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 $32.7\mathrm{\%}$.

Step-by-step solution

Given data

From the given table,

The saturated vapor pressure of water at $T_1=5^{\circ }\mathrm{C}$ is $P_1=872\mathrm{P}\mathrm{a}$.

The saturated vapor pressure of water at $T_2={20}^{\circ }\mathrm{C}$ is $P_2=2330\mathrm{P}\mathrm{a}$.

The saturated vapor pressure of water at $T_3={25}^{\circ }\mathrm{C}$ is $P_3=3170\mathrm{P}\mathrm{a}$.

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

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$\mathbf{r}\mathbf{e}\mathbf{l}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{v}\mathbf{e}\mathbf{h}\mathbf{u}\mathbf{m}\mathbf{i}\mathbf{d}\mathbf{i}\mathbf{t}\mathbf{y}=\frac{P_1}{P_4}\times \mathbf{100}\mathbf{\%}$.

Calculation

Consider that vapor pressure is increasing linearly between ${20}^{\circ }\mathrm{C}$ to ${25}^{\circ }\mathrm{C}$.

Thus,

$\begin{array}{rl}c\frac{P_3-P_2}{T_3-T_2}& =\frac{P_4-P_2}{T_4-T_2}\\ \frac{3170\mathrm{P}\mathrm{a}-2330\mathrm{P}\mathrm{a}}{{25}^{\circ }\mathrm{C}-{20}^{\circ }\mathrm{C}}& =\frac{P_4-2330\mathrm{P}\mathrm{a}}{{22}^{\circ }\mathrm{C}-{20}^{\circ }\mathrm{C}}\\ P_4& =2666\mathrm{P}\mathrm{a}\end{array}$

Now, the relative humidity at $T_4={22}^{\circ }\mathrm{C}$ is:

$\begin{array}{rl}c\mathrm{Re}lativehumidity& =\frac{P_1}{P_4}\times 100\mathrm{\%}\\ & =\frac{872\mathrm{P}\mathrm{a}}{2666\mathrm{P}\mathrm{a}}\times 100\mathrm{\%}\\ & =32.7\mathrm{\%}\end{array}$

Hence, the relative humidity is $32.7\mathrm{\%}$.