Question

Balance this chemical equation by inspection: \(\mathrm{Fe}_{2} \mathrm{O}_{3}+\mathrm{C} \rightarrow \mathrm{Fe}+\mathrm{CO}_{2}\)

Short answer

The balanced chemical equation is: \[3\mathrm{C} + 2\mathrm{Fe}_{2} \mathrm{O}_{3} \rightarrow 4\mathrm{Fe} + 3\mathrm{CO}_{2}\]

Step-by-step solution

Identify the reactants and products in the equation

In this exercise, the given chemical equation is \[\mathrm{Fe}_{2} \mathrm{O}_{3}+\mathrm{C} \rightarrow \mathrm{Fe}+\mathrm{CO}_{2}.\] The reactants are \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) (iron(III) oxide) and \(\mathrm{C}\) (carbon), and the products are \(\mathrm{Fe}\) (iron) and \(\mathrm{CO}_{2}\) (carbon dioxide).

Count the number of atoms for each element on both sides of the equation

For this step, let's make a table to keep track of the number of atoms for each element. Reactant side: \[\mathrm{Fe}: 2 \quad \mathrm{O}: 3 \quad \mathrm{C}: 1\] Product side: \[\mathrm{Fe}: 1 \quad \mathrm{O}: 2 \quad \mathrm{C}: 1\]

Balance the atoms of each element

Now, let's balance the atoms one at a time. Begin with the element that appears the fewest times in the equation, which is iron (Fe). Reactant side: \[\mathrm{Fe}: 2 \quad \mathrm{O}: 3 \quad \mathrm{C}: 1\] Product side: \[\mathrm{Fe}: 2 \quad \mathrm{O}: 2 \quad \mathrm{C}: 1\] We can achieve balancing iron by multiplying iron (Fe) on the product side by 2. Now, the equation looks like this: \[\mathrm{Fe}_{2} \mathrm{O}_{3}+\mathrm{C} \rightarrow 2\mathrm{Fe}+\mathrm{CO}_{2}.\] Next, let's move on to balance the atoms of oxygen (O). To balance the oxygen atoms, we can multiply carbon dioxide (CO2) by 3: \[\mathrm{Fe}_{2} \mathrm{O}_{3}+\mathrm{C} \rightarrow 2\mathrm{Fe}+3\mathrm{CO}_{2}.\] Reactant side: \[\mathrm{Fe}: 2 \quad \mathrm{O}: 3 \quad \mathrm{C}: 1\] Product side: \[\mathrm{Fe}: 2 \quad \mathrm{O}: 6 \quad \mathrm{C}: 3\] Finally, let's balance the atoms of carbon (C). We can see that there are 3 carbon atoms on the product side and only 1 on the reactant side. So, we can multiply the carbon atom (C) on the reactant side by 3: \[3(\mathrm{Fe}_{2} \mathrm{O}_{3}+\mathrm{C}) \rightarrow 2\mathrm{Fe}+3\mathrm{CO}_{2}.\] Reactant side: \[\mathrm{Fe}: 2 \quad \mathrm{O}: 3 \quad \mathrm{C}: 3\] Product side: \[\mathrm{Fe}: 2 \quad \mathrm{O}: 6 \quad \mathrm{C}: 3\] The equation is now balanced.

Write down the balanced equation

The balanced chemical equation is: \[3\mathrm{C} + 2\mathrm{Fe}_{2} \mathrm{O}_{3} \rightarrow 4\mathrm{Fe} + 3\mathrm{CO}_{2}\]

Key concepts

Reactants and Products

In any chemical equation, identifying the reactants and products is the first step toward understanding the transformation taking place. Reactants are the starting materials that undergo a chemical change, while products are the substances formed as a result of this change. In our example equation, \(\mathrm{Fe}_{2} \mathrm{O}_{3} + \mathrm{C} \rightarrow \mathrm{Fe} + \mathrm{CO}_{2}\), the reactants are iron(III) oxide (\(\mathrm{Fe}_{2} \mathrm{O}_{3}\)) and carbon (\(\mathrm{C}\)). These are the substances that will interact with each other.

Once the reaction occurs, these reactants transform into the products: iron (\(\mathrm{Fe}\)) and carbon dioxide (\(\mathrm{CO}_{2}\)). Identifying reactants and products is crucial as it helps us determine the changes that occur at the atomic level during the reaction.

Conservation of Mass

The law of conservation of mass states that matter is neither created nor destroyed in a chemical reaction. This means the mass of the reactants must equal the mass of the products. Therefore, each type of atom present must be balanced on both sides of the chemical equation.

In our example, this principle ensures that the same number of iron, oxygen, and carbon atoms exist in the reactants as in the products. Initially, we have 2 iron atoms, 3 oxygen atoms, and 1 carbon atom on the reactant side. For the products, we need the same counts of these atoms, even though they are arranged differently in new compound formats.

• Iron atoms: 2 in \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) must match with 2 in \(\mathrm{Fe}\).
• Oxygen atoms: 3 in \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) must spread across products, with necessary adjustments.
• Carbon atoms: Adjusted to match between \(\mathrm{C}\) and \(\mathrm{CO}_{2}\).

By balancing the equation, we respect this fundamental law, ensuring mass is conserved.

Atom Counting

Counting atoms is a pivotal step in balancing chemical equations. It involves assessing how many atoms of each element appear on both sides of the reaction arrow. This is similar to taking inventory before and after a chemical change. Let's use our example:
- The reactant side initially has 2 iron, 3 oxygen, and 1 carbon atoms. - Compare this with the initial product side, which has 1 iron, 2 oxygen, and 1 carbon atom.

To balance, we adjust coefficients in front of compounds and elements. For instance, we doubled the iron in the products and tripled the carbon dioxide to have a balanced number of oxygen atoms. Following this carefully ensures that we match equal numbers on both sides, corresponding to the law of conservation of mass.

This verification process of atom counting is iterative, often requiring multiple adjustments.

Chemical Equation

A chemical equation is a symbolic representation of a chemical reaction. It shows the types and numbers of atoms or molecules involved, as well as the way they transform from reactants to products. The format typically used is Reactants \(\rightarrow\) Products.

In our exercise, \(\mathrm{Fe}_{2} \mathrm{O}_{3} + \mathrm{C} \rightarrow \mathrm{Fe} + \mathrm{CO}_{2}\) initially stands for the reaction without balancing. By adjusting this equation to \(3\mathrm{C} + 2\mathrm{Fe}_{2} \mathrm{O}_{3} \rightarrow 4\mathrm{Fe} + 3\mathrm{CO}_{2}\), we ensured it properly represents the chemical process.

• Symbols stand for individual elements or compounds involved.
• The plus sign (+) separates different reactants or products.
• The arrow (\(\rightarrow\)) signifies the direction of the reaction.

A balanced chemical equation not only shows the transformation but also conveys stoichiometric relationships between substances. Thus, it's central to understanding chemical reactions.