Q60PE - College Physics (Urone)

Question

Question: If the relative humidity is 90.0% on a muggy summer morning when the temperature is 20.0ºC, what will it be later in the day when the temperature is 30.0ºC, assuming the water vapor density remains constant?

Step-by-Step Solution

Answer

Answer

The relative humidity is equal to 50.92%

Step 1: Definition of relative humidity

The relative humidity is the ratio of vapor density to the saturation vapor density expressed in percentage.

$\mathbf{Percent} \mathbf{RelativeHumidity} \mathbf{=} \frac{\mathbf{Vapor} \mathbf{Density}}{\mathbf{SaturationVapor} \mathbf{Density}}\mathbf{\times }\mathbf{100}\mathbf{\%}$

Here, vapor density is constant and saturation vapor density varies with the temperature. Saturation vapor density corresponding to 20.0°C is 17.2 g/m³ and 30.0°C is 30.4 g/m³.

Step 2: Find relative humidity at both the temperature

Relative humidity at 20.0°C is given as 90.0%. Substitute these values in the equation to find relative humidity at both temperatures.

$$\begin{gathered} 90.0\%=\frac{{\displaystyle vapor density}}{17.2 g/m^3}\times 100 \\ \text{Relative} \text{humidity} \text{at 30}.0°C=\frac{{\displaystyle vapor density}}{30.4g/m^3}\times 100 \end{gathered}$$

Dividing these two equations gives;

$$\begin{gathered} \frac{90.0\%}{\text{Relative} \text{humidity} \text{at 30}.0°C}=\frac{30.4g/m^3}{17.2g/m^3} \\ \text{Relative} \text{humidity} \text{at 30}.0°C=90.0\%\times \frac{{\displaystyle 17.2g/m^3}}{{\displaystyle 30.4g/m^3}} \\ ==50.92\% \end{gathered}$$

So, the relative humidity at 30.0°C is 50.92%.