Question

On entering a dwelling maintained at ${20}^{\circ }\mathrm{C}$ from the outdoors where the temperature is ${10}^{\circ }\mathrm{C}$, a person's eye-glasses are observed not to become fogged. A humidity gauge indicates that the relative humidity in the dwelling is $55\mathrm{\%}$. Can this reading be correct? Provide supporting calculations.

Short answer

No, the humidity reading is likely incorrect because glasses should fog if the reading is true.

Step-by-step solution

Understand the Problem

Determine whether the given humidity reading in the house is accurate by analyzing the conditions under which glasses do not fog.

Identify Relevant Information

Inside temperature: 20°C, Outside temperature: 10°C, Indoor relative humidity: 55%. Note that glasses fog when the temperature falls below the dew point of the air.

Calculate the Dew Point

Use the approximate formula to find the dew point temperature: $$T_d\approx T-\frac{(100-RH)}{5}$$ where $T$ is the indoor temperature and $RH$ is the relative humidity.

Plug in the Values

Use the provided values: $$T={20}^{\circ }C,RH=55\to T_d\approx 20-\frac{(100-55)}{5}=20-\frac{45}{5}=20-9={11}^{\circ }C$$

Analysis and Conclusion

Since the dew point at 20°C and 55% RH is approximately 11°C, and the outside temperature is 10°C which is below the dew point, the glasses would typically fog. As they do not fog, the indoor humidity reading of 55% may be incorrect.

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that studies the movement of energy and how it converts from one form to another. It's the science behind heating, cooling, and the state changes in matter. The core idea is that energy cannot be created or destroyed, only transformed. Thermodynamics applies to everything from engines to refrigerators, and even the air we breathe.
When we talk about dew point and relative humidity, we're dealing with thermodynamic principles. Understanding these can help us predict whether glasses will fog when moving between environments.
For instance, when warm air inside a house hits cold glasses, the energy transfer can cause the air's water vapor to condense, a process directly controlled by the laws of thermodynamics.

Relative Humidity

Relative humidity is a measure of how much moisture is in the air compared to the maximum amount of moisture the air can hold at that temperature. It is expressed as a percentage.
At a relative humidity of 55%, the air contains 55% of the moisture it could hold at that temperature. This reading helps us understand whether the air is dry or if it's more likely to cause condensation.
Relative humidity affects comfort levels and the likelihood of dew or frost forming on surfaces. To prevent condensation on glasses, you need to ensure that the air isn't overly saturated with moisture.

• 100% relative humidity: Air is fully saturated.
• 50% relative humidity: Air is half saturated with moisture.
• Relative humidity below 30% often feels dry and uncomfortable.

Fogging of Glasses

Fogging of glasses happens when warm, moist air comes into contact with a cooler surface, causing the moisture to condense into tiny droplets. This is a common issue when moving from a cold environment to a warmer one or vice versa.
In our scenario, when a person wearing glasses enters a house from outside, their glasses might fog if the indoor environment has a higher dew point temperature than the glasses' temperature.
To understand why this doesn't happen in the problem, we need to consider the dew point temperature. If the dew point is higher than the temperature of the glasses, you’ll likely see fogging on the lenses. However, if the dew point is lower, the glasses remain clear.

Dew Point Temperature

The dew point temperature is the temperature at which air becomes saturated with moisture and dew forms. It’s a critical concept for understanding humidity and condensation.
The dew point tells us how much moisture is in the air. For example, a high dew point indicates more moisture, leading to a higher chance of condensation.
In the given exercise, the dew point calculation is crucial. Given an indoor temperature of 20°C and relative humidity of 55%, the dew point is approximately 11°C. This implies that any surface cooler than 11°C could see condensation forming. Since the outside temperature is 10°C (below the dew point), glasses should fog.
However, since they don’t, the relative humidity reading of 55% is likely incorrect. This shows the importance of accurate measurements in thermodynamics and humidity control.