Problem 93 - Chemistry: An Atoms-Focused Approach

Question

Healthy Air for Submariners The CO, that builds up in the air of a submerged submarine can be removed by reacting it with an aqueous solution of 2 -aminoethanol: \(\mathrm{CO}_{2}(g)+2 \mathrm{HOCH}_{2} \mathrm{CH}_{2} \mathrm{NH}_{2}(a q) \rightarrow\) $$\mathrm{HOCH}_{2} \mathrm{CH}_{2} \mathrm{NH}_{3}^{+}(a q)+\mathrm{HOCH}_{2} \mathrm{CH}_{2} \mathrm{NHCO}_{2}^{-}(a q)$$ If a sailor exhales \(125 \mathrm{mL}\) of \(\mathrm{CO}_{2}\) per minute at \(23^{\circ} \mathrm{C}\) and1.02 atm, what volume of \(4.0 \mathrm{M}\) 2-aminocthanol is needed per sailor in a 24 -hour period?

Step-by-Step Solution

Answer

Answer: 71.6 L/day

Step 1: Calculate the moles of CO2 exhaled per minute

The amount of CO2 exhaled by a sailor per minute is given as 125 mL at 23°C and 1.02 atm. We'll use the molar volume of gas at STP (22.4 L/mol) and the given temperature and pressure to calculate moles of CO2 per minute. First convert the volume into liters: 125 mL * (1 L / 1000 mL) = 0.125 L Now, rearrange the ideal gas law to solve for n (moles) and apply the given conditions: PV = nRT => n = (PV)/(RT) = (1.02 atm * 0.125 L) / (0.0821 L atm/mol K * (23°C + 273.15 K)) = 0.00498 mol CO2/min

Step 2: Calculate the moles of CO2 exhaled in 24 hours

To find the moles of CO2 exhaled in 24 hours, simply multiply the moles per minute by the total minutes in 24 hours: 0.00498 mol/min * (24 hours/day * 60 min/hour) = 143.1 mol CO2/day

Step 3: Use stoichiometry to find moles of 2-aminoethanol needed

According to the balanced chemical equation, every 1 mole of CO2 reacts with 2 moles of 2-aminoethanol. Therefore, the moles of 2-aminoethanol required for the given moles of CO2 can be obtained as follows: (143.1 mol CO2/day) * (2 mol 2-aminoethanol / 1 mol CO2) = 286.2 mol 2-aminoethanol/day

Step 4: Calculate the volume of 4.0 M 2-aminoethanol needed

Since the 2-aminoethanol solution has a concentration of 4.0 M, we can now calculate the volume of the solution required using the moles of 2-aminoethanol needed: V = (286.2 mol 2-aminoethanol/day) / (4.0 mol/L) = 71.6 L/day So, per sailor, 71.6 liters of 4.0 M 2-aminoethanol is needed in a 24-hour period to remove the CO2 exhaled.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that helps us understand the behavior of gases. It connects four key properties of gases: pressure (P), volume (V), number of moles (n), and temperature (T), via the equation:

• \[ PV = nRT \]

This equation is very handy for calculating the number of moles of a gas, a common requirement in stoichiometric calculations. In the equation, R represents the ideal gas constant, which is typically 0.0821 L atm/mol K when pressure is in atmospheres and volume is in liters.
To use the Ideal Gas Law, it's crucial to ensure all units match the constant. Temperature must be in Kelvin (K), derived from Celsius by adding 273.15, and volume should be converted to liters if initially given in milliliters.
Applying the Ideal Gas Law helps in finding out how much of a gas is present under given conditions of temperature and pressure, as was done to determine the moles of CO2 exhaled by the sailor. By rearranging the equation to solve for moles,

• \[ n = \frac{PV}{RT} \]

we can plug in the known values to calculate the moles of CO2 breathed out every minute.

Chemical Reactions

Chemical reactions involve the transformation of reactants into products. They are represented by balanced chemical equations that ensure the conservation of mass, meaning the number of atoms for each element is the same on both sides.
In our exercise, the chemical reaction describes the removal of carbon dioxide (CO₂) from submariners’ air by reacting with 2-aminoethanol, forming two products in an aqueous solution. This reaction can be seen as:

• \[ \text{CO}_{2}(g) + 2 \text{HOCH}_{2} \text{CH}_{2} \text{NH}_{2}(aq) \rightarrow \text{HOCH}_{2} \text{CH}_{2} \text{NH}_{3}^{+}(aq) + \text{HOCH}_{2} \text{CH}_{2} \text{NHCO}_{2}^{-}(aq) \]

This balanced equation shows the stoichiometry, or the quantitative relationship between the reactants and the products in a chemical reaction. Key to stoichiometry is the use of mole ratios, derived directly from the coefficients of the balanced equation.
Understanding these ratios is important to determine how much of a reactant is needed to react completely with another, such as how much 2-aminoethanol is required to remove a specific amount of CO₂.

Mole Calculations

Moles are a basic unit in chemistry that measures amounts of substance. They allow for the quantification of chemical reactions using balanced equations. A mole corresponds to Avogadro's number, which is approximately

• \[ 6.022 \times 10^{23} \] particles (atoms, molecules, etc.).

In stoichiometry, calculating moles involves converting given masses or volumes into moles using the molar mass (for solids and liquids) or the Ideal Gas Law (for gases).
Once the moles of a substance are known, they can be used to calculate the amounts of other substances involved in a reaction by using the mole ratios from the balanced equation. This approach was used in the exercise to first find how many moles of CO₂ were exhaled by the sailor, and then to determine the moles and volume of 2-aminoethanol required for a reaction over a 24-hour period.
Helpful conversions also include using concentrations (like molarity, M) in solution reactions, which relates the number of moles of solute to the volume of solution. Understanding mole calculations is essential for effectively solving many chemistry problems and conducting chemical experiments.

• \[ \text{Molarity} = \frac{\text{moles of solute}}{\text{liters of solution}} \]