Question

The dew point of an unsaturated mixture of water vapour and air at constant temperature and pressure:

• A. Does not change with change in absolute humidity
• B. Increases with increase in absolute humidity
• C. Decreases with increase in absolute humidity
• D. Decreases linearly with increase in absolute humidity

Hint:

Dew point temperature dependency: - Dew Point (\( T_{dp} \)) is a direct measure of the moisture content in the air. - More moisture \( \rightarrow \) Higher absolute humidity (\( Y \)) \( \rightarrow \) Higher partial vapor pressure (\( p_v \)) \( \rightarrow \) Higher dew point temperature (\( T_{dp} \)).

Answer 1

The Correct Option is B

Concept: The dew point temperature (\( T_{dp} \)) of a moist air mixture is defined as the temperature to which the air must be cooled at a constant total pressure to reach full saturation (\( \text{Relative Humidity} = 100\% \)). At this point, the initial condensation of liquid water begins. Thermodynamically, the dew point temperature depends uniquely on the partial pressure of water vapor (\( p_v \)) present in the mixture.

Step 1: Linking absolute humidity to partial pressure.
Let us evaluate the relationship between absolute humidity (also called humidity ratio, denoted as \( Y \)) and the partial pressure of water vapor (\( p_v \)) in a gas mixture at a total system pressure \( P \): \[ Y = 0.622 \cdot \frac{p_v}{P - p_v} \] Rearranging this algebraic equation to isolate the partial pressure term \( p_v \) yields: \[ p_v = \frac{Y \cdot P}{0.622 + Y} \] This relation shows that at a constant total system operating pressure \( P \), the partial pressure of water vapor \( p_v \) is a monotonically increasing function of the absolute humidity \( Y \): \[ Y \uparrow \quad \Rightarrow \quad p_v \uparrow \]

Step 2: Connecting partial pressure to the dew point temperature.
By definition, when a gas mixture cools to its dew point temperature, the partial pressure of water vapor matches the saturation vapor pressure of pure water at that specific temperature: \[ p_v = p_{sat}(T_{dp}) \] According to the Clausius-Clapeyron equation, the saturation pressure of water increases monotonically with temperature: \[ T_{dp} \uparrow \quad \Longleftrightarrow \quad p_{sat}(T_{dp}) \uparrow \] Since an increase in absolute humidity (\( Y \)) increases the partial pressure of water vapor (\( p_v \)), it requires a higher saturation temperature to initiate condensation. Therefore, if the absolute humidity of an air-water vapor mixture increases, its dew point temperature increases accordingly.