Problem 143 - General Chemistry

Question

The carbon dioxide exhaled in the breath of astronauts is often removed from the spacecraft by reaction with lithium hydroxide. $$ 2 \mathrm{LiOH}(s)+\mathrm{CO}_{2}(g) \longrightarrow \mathrm{Li}_{2} \mathrm{CO}_{3}(s)+\mathrm{H}_{2} \mathrm{O}(l) $$ Estimate the grams of lithium hydroxide required per astronaut per day. Assume that each astronaut requires \(2.50 \times 10^{3}\) kcal of energy per day. 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 \(2.50 \times 10^{3}\) kcal of heat, calculate the amount of \(\mathrm{CO}_{2}\) produced and hence the amount of LiOH required. The \(\Delta H_{f}^{\circ}\) for glucose \((s)\) is \(-1273 \mathrm{~kJ} / \mathrm{mol}\).

Step-by-Step Solution

Answer

Approximately 2360 grams of LiOH are required per astronaut per day.

Step 1: Calculate the moles of glucose needed

First, determine the amount of glucose needed to provide the energy requirement of 2500 kcal/day per astronaut. Since 1 kcal = 4.184 kJ, convert the energy to kilojoules: \[ 2.50 \times 10^3 \text{ kcal/day} \times 4.184 \text{ kJ/kcal} = 10460 \text{ kJ/day}. \]Glucose combusts according to the following reaction: \[ 1 \text{ mol of glucose } (\mathrm{C}_6\mathrm{H}_{12}\mathrm{O}_6) \longrightarrow \mathrm{CO}_2(g) + \mathrm{H}_2\mathrm{O}(l). \]The enthalpy change \(\Delta H_f^\circ\) for the combustion of glucose is \(-1273 \text{ kJ/mol} \).Determine the moles of glucose needed to produce 10460 kJ of energy:\[ \text{moles of glucose} = \frac{10460 \text{ kJ}}{-1273 \text{ kJ/mol}} \approx 8.21 \text{ mol}. \]

Step 2: Calculate moles of CO₂ produced

From the balanced chemical equation:\[ \mathrm{C}_6\mathrm{H}_{12}\mathrm{O}_6 + 6 \mathrm{O}_2 \rightarrow 6 \mathrm{CO}_2 + 6 \mathrm{H}_2\mathrm{O}, \]it's clear that 1 mole of glucose produces 6 moles of \(\mathrm{CO}_2\).Thus, the amount of \(\mathrm{CO}_2\) produced from 8.21 mol of glucose is:\[ 8.21 \text{ mol glucose} \times 6 \frac{\text{mol } \mathrm{CO}_2}{\text{mol glucose}} = 49.26 \text{ mol of } \mathrm{CO}_2. \]

Step 3: Determine moles of LiOH required

According to the reaction:\[ 2 \mathrm{LiOH}(s) + \mathrm{CO}_{2}(g) \rightarrow \mathrm{Li}_{2}\mathrm{CO}_{3}(s) + \mathrm{H}_{2} \mathrm{O}(l), \]2 moles of \(\mathrm{LiOH}\) are required for every 1 mole of \(\mathrm{CO}_2\).Therefore, the moles of \(\mathrm{LiOH}\) required per day are:\[ 49.26 \text{ mol } \mathrm{CO}_2 \times 2 \frac{\text{mol } \mathrm{LiOH}}{\text{mol } \mathrm{CO}_2} = 98.52 \text{ mol } \mathrm{LiOH}. \]

Step 4: Calculate grams of LiOH required

The molar mass of LiOH is approximately 23.95 g/mol. Use this to find the mass of 98.52 moles:\[ \text{mass of LiOH} = 98.52 \text{ mol} \times 23.95 \text{ g/mol} = 2359.79 \text{ g}. \]Therefore, the amount of \(\mathrm{LiOH}\) required per astronaut per day is approximately 2360 grams (rounded to 3 significant figures).

Key Concepts

Calculation of Chemical Substance

Understanding how to calculate the amount of a chemical substance is essential in chemistry, especially when dealing with reactions. The basic calculation involves determining the moles of reactants or products in a chemical equation.
Begin by balancing the chemical equation to ensure that atoms are conserved during the reaction. Once balanced, use the stoichiometric coefficients for calculations.

• These coefficients indicate the relative number of moles of each substance.
• Convert the mass of a substance to moles using its molar mass.

In the spacecraft example, the calculation starts by determining the moles of glucose needed to meet the energy requirements, which then guides the amount of other substances involved in the reaction.

Energy Combustion of Glucose

Energy production through glucose combustion is a key process in biology and chemistry. When glucose combusts, it reacts with oxygen to release energy, carbon dioxide, and water.
The energy produced can be measured as enthalpy change, denoted as \( \Delta H_f^\circ \).
Factors affecting this energy release include the amount of glucose and the conditions under which combustion occurs. In the exercise:

• Glucose combustion releases heat, equivalent to \( \Delta H_f^\circ = -1273 \) kJ/mol.
• The amount of glucose combusted to produce a specific energy is calculated using the energy requirement (e.g., 2500 kcal per day).

This provides the basis for further calculations involving other substances like CO2 and LiOH.

Molar Mass in Chemistry

Molar mass is a fundamental concept in chemistry that represents the mass of one mole of a particular substance.
It is usually expressed in grams per mole (g/mol). To calculate the molar mass, sum up the atomic masses of all the atoms present in the molecular formula.
Knowing the molar mass is crucial for converting grams to moles and vice versa during calculations.

• For LiOH, the molar mass is approximately 23.95 g/mol.
• This converts moles of LiOH needed into grams to determine the mass required for the reaction, like in the spacecraft scenario.

Accurate molar mass calculations ensure precise stoichiometric computations, impacting various applications from industrial processes to space missions.

Thermochemistry

Thermochemistry involves the study of heat changes that occur during chemical reactions, integrating the concepts of enthalpy and energy transfer.
The \( \Delta H_f^\circ \), or standard enthalpy change, reflects the heat absorbed or released under standard conditions.
This concept is pivotal in determining the energy balance within reactions.

• In our context, the heat of combustion for glucose is measured to evaluate how much energy it releases.
• Precisely calculating this helps in understanding how much energy needs to be managed or transferred during a reaction.

Such analyses are vital when designing systems for energy-efficient chemical processes, including those utilized in confined environments like spacecraft.

Stoichiometry

Stoichiometry relates to the quantitative relationships between reactants and products in a chemical reaction, guided by balanced equations.
It allows chemists to predict the amounts of substances consumed and produced.

• Stoichiometry requires understanding the mole ratio of reactants to products as dictated by the balanced equation.
• For instance, 2 moles of LiOH react with 1 mole of CO2 to form lithium carbonate and water.

This ratio serves as the backbone for calculating the necessary amounts of reactants or products, as demonstrated in spacecraft reactions where LiOH removes CO2 to manage air quality for astronauts.

Chemical Reactions in Space

Chemical reactions in space must be carefully controlled and understood due to the unique environment of spacecraft.
Reactions like the one involving LiOH and CO2 are crucial for maintaining habitable conditions by controlling atmospheric compositions.

• These reactions often involve the removal or neutralization of potentially harmful compounds.
• The efficiency of such reactions directly impacts the sustainability of long-term missions.

Understanding the stoichiometry and thermodynamics involved helps engineers design systems that effectively manage resources, ultimately ensuring astronaut safety and mission success.