Question

Lithium hydroxide, \(\mathrm{LiOH}\), is used in spacecraft to recondition the air by absorbing the carbon dioxide exhaled by astronauts. The reaction is $$ 2 \mathrm{LiOH}(s)+\mathrm{CO}_{2}(g) \longrightarrow \mathrm{Li}_{2} \mathrm{CO}_{3}(s)+\mathrm{H}_{2} \mathrm{O}(l) $$ What volume of carbon dioxide gas at \(21^{\circ} \mathrm{C}\) and 781 \(\mathrm{mmHg}\) could be absorbed by \(327 \mathrm{~g}\) of lithium hydroxide?

Short answer

Around 93.86 liters of CO2 gas can be absorbed.

Step-by-step solution

Determine Moles of Lithium Hydroxide

First, we need to find out how many moles of lithium hydroxide (\(\mathrm{LiOH}\)) are present in 327 grams. The molar mass of \(\mathrm{LiOH}\) is 24.0 g/mol for Li, 16.0 g/mol for O, and 1.0 g/mol for H, making the total molar mass 24.0 + 16.0 + 1.0 = 41.0 g/mol. So, the number of moles is given by \(\frac{327 \, \text{g}}{41.0 \, \text{g/mol}} = 7.976\, \text{mol}.\)

Use Stoichiometry to Find Moles of CO2

According to the reaction equation, 2 moles of \(\mathrm{LiOH}\) react with 1 mole of \(\mathrm{CO}_2\). Therefore, the number of moles of \(\mathrm{CO}_2\) that can react with 7.976 moles of \(\mathrm{LiOH}\) is \(\frac{7.976 \, \text{mol LiOH}}{2} = 3.988\, \text{mol}.\)

Calculate Volume of CO2 at Given Conditions

Next, we calculate the volume using the ideal gas law equation \(PV = nRT\). We are given that \(n = 3.988\, \text{mol}\), \(T = 21^{\circ} \text{C} = 294\, \text{K}\), and \(P = 781\, \text{mmHg} = 1.028\, \text{atm}.\) Using the ideal gas constant \(R = 0.0821\, \text{L atm/mol K},\) we find the volume \(V\) as follows:\[ \begin{align*} V &= \frac{nRT}{P} \ &= \frac{3.988 \, \text{mol} \times 0.0821 \, \text{L atm/mol K} \times 294 \, \text{K}}{1.028 \, \text{atm}} \ &= 93.86 \, \text{L}. \end{align*} \]So, the volume of \(\mathrm{CO}_2\) gas absorbed is approximately \(93.86 \, \text{L}.\)

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental concept in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. It is expressed by the formula \[ PV = nRT \] where:

• \( P \) stands for pressure, usually measured in atmospheres (atm) or millimeters of mercury (mmHg).
• \( V \) is the volume of the gas, measured in liters (L).
• \( n \) is the number of moles of gas.
• \( R \) is the ideal gas constant, which is \(0.0821 \, \text{L atm/mol K}\) for these units.
• \( T \) is the temperature in Kelvin (K).

To solve problems using the Ideal Gas Law, ensure all units are consistent. For example, convert the pressure to atm if given in mmHg and temperature to Kelvin if given in Celsius. In our given problem, once the moles of CO₂ were determined, we used these variables to find the volume of gas involved. Make temperature conversions easy by adding 273 to the Celsius value to get Kelvin.

Lithium Hydroxide

Lithium hydroxide (\(\mathrm{LiOH}\)) is a chemical compound that is used for a variety of purposes, including the removal of carbon dioxide (\(\mathrm{CO}_2\)) from the air. This is especially useful in closed environments like space shuttles, where air needs to be continuously purified to maintain a livable atmosphere for astronauts.
The reaction of lithium hydroxide with carbon dioxide can be represented by the equation: \[ 2 \mathrm{LiOH}(s) + \mathrm{CO}_2(g) \rightarrow \mathrm{Li}_2\mathrm{CO}_3(s) + \mathrm{H}_2\mathrm{O}(l) \]Here, lithium hydroxide acts to both absorb harmful carbon dioxide and produce benign byproducts like lithium carbonate and water. It plays a crucial role in air purification systems because it efficiently traps CO₂ without generating additional harmful substances. The use of LiOH in space missions is indispensable, highlighting its capability in maintaining breathable air.

Molar Mass Calculation

Calculating the molar mass of a compound is crucial for converting between grams and moles during stoichiometric calculations. The molar mass of a compound is the sum of the atomic masses of all the atoms in a molecule. Let's break this down for lithium hydroxide (\(\mathrm{LiOH}\)):

• Lithium (Li) has an atomic mass of approximately 6.94 g/mol.
• Oxygen (O) has an atomic mass of 16.00 g/mol.
• Hydrogen (H) has an atomic mass of 1.01 g/mol.

Thus, the molar mass of lithium hydroxide is calculated as:\[ 6.94\, \text{g/mol} + 16.00\, \text{g/mol} + 1.01\, \text{g/mol} = 23.95\, \text{g/mol} \]However, during practical issues, it might get rounded to 24.00 g/mol, depending on the precision required. Molar mass helps convert the weight of a substance in grams to the amount of that substance in moles, facilitating the subsequent stoichiometric calculations needed for reaction predictions.

Chemical Reaction

Understanding chemical reactions is fundamental in stoichiometry. A chemical reaction denotes a process in which reactants are transformed into products. For lithium hydroxide reacting with carbon dioxide, the chemical equation is: \[ 2 \mathrm{LiOH}(s) + \mathrm{CO}_2(g) \rightarrow \mathrm{Li}_2\mathrm{CO}_3(s) + \mathrm{H}_2\mathrm{O}(l) \]This equation demonstrates the conversion of reactants \(2 \mathrm{LiOH}\) and \(\mathrm{CO}_2\) to products \( \mathrm{Li}_2\mathrm{CO}_3 \) and \(\mathrm{H}_2\mathrm{O}\). The equation also indicates that 2 moles of lithium hydroxide are needed to absorb 1 mole of carbon dioxide.In stoichiometry:

• The coefficients in a balanced chemical equation reflect the ratio of moles needed for the reaction. Here, the 2:1 ratio between LiOH and CO₂ provides critical insight for determining how much of each reactant is necessary, guiding the calculation of moles from given masses.
• Atoms are neither created nor destroyed, ensuring mass and atom balance across the reaction.

Understanding these relationships allows for the strategic use of resources and accurate prediction of reaction outcomes, which are pivotal in chemical engineering and various applied sciences.