Problem 49 - Chemistry The Molecular Science

Question

The build-up of excess carbon dioxide in the air of a submerged submarine is prevented by reacting \(\mathrm{CO}_{2}\) with sodium peroxide, \(\mathrm{Na}_{2} \mathrm{O}_{2}\) $$2 \mathrm{Na}_{2} \mathrm{O}_{2}(\mathrm{~s})+2 \mathrm{CO}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{Na}_{2} \mathrm{CO}_{3}(\mathrm{~s})+\mathrm{O}_{2}(\mathrm{~g})$$ Calculate the mass of \(\mathrm{Na}_{2} \mathrm{O}_{2}\) needed in a \(24.0-\mathrm{h}\) period per submariner if each exhales \(240 \mathrm{~mL} \mathrm{CO}_{2}\) per minute at \(21^{\circ} \mathrm{C}\) and \(1.02 \mathrm{~atm} .\)

Step-by-Step Solution

Answer

You need 860.1 grams of Na₂O₂ per submariner in 24 hours.

Step 1: Calculate Total Exhaled CO₂ Volume per Submariner

Each submariner exhales 240 mL of CO₂ per minute. To find the total volume exhaled in 24 hours, calculate:\[\text{Total CO}_2\text{ Volume} = 240 \text{ mL/min} \times 60 \text{ min/hr} \times 24 \text{ hr}\]This gives the total volume of CO₂ exhaled by one submariner in 24 hours.

Step 2: Convert the Volume of CO₂ to Moles Using Ideal Gas Law

To convert the volume of CO₂ to moles, use the ideal gas law equation, \[ PV = nRT \]where:- \(P\) is the pressure (1.02 atm)- \(V\) is the volume (convert mL to L by dividing by 1000)- \(R\) is the gas constant (0.0821 L·atm/mol·K)- \(T\) is the temperature in Kelvin (21°C = 294 K)Rearrange to find \(n\), the number of moles:\[ n = \frac{PV}{RT} \]

Step 3: Use Stoichiometry to Find Moles of Na₂O₂ Required

From the balanced reaction equation, 2 moles of Na₂O₂ are needed for 2 moles of CO₂. Determine the moles of Na₂O₂ required based on the moles of CO₂ found in Step 2. Since they are in a 1:1 molar ratio, the moles of Na₂O₂ needed will be equal to the moles of CO₂ exhaled.

Step 4: Convert Moles of Na₂O₂ to Grams

Convert the moles of Na₂O₂ needed to grams using its molar mass. The molar mass of Na₂O₂ is approximately 77.98 g/mol. Multiply the moles of Na₂O₂ by its molar mass to find the mass in grams:\[ \text{Mass of Na}_2\text{O}_2 = \text{moles of Na}_2\text{O}_2 \times 77.98 \text{ g/mol} \]

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. The equation is expressed as \( PV = nRT \), where:

• \( P \): Pressure of the gas
• \( V \): Volume of the gas
• \( n \): Number of moles of the gas
• \( R \): Gas constant, which is 0.0821 L·atm/mol·K
• \( T \): Temperature in Kelvin

To solve problems related to gas, like the amount of carbon dioxide exhaled in a submarine, you use the Ideal Gas Law to convert the given condition of the gas (e.g., pressure, volume, temperature) into a more manageable form such as moles. By rearranging the Ideal Gas Law, inversely for moles \( n = \frac{PV}{RT} \), we can understand how much of the gas is present in terms of the amount of substance. This formula is vital when we need to proceed with any chemical conversions or reactions involving gases.

Molar Mass Calculations

Molar Mass Calculations are essential for converting moles of a substance to its mass in grams, which is often needed in chemical reactions to ensure the accuracy of the reagents used. The molar mass of a compound is the sum of the atomic masses of all atoms in the molecule. For instance, to calculate the molar mass of sodium peroxide \( \text{Na}_2\text{O}_2 \), you find:

• Sodium (Na) has an atomic mass of approximately 23 g/mol, so two sodium atoms contribute \( 2 \times 23 = 46 \) g/mol.
• Oxygen (O) has an atomic mass of about 16 g/mol, so two oxygen atoms contribute \( 2 \times 16 = 32 \) g/mol.

This gives a total molar mass of \( 77.98 \) g/mol for \( \text{Na}_2\text{O}_2 \). In any stoichiometric calculation, once the moles of a compound have been determined (using the Ideal Gas Law or a balanced equation), you multiply by the compound's molar mass to convert to grams. This conversion is crucial to practically measure the exact amounts necessary for chemical reactions.

Chemical Reactions

Chemical Reactions involve the transformation of reactants into products, characterized by a chemical equation that depicts the reactants and products with their respective stoichiometric coefficients. In the conservation of mass, the amount of atoms for each element must be the same on both sides of the reaction. For example, consider the reaction between carbon dioxide and sodium peroxide:\[ 2 \text{Na}_2\text{O}_2\;(s) + 2\;\text{CO}_2\;(g) \rightarrow 2\;\text{Na}_2\text{CO}_3\;(s) + \text{O}_2\;(g) \]

• Reactants: \( \text{Na}_2\text{O}_2 \) and \( \text{CO}_2 \)
• Products: \( \text{Na}_2\text{CO}_3 \) and \( \text{O}_2 \)

The balanced equation shows that for every 2 moles of \( \text{Na}_2\text{O}_2 \) reacting, 2 moles of \( \text{CO}_2 \) are used, demonstrating a 1:1 molar ratio between them. Understanding these stoichiometric relationships helps determine the amount of each substance needed or produced in a reaction. This stoichiometry allows students to calculate the exact quantities needed for reactions, like how much \( \text{Na}_2\text{O}_2 \) is required to react with the \( \text{CO}_2 \) exhaled by submariners, ensuring optimal chemical use.