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Chapter 14: Problem 40

Consider a glass of water in a room at \(15^{\circ} \mathrm{C}\) 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

Expert verified
Answer: The mole fraction of water vapor in the air (Y_H2O) is 0.0175, and the mole fraction of air in the water (X_air) is \(1.26 \times 10^{-3}\).

Step by step solution

01

Determine the vapor pressure of water at the given temperature

The vapor pressure of water at a given temperature can be found using the Antoine equation or by referring to a vapor pressure table. At \(15^{\circ} \mathrm{C}\), the vapor pressure of water is approximately \(1.70\mathrm{kPa}\).
02

Calculate the mole fraction of water vapor in the air

To calculate the mole fraction of water vapor in the air, divide the vapor pressure of the water by the total pressure of the room: Mole fraction of water vapor (Y_H2O) = \(\frac{P_{H2O}}{P_{total}}\) Using the given values, we have: Y_H2O = \(\frac{1.70\mathrm{kPa}}{97\mathrm{kPa}}\) = 0.0175
03

Determine the partial pressure of air

Subtract the vapor pressure of the water from the total pressure to find the partial pressure of air: P_air = \(P_{total} - P_{H2O}\) = \(97\mathrm{kPa} - 1.70\mathrm{kPa}\) = \(95.3\mathrm{kPa}\)
04

Calculate the mole fraction of air in the water using Henry's law

Henry's law states that the solubility of a gas in a liquid is proportional to its partial pressure in the gas phase. For air in water, we can use a value for Henry's law constant (K_H) of approximately \(75440\mathrm{\frac{mol}{m^3\cdot Pa}}\). The mole fraction of air in the water (X_air) can be calculated as follows: X_air = \(\frac{P_{air}}{K_H}\) Substituting the values, we get: X_air = \(\frac{95.3\mathrm{kPa}}{75440\mathrm{\frac{mol}{m^3\cdot Pa}}}\) = \(1.26 \times 10^{-3}\)
05

Present the final results

The mole fraction of water vapor in the air (Y_H2O) is 0.0175, and the mole fraction of air in the water (X_air) is \(1.26 \times 10^{-3}\).

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Most popular questions from this chapter

When the density of a species \(A\) in a semi-infinite medium is known at the beginning and at the surface, explain how you would determine the concentration of the species \(A\) at a specified location and time.

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