Question

Using properties of saturated water, explain how you would determine the mole fraction of water vapor at the surface of a lake when the temperature of the lake surface and the atmospheric pressure are specified.

Short answer

Solution: 1. Use the Antoine equation to find the saturation vapor pressure of water at the given temperature. 2. Assume the partial pressure of water vapor is equal to the saturation vapor pressure. 3. Use the ideal gas law to calculate the molar composition of the atmosphere. 4. Calculate the mole fraction of water vapor by dividing the partial pressure of water vapor by the total atmospheric pressure.

Step-by-step solution

Define the problem and prepare the required data

We are given the temperature of the lake surface (T) and the atmospheric pressure (P). We need to calculate the mole fraction of water vapor (Y) present at the surface of the lake at equilibrium. To do this, first we need to find the saturation vapor pressure of water at the given temperature.

Find the saturation vapor pressure of water at the given temperature

To find the saturation vapor pressure of water at the given temperature, you can use standard steam tables or the Antoine equation. For this example, let's use the Antoine equation. The equation is: $$P_{sat} = 10^{A - \frac{B}{T + C}}$$ Where: - \(P_{sat}\) is the saturation vapor pressure (in kPa) - \(T\) is the temperature (in Celsius) - \(A\), \(B\), and \(C\) are constants specific to water, which are: - \(A = 8.07131\) - \(B = 1730.63\) - \(C = 233.426\) Plug the given temperature value into the equation to calculate \(P_{sat}\).

Calculate the partial pressure of water vapor

In this step, we will use the saturation vapor pressure calculated in the previous step to find the partial pressure of water vapor (\(P_{vapor}\)). Since the lake surface is in equilibrium with the atmosphere, we can assume that the partial pressure of water vapor is equal to the saturation vapor pressure: $$P_{vapor} = P_{sat}$$

Use the ideal gas law to calculate the mole fraction of water vapor

Now, we have the partial pressure of water vapor and the total atmospheric pressure. We can use the ideal gas law to calculate the molar composition of the atmosphere and ultimately determine the mole fraction of water vapor. The ideal gas law is given by: $$PV = nRT$$ Where: - \(P\) is the pressure (in kPa or any other consistent units) - \(V\) is the volume (in liters or m\(^3\)) - \(n\) is the number of moles - \(R\) is the ideal gas constant - \(T\) is the temperature (in Kelvin) Since we are interested in the mole fraction of water vapor (\(Y = \frac{n_{vapor}}{n_{total}}\)), we can divide the ideal gas law equation for water vapor by the equation for the total atmosphere: $$\frac{P_{vapor}V}{n_{vapor}RT} = \frac{P_{total}V}{n_{total}RT}$$ Notice that volume, the ideal gas constant, and temperature cancel out: $$\frac{P_{vapor}}{n_{vapor}} = \frac{P_{total}}{n_{total}}$$ Now we can solve for the mole fraction of water vapor as: $$Y = \frac{n_{vapor}}{n_{total}} = \frac{P_{vapor}}{P_{total}}$$ Substitute the values of \(P_{vapor}\) and \(P_{total}\) found in previous steps and calculate the mole fraction of water vapor at the surface of the lake.