Problem 91 - Chemistry Matter and Change

Question

Gaseous methane \(\left(\mathrm{CH}_{4}\right)\) undergoes complete combustion by reacting with oxygen gas to form carbon dioxide and water vapor. a. Write a balanced equation for this reaction. b. What is the volume ratio of methane to water in this reaction?

Step-by-Step Solution

Answer

The balanced equation is \( \text{CH}_4 + 2\text{O}_2 \rightarrow \text{CO}_2 + 2\text{H}_2\text{O} \). The volume ratio of methane to water is 1:2.

Step 1: Write the Unbalanced Equation

In chemical reactions, we must first write down the reactants and products. Methane (\(\text{CH}_4\)) reacts with oxygen gas (\(\text{O}_2\)) to form carbon dioxide (\(\text{CO}_2\)) and water (\(\text{H}_2\text{O}\)). The unbalanced equation is: \[ \text{CH}_4 + \text{O}_2 \rightarrow \text{CO}_2 + \text{H}_2\text{O} \]

Step 2: Balance the Equation

To balance the equation, adjust the coefficients to have the same number of each type of atom on both sides. Start by balancing the carbon atoms, then hydrogen, and finally oxygen. The balanced equation is: \[ \text{CH}_4 + 2\text{O}_2 \rightarrow \text{CO}_2 + 2\text{H}_2\text{O} \] This means one molecule of methane reacts with two molecules of oxygen to produce one molecule of carbon dioxide and two molecules of water.

Step 3: Determine Volume Ratio of Methane to Water

In a balanced chemical equation, coefficients represent the molar ratio of the reactants and products. Assuming ideal gas behavior and constant temperature and pressure, this also represents the volume ratio. From the balanced equation, one volume of methane (\(\text{CH}_4\)) yields two volumes of water (\(\text{H}_2\text{O}\)). Thus, the volume ratio is 1:2.

Key Concepts

Chemical Equation Balancing

Balancing a chemical equation is crucial because it reflects the conservation of mass. This means that the number of atoms for every element must be the same on both sides of the equation. For the combustion of methane (\[ \text{CH}_4 + \text{O}_2 \rightarrow \text{CO}_2 + \text{H}_2\text{O} \] ), begin by identifying all present elements: carbon (C), hydrogen (H), and oxygen (O). - Start by balancing one element at a time, preferably beginning with the least common atom. For this reaction, start with carbon.- Balance carbon by ensuring the number of carbon atoms in methane equals those in carbon dioxide: 1 methane = 1 carbon dioxide.- Next, balance hydrogen. Methane has 4 hydrogens, and water has 2 hydrogens per molecule, so you need two water molecules.- Finally, balance the oxygen atoms last. Calculate how many molecules of O₂ are needed to reach the total number of oxygen atoms used in CO₂ and H₂O.After these steps, the equation becomes balanced: \[ \text{CH}_4 + 2\text{O}_2 \rightarrow \text{CO}_2 + 2\text{H}_2\text{O} \] Each type of atom now coexists in equal numbers on both sides of the equation.

Volume Ratio

Understanding the volume ratio in the context of a chemical reaction involves recognizing the role played by the balanced equation. The coefficients in a balanced chemical equation not only tell us the molar ratio of reactants and products involved but, under certain conditions, such as ideal gas behavior, they also tell us the volume ratio. In our balanced equation for methane combustion, \[ \text{CH}_4 + 2\text{O}_2 \rightarrow \text{CO}_2 + 2\text{H}_2\text{O} \] we see that one mole (or volume) of methane produces two moles (or volumes) of water vapor. Under the assumption that gases behave ideally, this yields a simple volume ratio of 1:2 for methane to water. This means for every liter of methane burned, two liters of water vapor are generated, assuming the reaction occurs at a constant temperature and pressure. These conditions allow the use of molar and volume ratios interchangeably.

Ideal Gas Law

The ideal gas law is a key concept in understanding how gases behave under various conditions, particularly in chemical reactions. This law is often expressed with the equation:\[ PV = nRT \] where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume of the gas.
• \(n\) is the number of moles of the gas.
• \(R\) is the ideal gas constant.
• \(T\) is the temperature in Kelvin.

In the scope of chemical reactions and volume ratios, the behavior of gases as described by the ideal gas law supports why volume ratios derived from the balanced equation are accurate. Gases are assumed to expand and contract proportionally as temperatures and pressures are maintained uniformly. Because of this, the ideal gas law justifies why volume ratios can directly represent moles for gaseous reactants and products in a balanced equation, simplifying the calculation of reactant and product volumes in reactions such as the combustion of methane.