Question

A 7600 liter compartment in a space capsule, maintained at an internal temperature of \(27^{\circ} \mathrm{C}\), is designed to hold one astronaut. The human body discharges \(960 \mathrm{~g}\) of carbon dioxide gas ( \(\mathrm{CO}_{2}\), molecular weight \(=44 \mathrm{~g} / \mathrm{mole}\) ) each day. If the initial partial pressure of carbon dioxide in the compartment is zero, how much \(\mathrm{CO}_{2}\) must be pumped out the first day to maintain a partial pressure of no more than \(4.1\) torr?

Short answer

Moles of CO₂ = \(21.82 \mathrm{~moles}\) #tag_title#Step 2: Calculate the partial pressure of CO₂#tag_content# To find the partial pressure of CO₂ produced, we can use the ideal gas law: \(PV = nRT\) Where: - P is the partial pressure of CO₂, - V is the volume of the compartment (7600 L), - n is the number of moles of CO₂ (21.82 moles), - R is the ideal gas constant (\(0.0821 \mathrm{~L~atm/mol~K}}\)), - T is the temperature in Kelvin (27°C + 273.15 = 300.15 K). Rearranging the formula to find P, we get: P = \(\cfrac{nRT}{V}\) P = \(\cfrac{(21.82 \mathrm{~moles})(0.0821 \mathrm{~L~atm/mol~K})(300.15 \mathrm{~K})}{7600 \mathrm{~L}}\) #tag_title#Step 3: Calculate the amount of CO₂ that corresponds to 4.1 torr#tag_content# First, convert the partial pressure from torr to atm: 4.1 torr × \(\cfrac{1 \mathrm{~atm}}{760 \mathrm{~torr}}\) = 0.00539 atm Now, use the ideal gas law to determine the number of moles of CO₂ corresponding to this partial pressure: n = \(\cfrac{PV}{RT}\) n = \(\cfrac{(0.00539 \mathrm{~atm})(7600 \mathrm{~L)}}{(0.0821 \mathrm{~L~atm/mol~K})(300.15 \mathrm{~K})}\) n = 1.11 moles of CO₂ #tag_title#Step 4: Calculate how much CO₂ must be pumped out#tag_content# To find the amount of CO₂ that must be removed, subtract the number of moles of CO₂ corresponding to 4.1 torr from the total number of moles of CO₂ produced per day: Moles of CO₂ to be pumped out = 21.82 moles - 1.11 moles Moles of CO₂ to be pumped out = 20.71 moles To get the amount in grams, multiply the number of moles by the molecular weight of CO₂: Amount of CO₂ to be pumped out = (20.71 moles) × (44 g/mol) Amount of CO₂ to be pumped out = \(910.24 \mathrm{~g}\) #tag_title#Short Answer#tag_content# To maintain a partial pressure of no more than 4.1 torr, \(910.24 \mathrm{~g}\) of CO₂ must be pumped out of the compartment on the first day.

Step-by-step solution

Convert grams of CO₂ to moles

To calculate the partial pressure of CO₂ in the compartment, we need to convert the amount of CO₂ produced by the astronaut from grams to moles. We are given that the human body discharges 960 g of CO₂ per day, with a molecular weight of 44 g/mol. Therefore, we can use the following formula: Moles of CO₂ = (grams of CO₂) / (molecular weight of CO₂) Moles of CO₂ = (960 g) / (44 g/mol)

Key concepts

Carbon Dioxide Gas

Carbon dioxide (CO₂) is a colorless and odorless gas vital to life on Earth. It's produced during the process of respiration by humans and animals and also by combustion of organic materials. In the context of our exercise, the exhaled CO₂ from an astronaut becomes a critical factor within an enclosed space, like a space capsule. Excess CO₂ can lead to dangerous conditions, so it's necessary to regulate its concentration in the air to ensure the safety and comfort of astronauts.

Carbon dioxide is also used in the scientific world as a reference for understanding gas behavior and properties such as gas density, gas solubility, and the impact of gases in various chemical reactions. Understanding the way CO₂ behaves at different temperatures and pressures is essential for many fields, including environmental science, space exploration, and chemical engineering.

Molar Mass Calculation

The molar mass of a substance is its weight per mole, which corresponds to the mass of 6.022 x 10²³ (Avogadro's number) particles of that substance. The molar mass calculation is a fundamental practice in chemistry, as it bridges the gap between the macroscopic quantities we measure in the lab (like grams) and the microscopic quantities (like number of molecules or moles) used in chemical equations and reactions.

To find the molar mass of a molecule such as CO₂, we sum the atomic masses of each constituent atom. For CO₂, with one Carbon (C) atom and two Oxygen (O) atoms, the molar mass is calculated by adding the atomic mass of Carbon (approximately 12.01 g/mol) to the atomic mass of Oxygen (approximately 16.00 g/mol) multiplied by two:
Molar mass of CO₂ = (1 x 12.01 g/mol) + (2 x 16.00 g/mol) = 44.01 g/mol.

This calculation is crucial when converting between mass and moles, which allows us to apply the ideal gas law in various practical scenarios, such as determining the amount of gas that must be removed from a space capsule to maintain safe CO₂ levels.

Gas Laws

Gas laws are the cornerstones of understanding gas behavior under different conditions of temperature, volume, and pressure. They are typically represented by formulas that can predict how a gas will respond to changes in these conditions. For our exercise, the relevant gas laws include Boyle's Law, which describes the inverse relationship between pressure and volume, and Gay-Lussac's Law, which addresses the direct relationship between pressure and temperature.

The Ideal Gas Law, another fundamental concept, combines these relationships into one formula: PV = nRT, where P is pressure, V is volume, n is the number of moles of the gas, R is the universal gas constant, and T is the temperature in Kelvin. When we talk about maintaining a constant partial pressure of CO₂ within the space capsule, the Ideal Gas Law can be manipulated to solve for the number of moles of gas that must be removed to maintain desired conditions.

For practical applications, understanding these gas laws is essential. It's not just about ensuring astronauts have a breathable environment but also factors into other industries like scuba diving, manufacturing, and even in our kitchen with the use of pressurized containers.