Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

(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

Expert verified

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

Step by step solution

01

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 \({T_4} = {22^ \circ }{\rm{C}}\).

02

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\({\bf{relative}}\;{\bf{humidity}} = \frac{{{P_1}}}{{{P_4}}}{\bf{ \times 100\% }}\).

03

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 \(32.7\% \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Study anywhere. Anytime. Across all devices.

Sign-up for free