Question

Consider a glass of water in a room that is at \(20^{\circ} \mathrm{C}\) and 40 percent relative humidity. If the water temperature is \(15^{\circ} \mathrm{C},\) determine the vapor pressure \((a)\) at the free surface of the water and \((b)\) at a location in the room far from the glass.

Short answer

#tag_title#Solution (continued) #tag_content#For temperature T = 15°C: $P_{sat}(15^{\circ}\mathrm{C}) = 10^{8.07131 - \frac{1730.63}{15 + 233.426}}$ $P_{sat}(15^{\circ}\mathrm{C}) = 12.8\,\mathrm{mm\,Hg}$ For temperature T = 20°C: $P_{sat}(20^{\circ}\mathrm{C}) = 10^{8.07131 - \frac{1730.63}{20 + 233.426}}$ $P_{sat}(20^{\circ}\mathrm{C}) = 17.5\,\mathrm{mm\,Hg}$ #Step 2: Determine the vapor pressure at the free surface of the water (location A) As the water glass is at 15°C, the vapor pressure at the free surface (location A) will be equal to the saturation pressure at 15°C: $P_A = P_{sat}(15^{\circ}\mathrm{C}) = 12.8\,\mathrm{mm\,Hg}$ #Step 3: Determine the vapor pressure at the location in the room far from the glass (location B) To find the vapor pressure at location B, we first need to use the relative humidity. As the relative humidity is 40%, we know that the vapor pressure at location B is 40% of the saturation pressure at 20°C: $P_B = 0.40 * P_{sat}(20^{\circ}\mathrm{C}) = 0.40 * 17.5\,\mathrm{mm\,Hg} = 7.0\,\mathrm{mm\,Hg}$ #Summary# The vapor pressure at the free surface of the water in a glass (location A) is 12.8 mm Hg, and the vapor pressure at a location in the room far from the glass (location B) is 7.0 mm Hg.

Step-by-step solution

Calculate the saturation pressure of water vapor at the given temperatures

We can use the Antoine equation to calculate the saturation pressure of water vapor at a given temperature. The Antoine equation is: $$P_{sat} = 10^{A - \frac{B}{T + C}}$$ Where: \(P_{sat}\) - saturation pressure (mm Hg) \(T\) - temperature (in Celsius) \(A\), \(B\), and \(C\) - Antoine constants for water The Antoine constants for water are: \(A = 8.07131\) \(B = 1730.63\) \(C = 233.426\) Now, we can calculate the saturation pressure at \(15^{\circ}\mathrm{C}\) and \(20^{\circ}\mathrm{C}\).