Question

As early as \(1938,\) the use of NaOH was suggested as a means of removing \(\mathrm{CO}_{2}\) from the cabin of a spacecraft according to the following (unbalanced) reaction: \(\mathrm{NaOH}+\mathrm{CO}_{2} \longrightarrow \mathrm{Na}_{2} \mathrm{CO}_{3}+\mathrm{H}_{2} \mathrm{O}\) . a. If the average human body discharges 925.0 \(\mathrm{gCO}_{2}\) per day, how many moles of NaOH are needed each day for each person in the spacecraft? b. How many moles of each product are formed?

Short answer

42.04 moles of NaOH are needed. 21.02 moles of Na\textsubscript{2}CO\textsubscript{3} and 21.02 moles of H\textsubscript{2}O are formed.

Step-by-step solution

Write the Balanced Equation

First, balance the chemical equation: \[2\text{NaOH} + \text{CO}_2 \rightarrow \text{Na}_2\text{CO}_3 + \text{H}_2\text{O}\]

Calculate Moles of CO\textsubscript{2}

Convert the mass of \(\text{CO}_2\) to moles using its molar mass (44.01 g/mol): \[\text{Moles of CO}_2 = \frac{925.0 \text{ g}}{44.01 \text{ g/mol}} = 21.02 \text{ moles}\]

Determine Moles of NaOH

Using the balanced equation, note that 2 moles of NaOH react with 1 mole of CO\textsubscript{2}. Thus, \[\text{Moles of NaOH} = 2 \times 21.02 = 42.04 \text{ moles}\]

Calculate Moles of Each Product

The balanced equation shows that 1 mole of Na\textsubscript{2}CO\textsubscript{3} and 1 mole of H\textsubscript{2}O are produced for each mole of CO\textsubscript{2}. Therefore, \[\text{Moles of Na}_2\text{CO}_3 = 21.02 \text{ moles}\] and \[\text{Moles of H}_2\text{O} = 21.02 \text{ moles}\]

Key concepts

Molar Mass Calculation

To solve the given exercise, the first crucial step involves understanding molar mass calculation. Molar mass is the mass of one mole of a substance, typically measured in grams per mole (g/mol). For example, the molar mass of CO\textsubscript{2} is calculated by summing the atomic masses of one carbon atom and two oxygen atoms. Carbon has an atomic mass of approximately 12.01 g/mol, and oxygen has an atomic mass of about 16.00 g/mol. Hence, the molar mass of CO\textsubscript{2} is : \( 12.01 \text{ g/mol} + 2 \times 16.00 \text{ g/mol} = 44.01 \text{ g/mol} \) In this exercise, you convert the mass of CO\textsubscript{2} (925.0 g) into moles using its molar mass (44.01 g/mol). This conversion allows us to determine the number of moles needed for the subsequent stoichiometric calculations.

Stoichiometry

Stoichiometry is the part of chemistry that deals with the relative quantities of reactants and products in chemical reactions. It involves using the balanced chemical equation to understand the proportional relationships between different substances. In the provided exercise, after balancing the chemical equation, we utilize stoichiometry to determine the moles of NaOH needed. The balanced chemical equation is: \[ 2 \text{NaOH} + \text{CO}_2 \rightarrow \text{Na}_2\text{CO}_3 + \text{H}_2\text{O} \] According to the balanced equation, 2 moles of NaOH react with 1 mole of CO\textsubscript{2}. From the previous step, we know that the number of moles of CO\textsubscript{2} is 21.02. Thus, the moles of NaOH needed are: \( 2 \times 21.02 = 42.04 \text{ moles} \) Additionally, the stoichiometry of the reaction tells us that 1 mole of CO\textsubscript{2} produces 1 mole each of the products Na\textsubscript{2}CO\textsubscript{3} and H\textsubscript{2}O.

Chemical Equation

A chemical equation represents the substances involved in a chemical reaction and shows the reactants transforming into products. In the provided exercise, the initial unbalanced equation was: \( \text{NaOH} + \text{CO}_2 \rightarrow \text{Na}_2\text{CO}_3 + \text{H}_2\text{O} \) To solve the problem, we first need to balance this equation to accurately reflect the law of conservation of mass, which states that mass cannot be created or destroyed in a chemical reaction. Balancing ensures that the same number of each type of atom appears on both sides of the equation. The balanced equation in the exercise is: \[ 2 \text{NaOH} + \text{CO}_2 \rightarrow \text{Na}_2\text{CO}_3 + \text{H}_2\text{O} \] This indicates that for every 2 molecules of NaOH, 1 molecule of CO\textsubscript{2} is required, resulting in the formation of 1 molecule of Na\textsubscript{2}CO\textsubscript{3} and 1 molecule of H\textsubscript{2}O. The balanced equation is essential for performing stoichiometric calculations and determining the quantities of reactants and products involved. Understanding how to balance chemical equations is fundamental for anyone studying chemistry, as it enables accurate predictions and analyses of chemical reactions.