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Difficulty: HardVapour Pressure, Boiling, Evaporation, and Relative Humidity

A mass of air at 30∘C30^\circ\text{C} has a relative humidity of 50%50\%. The saturated vapour pressure of water is 30 mmHg30\text{ mmHg} at 30∘C30^\circ\text{C} and 9 mmHg9\text{ mmHg} at 10∘C10^\circ\text{C}. If the temperature of the air is lowered to 10∘C10^\circ\text{C}, what percentage of the initial water vapour condenses out?

  1. $40\%Answer
  2. B
    $60\%
  3. C
    $70\%
  4. D
    $30\%

Answer

The percentage of initial water vapour that condenses out when cooled to 10∘C10^\circ\text{C} is 40%40\%.
The correct answer is 40%40\%. Initially, the air contains water vapour exerting a partial pressure of 0.50×30 mmHg=15 mmHg0.50 \times 30\text{ mmHg} = 15\text{ mmHg}. Upon cooling to 10∘C10^\circ\text{C}, the air becomes saturated at 9 mmHg9\text{ mmHg}, causing 15 mmHg−9 mmHg=6 mmHg15\text{ mmHg} - 9\text{ mmHg} = 6\text{ mmHg} worth of vapour to condense into liquid. The condensed amount as a fraction of the initial vapour is 6/15=0.406 / 15 = 0.40, or 40%40\%.

Step-by-Step Solution

1
Calculate the initial partial vapour pressure of water at 30∘C30^\circ\text{C}.
Partial Vapour Pressure=Relative Humidity×SVP at 30∘C=0.50×30 mmHg=15 mmHg\text{Partial Vapour Pressure} = \text{Relative Humidity} \times \text{SVP at } 30^\circ\text{C} = 0.50 \times 30\text{ mmHg} = 15\text{ mmHg}.
Relative humidity is the ratio of actual partial vapour pressure to the saturated vapour pressure at that temperature.
2
Determine the amount of vapour pressure that must condense when cooled to 10∘C10^\circ\text{C}.
Vapour pressure condensed=15 mmHg−9 mmHg=6 mmHg\text{Vapour pressure condensed} = 15\text{ mmHg} - 9\text{ mmHg} = 6\text{ mmHg}.
At 10∘C10^\circ\text{C}, the air can hold at most its saturated vapour pressure of 9 mmHg9\text{ mmHg}, so any excess vapour above 9 mmHg9\text{ mmHg} condenses into liquid water.
3
Calculate the percentage of the initial water vapour that condenses out.
Percentage condensed=(6 mmHg15 mmHg)×100%=40%\text{Percentage condensed} = \left(\frac{6\text{ mmHg}}{15\text{ mmHg}}\right) \times 100\% = 40\%.
The question asks for the fraction of the initial vapour originally present that leaves the gaseous state.

Key Concept

Relative Humidity and Dew Point Condensation
Estimated Time:1m 30s
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Topics

Vapour Pressure, Boiling, Evaporation, and Relative Humidity

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