The relative humidity is the ratio (expressed as a percent)...

The relative humidity is the ratio (expressed as a percent) of the amount of water vapor in the air to the maximum amount that it can hold at a specific temperature. The relative humidity, $R,$ is found using the following formula: $$ R=10^{\frac{4221}{T+499.4}-\frac{4221}{D+459.4}+2} $$ where $T$ is the air temperature (in $\left.{ }^{\circ} \mathrm{F}\right)$ 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?

The relative humidity is the ratio (expressed as a percent) of the amount of water vapor in the air to the maximum amount that it can hold at a specific temperature. The relative humidity, $R,$ is found using the following formula:
$$
R=10^{\frac{4221}{T+499.4}-\frac{4221}{D+459.4}+2}
$$
where $T$ is the air temperature (in $\left.{ }^{\circ} \mathrm{F}\right)$ 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?

Key Concepts

Exponential Relationships in Saturation

Exponential relationships are often used in meteorological formulas to express how the saturation vapor pressure of water increases non-linearly with temperature. Such relationships capture the rapid changes in the air's capacity to hold moisture as temperature changes, which is fundamental to accurately determining relative humidity.

Dew Point Temperature

Dew point temperature is the temperature at which the air becomes saturated with water vapor and condensation begins, forming dew. It is a critical parameter in meteorology because it provides insight into the moisture content of the air and is a key variable in calculating relative humidity.

Relative Humidity

Relative humidity is the measure of the current water vapor concentration in the air relative to the maximum amount of water vapor the air can hold at a particular temperature, expressed as a percentage. It provides important information about human comfort, weather conditions, and the potential for precipitation by indicating how close the air is to saturation.

Air Temperature

Air temperature is the measure of how hot or cold the atmosphere is, and it plays a critical role in determining the air's ability to hold water vapor. In the context of humidity, higher air temperatures allow the air to hold more moisture, affecting the calculation of relative humidity.

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