Question

Dry air near sea level has the following composition by volume: \(\mathrm{N}_{2}, 78.08\) percent; \(\mathrm{O}_{2}, 20.94\) percent; \(\mathrm{Ar}, 0.93\) percent; \(\mathrm{CO}_{2}, 0.05\) percent. The atmospheric pressure is 1.00 atm. Calculate (a) the partial pressure of each gas in atmospheres and (b) the concentration of each gas in \(\mathrm{mol} / \mathrm{L}\) at \(0^{\circ} \mathrm{C}\). (Hint: Because volume is proportional to the number of moles present, mole fractions of gases can be expressed as ratios of volumes at the same temperature and pressure.)

Short answer

Partial pressures: \(N_2 = 0.7808\) atm, \(O_2 = 0.2094\) atm, \(Ar = 0.0093\) atm, \(CO_2 = 0.0005\) atm. Concentrations: \(N_2 = 0.0344\) mol/L, \(O_2 = 0.0093\) mol/L, \(Ar = 0.0004\) mol/L, \(CO_2 = 0.000022\) mol/L.

Step-by-step solution

Determine Partial Pressure of Each Gas

To find the partial pressure of each gas, we apply Dalton's Law of Partial Pressures. This law states that the partial pressure of a gas in a mixture is equal to the mole fraction of the gas times the total pressure. Given the percentage by volume, convert each percentage directly to mole fraction since \( P_{ ext{gas}} = \text{mole fraction} \times P_{\text{total}} \), with \(P_{\text{total}} = 1.00 \text{ atm}\). Calculate:\[ P_{N_2} = 0.7808 \times 1.00 = 0.7808 \,\text{atm}\]\[ P_{O_2} = 0.2094 \times 1.00 = 0.2094 \,\text{atm}\]\[ P_{Ar} = 0.0093 \times 1.00 = 0.0093 \,\text{atm}\]\[ P_{CO_2} = 0.0005 \times 1.00 = 0.0005 \,\text{atm}\]

Calculate Molar Concentration at 0°C

Use the ideal gas law \( PV = nRT \) to determine the molar concentration \( C = \frac{n}{V} \). At STP (0°C and 1 atm), \( R = 0.0821 \, \text{L atm mol}^{-1} \text{K}^{-1} \) and \( T = 273.15 \, \text{K} \), the concentration can be simplified to:\[ n/V = P/RT = \frac{P}{0.0821 \times 273.15} \]Calculate for each gas using its partial pressure:\[ C_{N_2} = \frac{0.7808}{0.0821 \times 273.15} \approx 0.0344 \, \text{mol/L} \]\[ C_{O_2} = \frac{0.2094}{0.0821 \times 273.15} \approx 0.0093 \, \text{mol/L} \]\[ C_{Ar} = \frac{0.0093}{0.0821 \times 273.15} \approx 0.0004 \, \text{mol/L} \]\[ C_{CO_2} = \frac{0.0005}{0.0821 \times 273.15} \approx 0.000022 \, \text{mol/L} \]

Key concepts

Partial Pressure

Dalton's Law of Partial Pressures is a fundamental principle used to calculate the pressure exerted by an individual gas in a mixture. It states that the total pressure exerted by a mixture of non-reactive gases is the sum of the partial pressures of each gas. Each gas in a mixture contributes to the total pressure based on its mole fraction.
To find the partial pressure of a gas, multiply its mole fraction by the total pressure. For example, if we have dry air at 1 atm total pressure, and nitrogen makes up 78.08% by volume, the partial pressure of nitrogen, \( P_{N_2} \), is found by:

• Calculating the mole fraction: \( 0.7808 \)
• Multiplying by the total pressure: \( P_{N_2} = 0.7808 \times 1.00 \text{ atm} = 0.7808 \text{ atm} \)

This method can be applied to calculate the partial pressures of oxygen, argon, and carbon dioxide in the mixture as shown. The sum of these partial pressures equals the total pressure of 1 atm.

Ideal Gas Law

The ideal gas law is a valuable tool for relating the properties of gases. Its formula, \( PV = nRT \), connects pressure \( P \), volume \( V \), the number of moles \( n \), the ideal gas constant \( R \), and temperature \( T \). Using the ideal gas law, we can solve for various properties when certain variables are known.
For determining molar concentration, \( C = \frac{n}{V} \), we rearrange the ideal gas law:

• \( C = \frac{P}{RT} \)
• \( R = 0.0821 \text{ L atm mol}^{-1} \text{ K}^{-1} \)
• For STP (0°C or 273.15 K), it simplifies calculations.

The formula helps determine how many moles of a gas exist in a specific volume under known conditions, such as 1 atm and 0°C.

Molar Concentration

Once the partial pressure is known, you can find the molar concentration of each gas using the relationship derived from the ideal gas law: \( C = \frac{P}{RT} \). This expression allows us to calculate how many moles of gas are present per liter of gas.
To find the molar concentration of nitrogen at 0°C and 1 atm:

• Calculate \( C_{N_2} = \frac{0.7808}{0.0821 \times 273.15} \)
• This results in approximately \( 0.0344 \text{ mol/L} \)

Repeat for other gases with their respective partial pressures. This interpretation helps understand the number of molecules within a specific volume, useful in many scientific calculations.

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of the air in the Earth's atmosphere. It varies depending on altitude, weather, and temperature. Near sea level, the standard atmospheric pressure is approximately 1 atm.
Understanding atmospheric pressure is crucial for applying Dalton's Law. It serves as the total pressure in these calculations. Therefore, knowing the standard atmospheric pressure allows scientists and students to predict how gases will behave under different conditions.

• Influences weather patterns and conditions.
• Essential in aviation and marine navigation.
• Critical for assessing how gases interact and perform experiments.

It acts as a foundational element in many chemistry equations and calculations.