, and $\text{CO}$ is oxidized to $\text{CO}_2$. Understanding how to balance these chemical equations is fundamental to calculating quantities of reactants and products in chemistry, a topic known as stoichiometry.

The state symbols (s) for solid and (g) for gas indicate the physical state of the substances under the reaction conditions. Including these symbols makes the chemical equation more informative.