Problem 144 - General Chemistry

Question

A rebreathing gas mask contains potassium superoxide, \(\mathrm{KO}_{2}\), which reacts with moisture in the breath to give oxygen. $$ 4 \mathrm{KO}_{2}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow 4 \mathrm{KOH}(s)+3 \mathrm{O}_{2}(g) $$ Estimate the grams of potassium superoxide required to supply a person's oxygen needs for one hour. Assume a person requires \(1.00 \times 10^{2}\) kcal of energy for this time period. Further assume that this energy can be equated to the heat of combustion of a quantity of glucose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\), to \(\mathrm{CO}_{2}(g)\) and \(\mathrm{H}_{2} \mathrm{O}(l) .\) From the amount of glucose required to give \(1.00 \times 10^{2} \mathrm{kcal}\) of heat, calculate the amount of oxygen consumed and hence the amount of \(\mathrm{KO}_{2}\) required. The \(\Delta H_{f}^{\circ}\) for glucose \((s)\) is \(-1273 \mathrm{~kJ} / \mathrm{mol}\).

Step-by-Step Solution

Answer

85.32 g of KO₂ is required.

Step 1: Calculate Heat of Combustion in kJ

First, convert the energy requirement from kcal to kJ. \(1.00 \times 10^2\) kcal is equivalent to \(1.00 \times 10^2 \times 4.184\) kJ = \(418.4\) kJ.

Step 2: Determine Moles of Glucose Needed

The standard enthalpy of formation for glucose is \(-1273 \text{ kJ/mol}\). To find the moles of glucose required to release \(418.4\) kJ of energy, use the equation: \( \text{Moles of glucose} = \frac{418.4 \text{ kJ}}{2803 \text{ kJ/mol}} \approx 0.15 \text{ mol} \). Note: The combustion reaction of glucose releases \(2803\) kJ/mol.

Step 3: Balance Glucose Combustion Reaction

The balanced equation for the combustion of glucose is: \[ \mathrm{C}_6\mathrm{H}_{12}\mathrm{O}_6(s) + 6\mathrm{O}_2(g) \rightarrow 6\mathrm{CO}_2(g) + 6\mathrm{H}_2\mathrm{O}(l) \]This shows that 1 mole of glucose consumes 6 moles of oxygen.

Step 4: Calculate Moles of Oxygen Required

From the balanced equation, \(0.15\) moles of glucose requires \(0.15 \times 6 = 0.9\) moles of \(\mathrm{O}_2\).

Step 5: Calculate Moles of KO₂ Required

From the reaction \(4 \mathrm{KO}_2 + 2 \mathrm{H}_2\mathrm{O} \rightarrow 4 \mathrm{KOH} + 3 \mathrm{O}_2\), 4 moles of KO₂ produce 3 moles of \(\mathrm{O}_2\). Therefore, to produce \(0.9\) moles of \(\mathrm{O}_2\), the moles of KO₂ required is \(\frac{0.9 \times 4}{3} = 1.2 \text{ moles}\).

Step 6: Calculate Mass of KO₂

The molar mass of KO₂ is \(39.1 \text{ (K)} + 2 \times 16.0 \text{ (O)} = 71.1 \text{ g/mol}\). Thus, the mass of KO₂ required is \(1.2 \text{ moles} \times 71.1 \text{ g/mol} = 85.32 \text{ g}\).

Key Concepts

Potassium Superoxide

Potassium superoxide (KO₂) is a chemical compound that plays a significant role in oxygen generation systems. It works efficiently in closed environments, such as submarines and space vehicles, where the renewal of breathable air is crucial. When this compound reacts with moisture, as found in exhaled breath, it produces oxygen gas (\(\mathrm{O}_2\)) and potassium hydroxide (\(\mathrm{KOH}\)). This reaction is represented by the equation: 4KO₂ + 2H₂O → 4KOH + 3O₂.
It is highly efficient as it not only provides oxygen but also scrubs out carbon dioxide by forming potassium carbonate after further reactions, keeping the air clean and refreshing. The reactivity with water makes it a powerful agent in emergency breathing apparatus. This compelling property is leveraged in rebreathing masks for safe and effective supplemental oxygen supply.
In our exercise, KO₂ is considered for producing the necessary oxygen for a person's metabolic needs by reacting exactly with the moisture in breath.

Glucose Combustion

Glucose combustion is a fundamental metabolic process that releases energy in living organisms. The chemical reaction involves glucose (\(\mathrm{C}_6\mathrm{H}_{12}\mathrm{O}_6\)) reacting with oxygen to produce carbon dioxide (\(\mathrm{CO}_2\)), water, and energy. The balanced chemical equation is \[ \mathrm{C}_6\mathrm{H}_{12}\mathrm{O}_6(s) + 6\mathrm{O}_2(g) \rightarrow 6\mathrm{CO}_2(g) + 6\mathrm{H}_2\mathrm{O}(l) \]
This reaction is exothermic, meaning it releases heat energy, which is used to fuel biological processes in organisms. The standard enthalpy of formation gives an indication of energy changes during the reaction and is \(-1273 \ \mathrm{kJ/mol}\) for glucose.
In the context of thermochemistry, understanding glucose combustion is essential because it demonstrates how biochemical processes and thermodynamic principles intersect to sustain life through energy conversion. The exercise uses this combustion to estimate how much oxygen a person needs to metabolize a certain amount of glucose to satisfy an energy demand of \(100\) kcal.

Oxygen Production

In many chemical and biological processes, the production of oxygen is a crucial step. Here, through the reaction of potassium superoxide (KO₂) with water, oxygen is generated. Potassium superoxide acts as a chemical oxygen generator. It is especially valuable in closed environments like spacecraft or rebreathers in mining, where oxygen levels need replenishing without external supply.
The reaction \[ 4\mathrm{KO}_2 + 2\mathrm{H}_2\mathrm{O} \rightarrow 4\mathrm{KOH} + 3\mathrm{O}_2 \]
illustrates how KO₂ effectively releases oxygen while simultaneously forming the by-product, potassium hydroxide. This dual action of oxygen release and carbon dioxide absorption exemplifies its utility in demanding environments.
In the exercise scenario, calculating the necessary oxygen involves understanding how many moles of oxygen are needed initially from glucose combustion. Potassium superoxide helps bridge the gap by enabling accurate calculations of oxygen production based on controlled chemical reactions.

Stoichiometry

Stoichiometry describes the quantitative relationship between reactants and products in a chemical reaction. It is the backbone of predicting the amount of substances consumed and produced in any chemical process. Through stoichiometric calculations, one can precisely determine the moles and mass of reactants needed to produce a desired quantity of products.
For the calculation in this exercise, stoichiometry helps us ascertain how much potassium superoxide is required to fulfill a person's oxygen needs. By leveraging the balanced chemical equation of both the KO₂ reaction and glucose combustion, the stoichiometry allows the conversion of energy requirements into specific chemical quantities.
In the context of the exercise, the calculations proceed as follows:

• Convert energy needs from kilocalories to kilojoules.
• Use glucose combustion enthalpy to find moles of glucose.
• Determine moles of oxygen required using stoichiometric ratios.
• Relate oxygen requirements back to moles of KO₂ using the balanced reaction equations.

Thus, stoichiometry provides a mathematical framework for transitioning from theoretical chemistry to practical applications, assisting in precise resource management in real-world scenarios.