Question

Consider a glass of water in a room at \(20^{\circ} \mathrm{C}\) and \(97 \mathrm{kPa}\). If the relative humidity in the room is 100 percent and the water and the air are in thermal and phase equilibrium, determine (a) the mole fraction of the water vapor in the air and \((b)\) the mole fraction of air in the water.

Short answer

(b) What is the mole fraction of air in the water? (a) The mole fraction of water vapor in the air is 0.0241. (b) The mole fraction of air (nitrogen) in the water is 1.238 x 10^-3.

Step-by-step solution

Calculate Partial Pressure of Water Vapor

Using the Antoine equation, we can find the saturation pressure of water vapor at the given temperature (20°C). The equation is: \(P_{sat}=10^{A-\frac{B}{T+C}}\) where \(P_{sat}\) is the saturation pressure, \(T\) is the temperature in Celsius, and \(A\), \(B\), and \(C\) are constants for water. For water, the constants are \(A = 8.07131\), \(B = 1730.63\), and \(C = 233.426\). Plug the temperature value to the Antoine equation: \(P_{sat} = 10^{8.07131 - \frac{1730.63}{20 + 233.426}}\) Calculate \(P_{sat}\): \(P_{sat} = 2.339\mathrm{kPa}\)

Find Partial Pressure of Air

To find the partial pressure of air, we can use Dalton's law of partial pressures: \(P_{total} = P_{H_2O} + P_{air}\) where \(P_{total}\) is the total pressure, \(P_{H_2O}\) is the partial pressure of water vapor, and \(P_{air}\) is the partial pressure of air. We are given the total pressure (\(97\mathrm{kPa}\)) and we calculated the partial pressure of water vapor in the previous step. So we can find the partial pressure of air: \(P_{air} = 97\mathrm{kPa} - 2.339\mathrm{kPa} = 94.661\mathrm{kPa}\)

Calculate Mole Fraction of Water Vapor in Air

Now we can find the mole fraction of water vapor in the air using the partial pressures we calculated: \(y_{H_2O} = \frac{P_{H_2O}}{P_{total}}\) Insert the known values: \(y_{H_2O} = \frac{2.339\mathrm{kPa}}{97\mathrm{kPa}}\) Calculate \(y_{H_2O}\): \(y_{H_2O} = 0.0241\) So the mole fraction of water vapor in the air is 0.0241.

Calculate Mole Fraction of Air in Water

Assuming that air is mostly nitrogen, we can use Henry's Law to find the mole fraction of air in the water. The equation is: \(P_{air} = K_{H} \times x_{air}\) where \(K_{H}\) is the Henry's Law constant for nitrogen in water, and \(x_{air}\) is the mole fraction of air (nitrogen) in water. The value of \(K_{H}\) for nitrogen in water at 20°C is \(76.48\frac{\mathrm{kPa}}{\mathrm{molality}}\). Rearranging the equation: \( x_{air} = \frac{P_{air}}{K_{H}}\) Insert the known values: \(x_{air} = \frac{94.661\mathrm{kPa}}{76.48\frac{\mathrm{kPa}}{\mathrm{molality}}}\) Calculate \(x_{air}\): \(x_{air} = 1.238\times10^{-3}\) So the mole fraction of air (nitrogen) in the water is \(1.238\times10^{-3}\). To summarize the results: (a) The mole fraction of water vapor in the air is 0.0241. (b) The mole fraction of air (nitrogen) in the water is \(1.238\times10^{-3}\).

Key concepts

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. It is the ratio of the number of moles of a particular substance to the total number of moles of all substances present in the mixture.

This measure is particularly useful in chemical dynamics and phase equilibrium problems.

The formula for calculating mole fraction (x or y) is given by: \( y_i = \frac{n_i}{n_{total}} \), where \( n_i \) is the number of moles of component \( i \), and \( n_{total} \) is the total number of moles of all components.

**Importance in Phase Equilibrium**:

• It is crucial for calculating other properties and behaviors of mixtures, including vapor pressure and solubility.
• In our problem, we used the mole fraction to assess the distribution of water vapor in a room's air and air (primarily nitrogen) in water.

Antoine Equation

The Antoine Equation is a mathematical expression used to estimate the vapor pressure of pure substances as a function of temperature. It is an empirical relation that simplifies the complex dependency of vapor pressure on temperature. This equation is particularly valuable in phase equilibrium calculations.

The general form of the Antoine Equation is: \( P_{sat} = 10^{A - \frac{B}{T + C}} \)Where,

• \( P_{sat} \): Saturation pressure of the vapor.
• \( T \): Temperature (usually in Celsius).
• \( A, B, C \): Component-specific constants.

**Usage in Example**:

• We calculated the saturation pressure of water vapor at \(20^{\circ}\text{C}\) using its known constants.
• This empirical equation allowed us to determine the partial pressure values necessary for subsequent mole fraction calculations.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of individual gases. Each gas in the mixture contributes to the total pressure according to its own pressure that it would exert if it occupied the entire volume alone.

The law is expressed as: \( P_{total} = P_1 + P_2 + \ldots + P_n \), where \( P_1, P_2, \ldots, P_n \) are the partial pressures of each gas.

**Application**:

• By knowing the total pressure and the saturation pressure of water vapor, we found the partial pressure exerted by air.
• Understanding this breakdown is vital for calculating the individual mole fractions in the mixture, as shown in our exercise example.

Henry's Law

Henry's Law is fundamental in understanding how gases dissolve in liquids. It describes the proportionality between the concentration of a dissolved gas in a liquid and its partial pressure above the liquid, maintaining a factor known as the Henry's Law constant. This relationship is often stated as: \( P = K_H \cdot x \), where:

• \( P \): Partial pressure of the gas.
• \( x \): Mole fraction of the gas in the liquid.
• \( K_H \): Henry's Law constant for the specific gas-liquid system.

**Role in Solution**:

• Henry's Law helped us calculate the mole fraction of air in the water.
• The constant varies with temperature and is specific for different gases and solutes, such as nitrogen's value used here to establish how air dissolves in the water at \(20^{\circ}\text{C}\).