Problem 181 - Introductory Chemistry Atoms First

Question

A mixture of methane \(\left(\mathrm{CH}_{4}\right)\) and ethane \(\left(\mathrm{C}_{2} \mathrm{H}_{6}\right)\) is \(25.6 \%\) methane by mass. When \(1.50 \mathrm{~g}\) of this mixture is burned, what is the mass of carbon dioxide produced?

Step-by-Step Solution

Answer

The mass of CO₂ produced when 1.50 g of a mixture containing 25.6% methane by mass and the rest ethane is burned can be found by following these steps: 1. Calculate the mass of each component in the mixture: Mass of CH₄ = \((25.6/100) \times 1.50 \mathrm{~g}\) and Mass of C₂H₆ = \(1.50 \mathrm{~g}\) - Mass of CH₄ 2. Convert the mass of each component to moles: Moles of CH₄ = \(\dfrac{\text{Mass of CH₄}}{16.04 \mathrm{ g/mol}}\) and Moles of C₂H₆ = \(\dfrac{\text{Mass of C₂H₆}}{30.07 \mathrm{ g/mol}}\) 3. Determine the moles of CO₂ produced from both CH₄ and C₂H₆: Moles of CO₂ from CH₄ = Moles of CH₄, Moles of CO₂ from C₂H₆ = 2 × Moles of C₂H₆ 4. Calculate the mass of CO₂ produced: Mass of CO₂ = Total moles of CO₂ × 44.01 g/mol, where Total moles of CO₂ = Moles of CO₂ from CH₄ + Moles of CO₂ from C₂H₆

Step 1: Calculate the mass of each component in the 1.50 g mixture

The mixture is 25.6% methane by mass. Therefore, the mass of methane in the 1.50 g mixture is: Mass of CH₄ = \((25.6/100) \times 1.50 \mathrm{~g}\) Since the rest of the mixture is ethane (due to the fact that it only consists of CH₄ and C₂H₆) Mass of C₂H₆ = \(1.50 \mathrm{~g}\) - Mass of CH₄

Step 2: Calculate the moles of each component in the mixture

Next, we need to convert the mass of each component to moles. The molecular weights of CH₄ and C₂H₆ are 16.04 g/mol and 30.07 g/mol, respectively. Moles of CH₄ = \(\dfrac{\text{Mass of CH₄}}{16.04 \mathrm{ g/mol}}\) Moles of C₂H₆ = \(\dfrac{\text{Mass of C₂H₆}}{30.07 \mathrm{ g/mol}}\)

Step 3: Use stoichiometry to calculate the moles of CO₂ produced

Now, we'll use stoichiometry to determine the moles of CO₂ produced when each component is burned. For methane: CH₄ + 2O₂ → CO₂ + 2H₂O This means 1 mole of CH₄ will produce 1 mole of CO₂. Moles of CO₂ produced from CH₄ = Moles of CH₄ For ethane: C₂H₆ + O₂ → 2CO₂ + 3H₂O + O₂ This means 1 mole of C₂H₆ will produce 2 moles of CO₂. Moles of CO₂ produced from C₂H₆ = 2 × Moles of C₂H₆

Step 4: Determine the mass of CO₂ produced

To determine the mass of CO₂ produced, we first need to find the total moles of CO₂ produced by both CH₄ and C₂H₆. The total moles of CO₂ produced is: Total moles of CO₂ = Moles of CO₂ from CH₄ + Moles of CO₂ from C₂H₆ The molecular weight of CO₂ is 44.01 g/mol. We can now determine the mass of CO₂ produced: Mass of CO₂ = Total moles of CO₂ × 44.01 g/mol Once we have calculated the mass of CO₂ produced, the problem is solved.

Key Concepts

Methane

Methane, or \( ext{CH}_4\), is a simple alkane and the main component of natural gas. Often referred to as marsh gas, it's a colorless, odorless gas at standard temperature and pressure. Understanding methane's role in chemical reactions, such as combustion, is important in stoichiometry.
When methane burns in oxygen, it reacts to form carbon dioxide and water. The balanced chemical equation for this process is:

• \( ext{CH}_4 + 2 ext{O}_2 \rightarrow ext{CO}_2 + 2 ext{H}_2 ext{O}\)

This equation indicates that one mole of methane reacts with two moles of oxygen to produce one mole of carbon dioxide and two moles of water. So, when calculating the amount of carbon dioxide produced when a specific amount of methane is burned, it's straightforward to see the 1:1 ratio between methane and carbon dioxide. Methane’s combustion is a key process in energy production and contributes to both heating homes and generating electricity.

Ethane

Ethane, \( ext{C}_2 ext{H}_6\), is another simple alkane present in natural gas, typically second to methane in quantity. It's colorless, odorless, and flammable, playing a significant role in both the chemical industry and as a fuel source.
When ethane combusts, it combines with oxygen to form carbon dioxide and water. The balanced chemical equation is:

• \( ext{C}_2 ext{H}_6 + rac{7}{2} ext{O}_2 \rightarrow 2 ext{CO}_2 + 3 ext{H}_2 ext{O}\)

This equation demonstrates that one mole of ethane yields two moles of carbon dioxide. Therefore, for stoichiometry calculations, it's crucial to remember this 1:2 relationship between ethane and carbon dioxide. Understanding these types of reactions helps explain how different hydrocarbon fuels produce varying amounts of carbon dioxide when burned, impacting energy output and emissions.

Carbon Dioxide

Carbon dioxide, or \( ext{CO}_2\), is a colorless and odorless gas inherent to combustion reactions involving hydrocarbons. Its molecular weight is 44.01 g/mol, a fact that's key when calculating the mass of \( ext{CO}_2\) produced in chemical reactions like burning methane and ethane.
In stoichiometry, carbon dioxide production is often a primary focus, particularly because \( ext{CO}_2\) emissions are a major concern in environmental science due to their role in climate change and global warming. Calculations involving \( ext{CO}_2\) help in determining the efficiency and impact of different fuels.
For example, in the combustion equations provided earlier:

• For methane: one mole of \( ext{CH}_4\) produces one mole of \( ext{CO}_2\).
• For ethane: one mole of \( ext{C}_2 ext{H}_6\) produces two moles of \( ext{CO}_2\).

These reactions underline the importance of calculating precise \( ext{CO}_2\) outputs when fuel mixtures are burned, helping to assess their environmental impact.