Problem 85 - Chemistry The Science in Context

Question

The CO \(_{2}\) that builds up in the air of a submerged submarine can be removed by reacting it with sodium peroxide: $$ 2 \mathrm{Na}_{2} \mathrm{O}_{2}(s)+2 \mathrm{CO}_{2}(g) \rightarrow 2 \mathrm{Na}_{2} \mathrm{CO}_{3}(s)+\mathrm{O}_{2}(g) $$ If a sailor exhales \(150.0 \mathrm{mL}\) of \(\mathrm{CO}_{2}\) per minute at \(20^{\circ} \mathrm{C}\) and 1.02 atm, how much sodium peroxide is needed per sailor in a 24 -hr period?

Step-by-Step Solution

Answer

Answer: 626.9 grams of sodium peroxide is needed per sailor in a 24-hour period.

Step 1: Calculate the total volume of CO2 exhaled by a sailor in a day.

We are given that the volume of CO2 exhaled by a sailor per minute is 150.0 mL. We need to find the total volume for 24 hours. There are 60 minutes in an hour and 24 hours in a day. Therefore: Total volume of CO\(_2\) = 150.0 mL/minute × 60 minutes/hour × 24 hours/day = 216000 mL

Step 2: Calculate the moles of CO\(_2\)

We are given the conditions at which the sailor exhales the CO2: a temperature of 20°C and a pressure of 1.02 atm. We can use the Ideal Gas Law, PV = nRT, to find the moles of CO2. First, convert the temperature to Kelvin: Temperature in Kelvin (K) = 20°C + 273.15 = 293.15 K Now, use the Ideal Gas Law: n = PV/RT Where, P = pressure = 1.02 atm V = volume = 216000 mL (converted to L) = 216 L R = gas constant = 0.0821 L atm K\(^{-1}\)mol\(^{-1}\) T = temperature = 293.15 K n = (1.02 atm × 216 L) / (0.0821 L atm K\(^{-1}\)mol\(^{-1}\) × 293.15 K) = 8.040 mol So, there are 8.040 moles of CO\(_2\) exhaled by a sailor in a day.

Step 3: Calculate the moles of sodium peroxide required

According to the stoichiometry of the balanced chemical equation, 2 moles of Na\(_2\)O\(_2\) are needed to react with 2 moles of CO\(_2\). This means that 1 mole of Na\(_2\)O\(_2\) reacts with 1 mole of CO\(_2\). The moles of sodium peroxide required will thus be: Moles of Na\(_2\)O\(_2\) = Moles of CO\(_2\) = 8.040 mol

Step 4: Convert moles of sodium peroxide into grams

To find how much sodium peroxide in grams is required, multiply the moles by the molar mass of Na\(_2\)O\(_2\). The molar mass of Na\(_2\)O\(_2\) is: (2 × 22.99 g/mol for Na) + (2 × 16.00 g/mol for O) = 77.98 g/mol Now, we can find the mass of sodium peroxide needed: Mass of Na\(_2\)O\(_2\) = moles × molar mass = 8.040 mol × 77.98 g/mol = 626.9 g So, 626.9 grams of sodium peroxide is needed per sailor in a 24-hour period.

Key Concepts

Understanding the Ideal Gas Law

The Ideal Gas Law is essential for understanding the behavior of gases in a variety of situations. It relates the pressure, volume, temperature, and number of moles of a gas using the formula: \[ PV = nRT \]where:

• \( P \) stands for pressure, measured typically in atmospheres (atm).
• \( V \) represents volume, often in liters (L).
• \( n \) is the number of moles.
• \( R \) is the ideal gas constant, nuanced by the units of pressure and volume, here as 0.0821 L atm K\(^{-1}\)mol\(^{-1}\).
• \( T \) is the temperature in Kelvin (K).

To convert Celsius to Kelvin, you simply add 273.15 to the Celsius temperature. The Ideal Gas Law allows us to calculate how much space a certain amount of gas will occupy at a given temperature and pressure. This calculation is crucial when trying to understand reactions that involve gaseous substances, like in the decomposition or consumption of carbon dioxide (\( \mathrm{CO_2} \)).

Chemical Reactions and Stoichiometry

Chemical reactions describe the transformation of reactants into products. Each reaction can be expressed as a balanced equation, which tells us the ratio of molecules involved. In this exercise, the reaction is:\[ 2 \mathrm{Na_{2}O_{2}}(s) + 2 \mathrm{CO_{2}}(g) \rightarrow 2 \mathrm{Na_{2}CO_{3}}(s) + \mathrm{O_{2}}(g) \] This balanced equation reveals the mole-to-mole ratio of reactants and products. The reaction shows that 2 moles of \( \mathrm{Na_{2}O_{2}} \) react with 2 moles of \( \mathrm{CO_{2}} \), indicating a 1:1 ratio for sodium peroxide to carbon dioxide. Stoichiometry is the calculation of reactants and products in chemical reactions, based on the balanced equation. It ensures that the amount of reactants needed and products formed are both accurately calculated. Stoichiometry allows chemists to predict yields and determine the quantities required or produced in chemical reactions.

Molar Calculations Simplified

Molar calculations are a cornerstone of stoichiometry and involve finding how many moles of a substance are involved in a reaction. To determine the moles, you can rearrange the Ideal Gas Law as follows: \[ n = \frac{PV}{RT} \] Here, using the conditions provided (volume in liters, pressure in atm, and temperature in Kelvin), moles of gaseous carbon dioxide can be calculated. This amount is crucial for finding out quantities of other substances via stoichiometry. Once the moles of \( \mathrm{CO_2} \) are known, they can be directly related to the moles of sodium peroxide (\( \mathrm{Na_{2}O_{2}} \)) using the stoichiometric coefficients. Remember, the relationship is direct because of the 1:1 mole ratio. Understanding and performing molar calculations means you can predict how much of each substance is used up or formed in a reaction, essential for any chemical process.