Question

Determine the mole fraction of dry air at the surface of a lake whose temperature is \(15^{\circ} \mathrm{C}\). Take the atmospheric pressure at lake level to be \(100 \mathrm{kPa}\).

Short answer

Question: Calculate the mole fraction of dry air at the surface of a lake with a temperature of 15°C and a total atmospheric pressure of 100 kPa. Answer: To calculate the mole fraction of dry air at the lake surface, follow these steps: 1. List the major constituent gases in dry air (N2, O2, and Ar) and their approximate percentages by volume. 2. Convert the percentages by volume to partial pressures using the given total pressure. 3. Use the Ideal Gas Law to find the moles of each gas at the given temperature and pressure. 4. Calculate the total moles of dry air by adding the moles of the constituent gases. 5. Determine the mole fraction of dry air by dividing the moles of each constituent gas by the total moles of dry air, and then summing the individual mole fractions. Byfollowing these steps, you will calculate the mole fraction of dry air at the lake surface.

Step-by-step solution

List the constituent gases in dry air

Dry air is generally considered to be a composition of certain gases with the following approximate percentages by volume: Nitrogen (N2) - 78%, Oxygen (O2) - 21%, Argon (Ar) - 1%, and trace amounts of other gases. In this problem, we will focus on the major components (N2, O2, and Ar) and ignore the trace amounts of other gases.

Convert percentages by volume to partial pressures

Given the total atmospheric pressure at the lake surface is 100 kPa, we can calculate the partial pressures of the major constituents of dry air as follows:$$ P_{N2} = 0.78 * 100 \, \text{kPa} = 78 \, \text{kPa} \\ P_{O2} = 0.21 * 100 \, \text{kPa} = 21 \, \text{kPa} \\ P_{Ar} = 0.01 * 100 \, \text{kPa} = 1 \, \text{kPa} $$

Use the Ideal Gas Law to find the moles of each gas

We will use the Ideal Gas Law to convert the partial pressures of each gas to moles. The Ideal Gas Law is given by: $$ PV = nRT $$ where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin. We can rearrange the Ideal Gas Law to solve for n: $$ n = \frac{PV}{RT} $$ First, convert the temperature to Kelvin: $$ T = 15 + 273.15 = 288.15 \, \text{K} $$ Next, use the ideal gas constant R = 8.314 J/(mol·K), and assume a volume of 1 m³ since we want to find the mole fraction and not the absolute number of moles (the volume will cancel out). Calculate the moles for each gas:$$ n_{N2} = \frac{78 \, \text{kPa} \cdot 1 \, \text{m}^3}{8.314 \, \text{J/(mol·K)} \cdot 288.15 \, \text{K}} \\ n_{O2} = \frac{21 \, \text{kPa} \cdot 1 \, \text{m}^3}{8.314 \, \text{J/(mol·K)} \cdot 288.15 \, \text{K}} \\ n_{Ar} = \frac{1 \, \text{kPa} \cdot 1 \, \text{m}^3}{8.314 \, \text{J/(mol·K)} \cdot 288.15 \, \text{K}} $$

Calculate the total moles

Add up the moles of each constituent gas to find the total moles of dry air:$$ n_\text{total} = n_{N2} + n_{O2} + n_{Ar} $$

Calculate the mole fraction of dry air

Calculate the mole fraction of each constituent gas by dividing each component's moles by the total moles:$$ \text{mole fraction}_{N2} = \frac{n_{N2}}{n_\text{total}} \\ \text{mole fraction}_{O2} = \frac{n_{O2}}{n_\text{total}} \\ \text{mole fraction}_{Ar} = \frac{n_{Ar}}{n_\text{total}} $$ Since the mole fractions sum up to 1, we can say the mole fraction of dry air is equal to the sum of the mole fractions of its major components:$$ \text{mole fraction}_\text{dry air} = \text{mole fraction}_{N2} + \text{mole fraction}_{O2} + \text{mole fraction}_{Ar} $$

Key concepts

Ideal Gas Law

, where \(P\) represents the pressure of the gas, \(V\) is the volume it occupies, \(n\) is the number of moles of gas present, \(R\) is the ideal gas constant, and \(T\) is the temperature measured in Kelvin. This law allows us to determine one of these variables if we know the values for the other three. In the context of this exercise, we use the Ideal Gas Law to calculate the amount in moles of gases in dry air, using the known values for pressure, temperature (converted to Kelvin), and assuming a fixed volume.

The versatility of the Ideal Gas Law makes it an essential tool for scientists and engineers, especially when dealing with gaseous reactions and processes. By isolating and solving for \(n\), the number of moles, one can also derive the mole fraction when given the appropriate information about the gas mixture.

Partial Pressure

Partial pressure is the pressure contributed by a single gas in a mixture of gases. Each gas in a mixture exerts pressure independently, as if it were the only gas present. The partial pressure of a gas can be calculated by using the total pressure of the gas mixture and the mole fraction of the gas in question. In the step-by-step solution presented, the partial pressures of nitrogen (\(N_2\)), oxygen (\(O_2\)), and argon (\(Ar\)) were computed based on their respective volume percentages in dry air.

Understanding partial pressures is crucial because it directly relates to the mole fraction, which is a measure of the abundance of a specific gas relative to the total number of moles of all gases present. For a given total pressure, knowing the mole fractions allows for the calculation of each gas's partial pressure, aiding in a deeper understanding of gas behavior.

Composition of Dry Air

Dry air is primarily composed of nitrogen, oxygen, and argon, with small amounts of other gases. For simplicity and practical purposes, many calculations involving air's properties consider only the major components, which is the case in the exercise provided. Knowing the composition of dry air is fundamental when we are applying the Ideal Gas Law to atmospheric conditions or working out gas-related calculations, such as determining mole fractions or partial pressures. Essentially, having an accurate model of air composition allows for more precise and applicable results in scientific analysis.

Converting Temperatures to Kelvin

Temperature conversions are essential when dealing with gas laws because the Ideal Gas Law requires temperature to be in Kelvin. To convert Celsius to Kelvin, you add 273.15 to the Celsius temperature. This step is crucial because the Kelvin scale is an absolute temperature scale where 0 Kelvin represents absolute zero - the theoretical point where particles have minimal kinetic energy.

Always remember to use the Kelvin scale in gas-related calculations to ensure accuracy. Incorrect temperature units can lead to erroneous results, potentially impacting any related experiments or studies.

Calculating Moles of Gas

\(n = \frac{PV}{RT}\).

By calculating the individual moles of each gas in a mixture and knowing their total, we can work out the mole fraction of each component. The mole fraction is a dimensionless number that represents the ratio of the number of moles of a component to the total number of moles in the mixture. This concept is particularly useful in chemistry and physics to describe the concentration of each component in a mixture.