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Chapter 14: Problem 24

Consider the free surface of a lake exposed to the atmosphere. If the air at the lake surface is saturated, will the mole fraction of water vapor in air at the lake surface be the same as the mole fraction of water in the lake (which is nearly 1)?

Short Answer

Expert verified
Answer: No, the mole fraction of water vapor in saturated air at the lake surface will not be the same as the mole fraction of water in the lake (which is nearly 1). This is due to the presence of other components in the air, such as oxygen and nitrogen, which limit the mole fraction of water vapor even if it reaches its maximum concentration for a given temperature.

Step by step solution

01

Understanding mole fraction

Mole fraction is the ratio of the number of moles of a particular component in a solution to the total number of moles of all components in the solution. Mathematically, it can be represented as: Mole fraction (X) = \(\frac{\text{Moles of Component}}{\text{Total Moles in Solution}}\)
02

Comparing water mole fraction in the lake and air

In the lake, water is the primary component, and its mole fraction is close to 1, as the given exercise states. However, in the air above the lake, there are several components present, such as oxygen, nitrogen, and other gases, in addition to the water vapor. The air at the lake surface is saturated, meaning the water vapor is at its highest possible concentration for a given temperature.
03

Understanding saturation

When air is saturated, it means that the air contains the maximum amount of water vapor that it can hold at a particular temperature. Thetemperature at the lake surface determines the maximum water vapor concentration in the air.
04

Determine water vapor mole fraction in saturated air

Although the air at the lake surface is saturated with water vapor, its maximum concentration will still be limited by the other components in the air. Since the air consists of many other components, like oxygen and nitrogen, the mole fraction of water vapor will be less than 1, even if it is at its saturation point.
05

Conclusion

In conclusion, the mole fraction of water vapor in saturated air at the lake surface will not be the same as the mole fraction of water in the lake (which is nearly 1). This is because the air contains many other components such as oxygen and nitrogen, limiting the mole fraction of water vapor in the air even if it reaches its maximum concentration at a given temperature.

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Most popular questions from this chapter

During a hot summer day, a \(2-L\) bottle drink is to be cooled by wrapping it in a cloth kept wet continually and blowing air to it with a fan. If the environment conditions are \(1 \mathrm{~atm}, 80^{\circ} \mathrm{F}\), and 30 percent relative humidity, determine the temperature of the drink when steady conditions are reached.

The roof of a house is \(15 \mathrm{~m} \times 8 \mathrm{~m}\) and is made of a 20 -cm-thick concrete layer. The interior of the house is maintained at \(25^{\circ} \mathrm{C}\) and 50 percent relative humidity and the local atmospheric pressure is \(100 \mathrm{kPa}\). Determine the amount of water vapor that will migrate through the roof in \(24 \mathrm{~h}\) if the average outside conditions during that period are \(3^{\circ} \mathrm{C}\) and 30 percent relative humidity. The permeability of concrete to water vapor is \(24.7 \times 10^{-12} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m} \cdot \mathrm{Pa}\).

A sphere of ice, \(5 \mathrm{~cm}\) in diameter, is exposed to \(50 \mathrm{~km} / \mathrm{h}\) wind with 10 percent relative humidity. Both the ice sphere and air are at \(-1^{\circ} \mathrm{C}\) and \(90 \mathrm{kPa}\). Predict the rate of evaporation of the ice in \(\mathrm{g} / \mathrm{h}\) by use of the following correlation for single spheres: Sh \(=\left[4.0+1.21(\mathrm{ReSc})^{2 / 3}\right]^{0.5}\). Data at \(-1^{\circ} \mathrm{C}\) and \(90 \mathrm{kPa}: D_{\text {air- } \mathrm{H}, \mathrm{O}}=2.5 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}^{3}\), kinematic viscosity (air) \(=1.32 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\), vapor pressure \(\left(\mathrm{H}_{2} \mathrm{O}\right)=\) \(0.56 \mathrm{kPa}\) and density (ice) \(=915 \mathrm{~kg} / \mathrm{m}^{3}\).

What is the physical significance of the Schmidt number? How is it defined? To what dimensionless number does it correspond in heat transfer? What does a Schmidt number of 1 indicate?

For the absorption of a gas (like carbon dioxide) into a liquid (like water) Henry's law states that partial pressure of the gas is proportional to the mole fraction of the gas in the liquid-gas solution with the constant of proportionality being Henry's constant. A bottle of soda pop \(\left(\mathrm{CO}_{2}-\mathrm{H}_{2} \mathrm{O}\right)\) at room temperature has a Henry's constant of \(17,100 \mathrm{kPa}\). If the pressure in this bottle is \(120 \mathrm{kPa}\) and the partial pressure of the water vapor in the gas volume at the top of the bottle is neglected, the concentration of the \(\mathrm{CO}_{2}\) in the liquid \(\mathrm{H}_{2} \mathrm{O}\) is (a) \(0.003 \mathrm{~mol}-\mathrm{CO}_{2} / \mathrm{mol}\) (b) \(0.007 \mathrm{~mol}-\mathrm{CO}_{2} / \mathrm{mol}\) (c) \(0.013 \mathrm{~mol}-\mathrm{CO}_{2} / \mathrm{mol}\) (d) \(0.022 \mathrm{~mol}-\mathrm{CO}_{2} / \mathrm{mol}\) (e) \(0.047 \mathrm{~mol}-\mathrm{CO}_{2} / \mathrm{mol}\)
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