Question

The composition of dry air by volume is \(78.1 \% \mathrm{~N}_{2}, 20.9 \% \mathrm{O}_{2}\) and \(1 \%\) other gases. Calculate the partial pressures, in atmospheres, in a tank of dry air compressed to \(10.0\) atmospheres.

Short answer

The partial pressures of each gas in the tank are: Nitrogen (N₂): \(7.81~atm\), Oxygen (O₂): \(2.09~atm\), and Other gases: \(0.10~atm\).

Step-by-step solution

Understand the mole fractions of gases in the dry air.

The composition of dry air by volume is given as 78.1% of nitrogen (N₂), 20.9% of oxygen (O₂), and 1% of other gases. From this, we can find the mole fractions for each gas. For nitrogen (N₂): Mole fraction = (Volume fraction of nitrogen) / (Total volume fraction) = 78.1% = \(0.781\) For oxygen (O₂): Mole fraction = (Volume fraction of oxygen) / (Total volume fraction) = 20.9% = \(0.209\) For other gases: Mole fraction = (Volume fraction of other gases) / (Total volume fraction) = 1% = \(0.01\)

Calculate the partial pressure of each gas using the mole fractions and total pressure.

The formula for partial pressure is given by: Partial pressure = (Mole fraction) x (Total pressure) Given total pressure in the tank = 10.0 atmospheres. For nitrogen (N₂): Partial pressure = (0.781) x (10.0 atmospheres) = \(7.81~atm\) For oxygen (O₂): Partial pressure = (0.209) x (10.0 atmospheres) = \(2.09~atm\) For other gases: Partial pressure = (0.01) x (10.0 atmospheres) = \(0.10~atm\)

Report the partial pressures for each gas in the tank.

The partial pressures of each gas in the tank are as follows: 1. Nitrogen (N₂): \(7.81~atm\) 2. Oxygen (O₂): \(2.09~atm\) 3. Other gases: \(0.10~atm\)

Key concepts

Mole Fraction

The idea of mole fraction is a fundamental concept in chemistry and is particularly important when discussing gases. The mole fraction of a gas is a way to express the concentration of that gas in a mixture, like air. It is calculated by taking the number of moles of a particular component and dividing it by the total number of moles in the mixture.
For our specific case of dry air, the composition by volume is:

• 78.1% nitrogen ( ₂)
• 20.9% oxygen ( ₂)
• 1% other gases

This percentage is directly used as the mole fraction for each gas as each percentage represents a ratio to the total. For instance, the mole fraction of nitrogen ( ₂) is 0.781, oxygen ( ₂) is 0.209, and for other gases, it is 0.01.
These mole fractions are crucial as they are used to calculate partial pressures.

Dry Air Composition

Dry air is a mixture of gases with nitrogen and oxygen being the most abundant elements. Understanding its composition helps us predict how different gases interact and behave under various conditions.
The composition by volume is approximately:

• 78.1% nitrogen ( ₂)
• 20.9% oxygen ( ₂)
• 1% of other trace gases, such as argon, carbon dioxide, neon, etc.

Since water vapor is excluded in dry air, this composition remains constant for calculations involving atmospheric pressure and gas laws. This steadiness is why we can use volume fraction to simplify our calculations with mole fractions when calculating things like partial pressures.

Total Pressure

Total pressure refers to the pressure exerted by a mixture of gases in a closed container. In the context of our exercise, it is the pressure inside the tank.
When you have a mixture of gases, like in a tank of dry air, the total pressure is simply the sum of the partial pressures of all individual gases combined.
In our exercise, the tank was compressed to 10.0 atmospheres. Knowing the mole fractions for each gas, total pressure allows calculation of each gas's contribution to this overall pressure via the formula:

• Partial Pressure = Mole Fraction × Total Pressure

This principle means the total pressure gives us a useful starting point to find detailed insights about each gas in the mixture.

Atmospheric Pressure

Atmospheric pressure is the pressure exerted by the weight of air in the atmosphere on any surface. It is usually measured in atmospheres, and at sea level, it is approximately 1 atmosphere.
Knowing atmospheric pressure is useful because it serves as a reference point for many scientific activities and calculations.
For instance, in our exercise, the tank's total pressure is 10 atmospheres, significantly higher than normal atmospheric pressure. This higher pressure changes the behavior of gases and makes their partial pressures important for calculations. It illustrates how deviations from standard atmospheric pressure require adjustments in calculations for partial pressures of gases.