Question

The relative humidity is the ratio (expressed as a percent) of the amount of water vapor in the air to the maximum amount that the air can hold at a specific temperature. The relative humidity, \(R\), is found using the following formula: $$ R=10^{\left(\frac{4221}{T+459.4}-\frac{4221}{D+459.4}+2\right)} $$ where \(T\) is the air temperature (in \({ }^{\circ} \mathrm{F}\) ) and \(D\) is the dew point temperature (in \(\left.^{\circ} \mathrm{F}\right)\) (a) Determine the relative humidity if the air temperature is \(50^{\circ}\) Fahrenheit and the dew point temperature is \(41^{\circ}\) Fahrenheit. (b) Determine the relative humidity if the air temperature is \(68^{\circ}\) Fahrenheit and the dew point temperature is \(59^{\circ}\) Fahrenheit. (c) What is the relative humidity if the air temperature and the dew point temperature are the same?

Short answer

For (a) the relative humidity is approximately 71.34%. For (b) it is approximately 72.17%. For (c) it is 100%.

Step-by-step solution

Understand the Formula

The formula for relative humidity is given by \[ R=10^{\left(\frac{4221}{T+459.4}-\frac{4221}{D+459.4}+2\right)} \]where:- \(T\) is the air temperature in Fahrenheit- \(D\) is the dew point temperature in Fahrenheit

Calculate Relative Humidity for (a)

For part (a):- \(T = 50^{\circ} F\)- \(D = 41^{\circ} F\)Substitute these values into the formula:\[ R = 10^{\left(\frac{4221}{50+459.4}-\frac{4221}{41+459.4}+2\right)}\]First, calculate the terms inside the exponents:\[\frac{4221}{50+459.4} = \frac{4221}{509.4} \approx 8.286\]\[\frac{4221}{41+459.4} = \frac{4221}{500.4} \approx 8.434\]Then, calculate the exponent:\[8.286 - 8.434 + 2 = \approx 1.852\]Finally, find the relative humidity:\[ R = 10^{1.852} \approx 71.34\% \]

Calculate Relative Humidity for (b)

For part (b):- \(T = 68^{\circ} F\)- \(D = 59^{\circ} F\)Substitute these values into the formula:\[ R = 10^{\left(\frac{4221}{68+459.4}-\frac{4221}{59+459.4}+2\right)}\]First, calculate the terms inside the exponents:\[\frac{4221}{68+459.4} = \frac{4221}{527.4} \approx 8.003\]\[\frac{4221}{59+459.4} = \frac{4221}{518.4} \approx 8.144\]Then, calculate the exponent:\[8.003 - 8.144 + 2 = \approx 1.859\]Finally, find the relative humidity:\[ R = 10^{1.859} \approx 72.17\% \]

Calculate Relative Humidity for (c)

For part (c):- \(T = D\)If the air temperature and the dew point temperature are the same, then the exponents in the formula will cancel each other out:\[ R = 10^{\left(\frac{4221}{T+459.4}-\frac{4221}{T+459.4}+2\right)} = 10^2 = 100\% \]

Key concepts

relative humidity formula

Relative humidity is a way to describe the amount of moisture in the air relative to the maximum amount the air can hold at a specific temperature. It is expressed as a percentage. The formula to calculate relative humidity, denoted as \(R\), is given by:
\[ R=10^{\left(\frac{4221}{T+459.4}-\frac{4221}{D+459.4}+2\right)} \]
In this formula:

• \(T\) represents the air temperature in degrees Fahrenheit (\(^\circ F\)).
• \(D\) represents the dew point temperature, also in degrees Fahrenheit (\(^\circ F\)).

This formula takes into account the air's capability to hold water vapor at different temperatures. Higher air temperatures can hold more water vapor, whereas lower dew point temperatures indicate less moisture in the air.

dew point temperature

Dew point temperature is an important concept in understanding humidity. It is the temperature at which air becomes saturated with moisture, leading to dew formation if cooled further. When the air temperature reaches the dew point, the relative humidity becomes 100%. Imagine it as the threshold beyond which condensation occurs.
This is significant because:

• It helps in predicting weather, like potential fog or precipitation.
• It influences comfort levels; higher dew points can make the air feel muggier.

For example, if the dew point temperature is close to the current air temperature, the air is near saturation. If they are equal, the relative humidity is at its maximum, 100%.

air temperature

Air temperature refers to how hot or cold the air is around us, typically measured in degrees Fahrenheit (\(^\circ F\)) or degrees Celsius (\(^\circ C\)). This is the primary factor in determining how much water vapor the air can hold. Higher temperatures allow air to hold more moisture. Conversely, at lower temperatures, the air can hold less moisture before becoming saturated.
Here's why it's crucial:

• Influences our daily comfort and health.
• Affects everything from weather patterns to the heating and cooling requirements of buildings.

In the context of relative humidity, the air temperature helps us understand the moisture percentage relative to its capacity. For instance, on a hot day, even a moderate amount of moisture can lead to high relative humidity, making it feel much hotter.