On a certain day in Orlando, Florida, the temperature of the air and dew point were 16°C and 7.6°C, respectively. Find the relative humidity of the air on that day. Assume the saturated vapor pressure of water at 7°C, 8°C, and 16°C are 7.5 mm Hg, 8.0 mm Hg, and 12.0 mm Hg, respectively.
The Correct Answer and Explanation is :
To find the relative humidity, we use the following formula:
[
\text{Relative Humidity} = \left( \frac{\text{Actual Vapor Pressure}}{\text{Saturated Vapor Pressure}} \right) \times 100
]
Step 1: Identify Given Values
- Temperature of air (Tₐ): 16°C
- Dew Point (Tₑ): 7.6°C
- Saturated Vapor Pressure at 7°C: 7.5 mm Hg
- Saturated Vapor Pressure at 8°C: 8.0 mm Hg
- Saturated Vapor Pressure at 16°C: 12.0 mm Hg
Step 2: Find the Saturated Vapor Pressure at the Dew Point (7.6°C)
Since 7.6°C is between 7°C and 8°C, we will estimate the saturated vapor pressure at this temperature by interpolating between 7°C (7.5 mm Hg) and 8°C (8.0 mm Hg).
The formula for linear interpolation is:
[
P_{s}(Tₑ) = P_{s}(T₁) + \left( \frac{(Tₑ – T₁)}{(T₂ – T₁)} \right) \times (P_{s}(T₂) – P_{s}(T₁))
]
Where:
- (T₁ = 7°C), (P_{s}(T₁) = 7.5 \, \text{mm Hg})
- (T₂ = 8°C), (P_{s}(T₂) = 8.0 \, \text{mm Hg})
- (Tₑ = 7.6°C)
Substitute the values:
[
P_{s}(7.6°C) = 7.5 + \left( \frac{(7.6 – 7)}{(8 – 7)} \right) \times (8.0 – 7.5)
]
[
P_{s}(7.6°C) = 7.5 + (0.6) \times (0.5) = 7.5 + 0.3 = 7.8 \, \text{mm Hg}
]
So, the saturated vapor pressure at the dew point (7.6°C) is 7.8 mm Hg.
Step 3: Calculate the Relative Humidity
Now, we can calculate the relative humidity using the formula:
[
\text{Relative Humidity} = \left( \frac{\text{Actual Vapor Pressure}}{\text{Saturated Vapor Pressure}} \right) \times 100
]
The actual vapor pressure of the air is equal to the saturated vapor pressure at the dew point, so:
[
\text{Relative Humidity} = \left( \frac{7.8}{12.0} \right) \times 100 = 65\%
]
Conclusion:
The relative humidity of the air on that day in Orlando, Florida was 65%.
Explanation:
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 a given temperature. When the air cools and reaches the dew point, the air becomes saturated with water vapor, meaning it can no longer hold any more moisture. By comparing the actual vapor pressure at the dew point to the saturated vapor pressure at the air temperature, we can determine the relative humidity. This value is important because it affects human comfort, weather patterns, and the likelihood of precipitation.