Question

The relative humidity of ambient air at \(300\text{ K}\) is \(50\%\) with a partial pressure of water vapour equal to \(P_{\text{v}}\). The saturation pressure of water at \(300\text{ K}\) is \(p_{\text{sat}}\). The correct relation for the air-water mixture is

• A. \(P_{\text{v}} = p_{\text{sat}}\)
• B. \(P_{\text{v}} = 0.622 p_{\text{sat}}\)
• C. \(P_{\text{v}} = 0.5 p_{\text{sat}}\)
• D. \(P_{\text{v}} = 2 p_{\text{sat}}\)

Hint:

Relative humidity is always expressed as a fraction or percentage of the saturation pressure.
Simply multiply the percentage (expressed as a decimal) by $p_{\text{sat}}$ to get the partial pressure of the vapor: $P_{\text{v}} = \phi \cdot p_{\text{sat}}$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks for the mathematical relationship between the partial pressure of water vapor (\(P_{\text{v}}\)) and the saturation pressure of water (\(p_{\text{sat}}\)) when the relative humidity is given as \(50\%\).

Step 2: Key Formula or Approach:
Relative humidity (\(\phi\)) is defined as the ratio of the partial pressure of water vapor in the mixture to the saturation pressure of pure water vapor at the same temperature:
\[ \phi = \frac{P_{\text{v}}}{p_{\text{sat}}} \]

Step 3: Detailed Explanation:

• Relative humidity acts as a measure of the amount of moisture in the air compared to the maximum amount of moisture the air can hold at that specific temperature.

• Given in the problem, the relative humidity is \(50\%\), which is represented decimally as:
\[ \phi = 0.50 \]

• Substitute this value of \(\phi\) into the relative humidity definition:
\[ 0.5 = \frac{P_{\text{v}}}{p_{\text{sat}}} \]

• Rearranging this equation to express the partial pressure of water vapor \(P_{\text{v}}\) in terms of the saturation pressure \(p_{\text{sat}}\) yields:
\[ P_{\text{v}} = 0.5 p_{\text{sat}} \]

Step 4: Final Answer:
The correct relationship representing the state of the mixture is \(P_{\text{v}} = 0.5 p_{\text{sat}}\).