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 \(1.00 \mathrm{~atm} .\) Calculate (a) the partial pressure of each gas in atm and (b) the concentration of each gas in moles per liter at \(0^{\circ} \mathrm{C}\).

Short answer

The partial pressure and concentration of each gas: for N2 are approximately 0.780 atm and \(0.033 mol/L\), for O2 are approximately 0.209 atm and \(0.009 mol/L\), for Ar are approximately 0.009 atm and \(0.0004 mol/L\), and for CO2 are approximately 0.0005 atm and \(0.00002 mol/L\).

Step-by-step solution

Calculating the Partial Pressure

The atmospheric pressure is given as 1.00 atm. Knowing that each gas contributes to the total pressure in the ratio of its volume percent, the partial pressure of each gas can be calculated by multiplying the atmospheric pressure by their individual volume percentage divided by 100. For instance, the partial pressure of N2 can be calculated as \(P_{N_2} = 1.00 atm * 78.08/100\). Similar computations will be done for O2, Ar, and CO2.

Determine the Temperature in Kelvin

Conversion of the temperature from Celsius to Kelvin is essential when calculating the number of moles in the Ideal Gas Law. The temperature in Kelvin can be determined by adding 273.15 to the Celsius temperature, so \(T = 0^{\circ}C + 273.15 = 273.15K\).

Calculate the Concentrations

The concentration of each gas in moles per liter can be calculated by substituting the partial pressure of that specific gas and the temperature into the Ideal Gas Law, \(PV = nRT\), and solving for n/V which gives us the concentration (C). The molar gas constant (R) is \(0.0821 L.atm/K.mol\). For example, the concentration of N2 is calculated as following: \(C_{N_2} = P_{N_2}/(RT)\). Similar computations will be done for O2, Ar, and CO2.

Key concepts

Composition of Air

Air is a mixture of different gases. Near sea level, dry air primarily consists of the following four gases:

• Nitrogen (\(\mathrm{N}_2\)): 78.08%
• Oxygen (\(\mathrm{O}_2\)): 20.94%
• Argon (\(\mathrm{Ar}\)): 0.93%
• Carbon Dioxide (\(\mathrm{CO}_2\)): 0.05%

These percentages represent the volume fraction of each gas in the air. Each gas contributes to the total atmospheric pressure, which is the sum of the partial pressures of all gases present. By knowing the composition by volume, we can calculate the partial pressure of each gas, allowing us to understand how much each gas contributes to the atmosphere's overall pressure.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics used to relate the pressure, volume, temperature, and number of moles of a gas sample. The law is expressed as:\[PV = nRT\]where:

• \(P\) = pressure of the gas
• \(V\) = volume occupied by the gas
• \(n\) = number of moles of gas
• \(R\) = ideal gas constant (\(0.0821\ \text{L.atm/K.mol}\))
• \(T\) = temperature in Kelvin

When working with gases, it is important to always convert temperatures to Kelvin by adding 273.15 to the Celsius temperature. This ensures accurate calculations of the other variables. By rearranging the Ideal Gas Law, we can solve for the concentration of gas, which is the number of moles per liter:\[C = \frac{n}{V} = \frac{P}{RT}\]This formula allows us to determine how concentrated a gas is within a mixture.

Moles per Liter

Concentration in chemistry often refers to how many moles of a substance are present in one liter of solution or mixture. When we talk about concentrations of gases, we typically calculate how many moles of the gas can be found in a liter of air under specific conditions.To find the concentration of a particular gas in moles per liter, use the formula derived from the Ideal Gas Law:\[C = \frac{P}{RT}\]where:

• \(C\) is the concentration in moles per liter.
• \(P\) is the partial pressure of the gas.
• \(R\) is the ideal gas constant (\(0.0821\ \text{L.atm/K.mol}\)).
• \(T\) is the absolute temperature in Kelvin.

Calculating the concentration of each gas in air using this method provides insight into how significantly each gas is present in terms of its amount of substance, rather than just its volume fraction, which can influence various chemical and physical properties of the air.