Calculating Oxygen Mass for Carbon Combustion

The question asks for the mass of oxygen required to react completely with 1 kg of carbon to produce \(\frac{11}{3}\) kg of carbon dioxide (\(\text{CO}_2\)). This is a stoichiometry problem based on the balanced chemical equation for the combustion of carbon.

Chemical Reaction

Carbon reacts with oxygen to form carbon dioxide according to the following balanced chemical equation:

\( \text{C} + \text{O}_2 \rightarrow \text{CO}_2 \)

Molar Masses of Reactants and Products

To understand the mass relationships in the reaction, we need the molar masses of carbon (C), oxygen (\(\text{O}_2\)), and carbon dioxide (\(\text{CO}_2\)).

• Molar mass of Carbon (C) = 12 g/mol
• Molar mass of Oxygen (\(\text{O}_2\)) = 2 \(\times\) 16 g/mol = 32 g/mol
• Molar mass of Carbon Dioxide (\(\text{CO}_2\)) = Molar mass of C + 2 \(\times\) Molar mass of O = 12 g/mol + 2 \(\times\) 16 g/mol = 12 + 32 = 44 g/mol

Stoichiometry by Mass

The balanced equation tells us that 1 mole of carbon reacts with 1 mole of oxygen gas to produce 1 mole of carbon dioxide. Using the molar masses, we can find the mass ratio:

1 mole C (12 g) reacts with 1 mole \(\text{O}_2\) (32 g) to produce 1 mole \(\text{CO}_2\) (44 g).

The mass ratio of C : \(\text{O}_2\) : \(\text{CO}_2\) is 12 : 32 : 44.

This ratio can be simplified by dividing each number by 4:

\( \frac{12}{4} : \frac{32}{4} : \frac{44}{4} = 3 : 8 : 11 \)

This means that for every 3 parts by mass of carbon, 8 parts by mass of oxygen are required to produce 11 parts by mass of carbon dioxide.

Calculating Required Oxygen Mass

We are given that 1 kg of carbon reacts to produce \(\frac{11}{3}\) kg of \(\text{CO}_2\). Let's verify if this fits the 3:11 carbon to carbon dioxide mass ratio:

• Ratio of C : \(\text{CO}_2\) by mass is 3 : 11.
• Given mass of C = 1 kg.
• Expected mass of \(\text{CO}_2\) = \( (\text{Ratio of } \text{CO}_2 / \text{Ratio of C}) \times \text{Mass of C} \)
• Expected mass of \(\text{CO}_2\) = \( (\frac{11}{3}) \times 1 \text{ kg} = \frac{11}{3} \text{ kg} \).

This matches the given information, confirming the reaction proceeds as expected based on stoichiometry.

Now, we need to find the mass of oxygen required for 1 kg of carbon using the carbon to oxygen mass ratio, which is 3:8.

• Ratio of C : \(\text{O}_2\) by mass is 3 : 8.
• Given mass of C = 1 kg.
• Required mass of \(\text{O}_2\) = \( (\text{Ratio of } \text{O}_2 / \text{Ratio of C}) \times \text{Mass of C} \)
• Required mass of \(\text{O}_2\) = \( (\frac{8}{3}) \times 1 \text{ kg} = \frac{8}{3} \text{ kg} \).

Therefore, the required mass of oxygen to convert 1 kg of carbon into \(\frac{11}{3}\) kg of \(\text{CO}_2\) is \(\frac{8}{3}\) kg.

Summary of Mass Relationships

The calculation confirms that \(\frac{8}{3}\) kg of oxygen is needed.