Problem 20 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

The atmosphere of a spacecraft with volume \(27 \mathrm{m}^{3}\) consists of \(80 \%\) helium and \(20 \%\) oxygen by volume. The gases continually escape by effusion through small leaks in the walls. The leak amounts to 1000 Pa per day. The temperature inside the spacecraft is \(20^{\circ} \mathrm{C}\). What masses of helium and oxygen must be carried to replace the gas that leaks during a 10 -day mission? (Section 8.5)

Step-by-Step Solution

Answer

35.28 g of helium and 70.40 g of oxygen are needed for a 10-day mission.

Step 1: Convert Temperature and Determine Moles of Gas Lost

First, convert the given temperature from Celsius to Kelvin by adding 273.15: \(20^{\circ} \mathrm{C} = 293.15 \mathrm{K}\). Each leak causes a pressure drop of 1000 Pa per day, or 10,000 Pa in 10 days. We use the ideal gas law \(PV = nRT\) to determine the moles of gas lost, where \(R = 8.314 \frac{J}{mol \, K}\) and \(V = 27 \, \mathrm{m}^{3}\). \(n = \frac{\Delta P \cdot V}{R \cdot T} = \frac{10000 \times 27}{8.314 \times 293.15}\). This calculation gives \(n \approx 11.02 \, \mathrm{moles}\).

Step 2: Calculate Individual Moles of Helium and Oxygen

The composition of the atmosphere is \(80\%\) helium and \(20\%\) oxygen. Calculate the number of moles of each gas: \(n_{He} = 0.8 \times 11.02 \approx 8.82\) moles and \(n_{O_2} = 0.2 \times 11.02 \approx 2.20\) moles.

Step 3: Convert Moles to Mass for Helium

Find the molar mass of helium (\(4.00 \, \mathrm{g/mol}\)) and convert the moles of helium to mass: \(m_{He} = 8.82 \times 4.00 = 35.28 \, \mathrm{g}\).

Step 4: Convert Moles to Mass for Oxygen

Find the molar mass of oxygen (\(32.00 \, \mathrm{g/mol}\)) and convert the moles of oxygen to mass: \(m_{O_2} = 2.20 \times 32.00 = 70.40 \, \mathrm{g}\).

Key Concepts

Effusion

Effusion is a process where gas molecules escape through small openings in a container. It occurs due to the random motion and kinetic energy of gas particles. The rate of effusion is influenced by the size of the opening and the speed of the gas molecules.
In a spacecraft, maintaining atmospheric pressure is crucial. The atmosphere is typically composed of gases like helium and oxygen. Over time, these gases can be lost through leaks by effusion. This makes it necessary to account for and replace the air that is lost, especially on long missions.
Effusion is described by Graham's law, which states that the rate of effusion is inversely proportional to the square root of the gas's molar mass. This means lighter gases, like helium, effuse more quickly than heavier gases, such as oxygen.

Mole Calculations

Mole calculations involve determining the number of particles in a given sample of a substance. This is where Avogadro's number (approximately \(6.022 \times 10^{23}\) molecules/mol) comes into play, as it relates the amount of substance to the number of constituent particles.
In the context of the exercise, we used the ideal gas law to find out how many moles of gas were lost. This law is expressed by the equation \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is moles, \(R\) is the ideal gas constant, and \(T\) is temperature in Kelvin.
Once we calculated the total moles of gas lost, we distributed them according to the composition of the gas mixture: 80% helium and 20% oxygen. Using a simple proportion, we calculated the moles of each gas separately, which then allowed us to convert these moles into masses by multiplying by their respective molar masses.

Temperature Conversion

Temperature conversion is crucial in scientific calculations, especially when using the ideal gas law. The standard unit for temperature in these equations is Kelvin, not Celsius or Fahrenheit.
To convert from Celsius to Kelvin, simply add 273.15 to the Celsius temperature. So, for a temperature of \(20^{\circ} \text{C}\), converting to Kelvin would yield \(293.15 \text{K}\). This conversion is essential as it ensures that all units are compatible when plugging values into the ideal gas law, thereby enabling accurate calculations.

Atmospheric Composition

Atmospheric composition refers to the different gases that make up the atmosphere of an environment, like a spacecraft. In the exercise, the spacecraft's atmosphere was primarily helium (80%) and oxygen (20%). These proportions significantly affect calculations related to pressure and volume.
The composition not only impacts how we calculate the mole fractions of gases but also the mass needed to compensate for those lost to effusion. For mission planning, it's essential to know each component's percentage, so resources, like oxygen and helium, can be carefully managed over the mission's duration.
Understanding the atmosphere's composition allows us to determine how much of each gas is necessary to maintain a breathable atmosphere, ensuring the safety and health of any personnel on board.