Question

(III) Air that is at its dew point of 5°C is drawn into a building where it is heated to 22°C. What will be the relative humidity at this temperature? Assume constant pressure of 1.0 atm. Take into account the expansion of the air.

Short answer

The relative humidity is \(32.7\% \).

Step-by-step solution

Given data

From the given table,

The saturated vapor pressure of water at \({T_1} = {5^ \circ }{\rm{C}}\) is \({P_1} = 872\;{\rm{Pa}}\).

The saturated vapor pressure of water at \({T_2} = {20^ \circ }{\rm{C}}\) is \({P_2} = 2330\;{\rm{Pa}}\).

The saturated vapor pressure of water at \({T_3} = {25^ \circ }{\rm{C}}\) is \({P_3} = 3170\;{\rm{Pa}}\).

Let \({P_4}\) be the saturated vapor pressure of water at \({T_4} = {22^ \circ }{\rm{C}}\).

Concepts

The relative humidity is the ratio of the partial pressure of water vapor to the saturated vapor pressure at a given temperature.

Here, you have to use the equation\({\bf{relative}}\;{\bf{humidity}} = \frac{{{P_1}}}{{{P_4}}}{\bf{ \times 100\% }}\).

Calculation

Consider that vapor pressure is increasing linearly between \({20^ \circ }{\rm{C}}\) to \({25^ \circ }{\rm{C}}\).

Thus,

\(\begin{aligned}{c}\frac{{{P_3} - {P_2}}}{{{T_3} - {T_2}}} &= \frac{{{P_4} - {P_2}}}{{{T_4} - {T_2}}}\\\frac{{3170\;{\rm{Pa}} - 2330\;{\rm{Pa}}}}{{{{25}^ \circ }{\rm{C}} - {{20}^ \circ }{\rm{C}}}} &= \frac{{{P_4} - 2330\;{\rm{Pa}}}}{{{{22}^ \circ }{\rm{C}} - {{20}^ \circ }{\rm{C}}}}\\{P_4} &= 2666\;{\rm{Pa}}\end{aligned}\)

Now, the relative humidity at \({T_4} = {22^ \circ }{\rm{C}}\) is:

\(\begin{aligned}{c}{\mathop{\rm Re}\nolimits} lative\;humidity &= \frac{{{P_1}}}{{{P_4}}} \times 100\% \\ &= \frac{{872\;{\rm{Pa}}}}{{2666\;{\rm{Pa}}}} \times 100\% \\ &= 32.7\% \end{aligned}\)

Hence, the relative humidity is \(32.7\% \).