Chapter 13, Problem 70P

To determine

To calculate:The relative humidity of the air when the temperature is changes from $5^0\text{C}$ to ${25}^0\text{C}$

Answer to Problem 70P

The relative humidity of the water vapor is $27.5\%$ .

Explanation of Solution

Given:

Pressure of the air= $1\text{atm}$

  $\begin{array}{l}{T}_1=5^{\text{0}}\text{C}\\ T_2={25}^{\text{0}}\text{C}\end{array}$

Formula used:

Ideal gas equation:

  $\begin{array}{l}PV=nRT\\ n=\frac{PV}{RT}\end{array}$

Here, $P$ is the pressure,

  $V$ is the volume,

  $T$ is the temperature,

  $R$ is the universal gas constant,

  $n$ is the no of moles.

Calculation:

At lower temperature the air is saturated. The vapor pressure of the water is $872\text{Pa}$ .

Ideal gas equation for the change in the volume

  $\begin{array}{l}PV=nRT\\ \frac{P}{nR}=\frac{T}{V}=\text{cont}\text{.}\\ \frac{T_1}{V_1}=\frac{T_2}{V_2}\end{array}$

And the value of the vapor pressure when the air is moves from outside to the inside

  $\begin{array}{l}{P}_{\text{inside}}=\frac{nR{T}_{\text{in}}}{V_{\text{in}}}=\frac{nR{T}_{\text{out}}}{V_{\text{out}}}=P_{\text{out}}\\ =872\text{Pa}\end{array}$

Vapor pressure at the inside temperature $3170\text{Pa}$

  $\begin{array}{l}\text{relative humidity}=\frac{872\text{Pa}}{3170\text{Pa}}=0.275\\ =27.5\%\end{array}$

Conclusion:

The relative humidity of the water vapor is $27.5\%$ .