A closed rigid container holds a mixture of dry air and water vapour at a temperature of 2… — JAMB UTME

A closed rigid container holds a mixture of dry air and water vapour at a temperature of $27^\circ\text{C}$ under a total pressure of $740\text{ mmHg}$ . The relative humidity of the air inside the container is $80\%$ , and the saturated vapour pressure of water at $27^\circ\text{C}$ is $25\text{ mmHg}$ . If the container is heated at constant volume to $127^\circ\text{C}$ , what is the partial pressure of the dry air in $\text{mmHg}$ at this higher temperature?

Answer: 960 mmHg

Answer

The partial pressure of dry air inside the container at

127^\circ\text{C}

960\text{ mmHg}

First, the partial pressure of water vapour at

27^\circ\text{C}

is determined by multiplying relative humidity (

80\%

) by the saturated vapour pressure (

25\text{ mmHg}

), yielding

20\text{ mmHg}

. Next, subtracting this vapour pressure from the total pressure of

740\text{ mmHg}

gives the partial pressure of dry air alone as

720\text{ mmHg}

27^\circ\text{C}

(

300\text{ K}

). Finally, applying the Pressure Law (

\frac{P_1}{T_1} = \frac{P_2}{T_2}

) for the dry air between

300\text{ K}

and

400\text{ K}

(

127^\circ\text{C}

) yields

P_2 = 720 \times \frac{400}{300} = 960\text{ mmHg}

Step-by-Step Solution

Calculate the partial pressure of water vapour at

27^\circ\text{C}

P_{\text{vapour}, 1} = 0.80 \times 25\text{ mmHg} = 20\text{ mmHg}

Relative humidity is the ratio of actual partial vapour pressure to the saturated vapour pressure at that temperature.

Determine the initial partial pressure of the dry air at

27^\circ\text{C}

using Dalton's Law

P_{\text{dry air}, 1} = 740\text{ mmHg} - 20\text{ mmHg} = 720\text{ mmHg}

Total pressure of a gas mixture is the sum of the partial pressures of its individual components.

Convert temperatures to Kelvin and apply Gay-Lussac's Pressure Law for dry air at constant volume

T_1 = 27 + 273 = 300\text{ K}

T_2 = 127 + 273 = 400\text{ K}

P_{\text{dry air}, 2} = 720\text{ mmHg} \times \left(\frac{400\text{ K}}{300\text{ K}}\right) = 960\text{ mmHg}

For a fixed mass of gas at constant volume, pressure is directly proportional to absolute temperature.

Key Concept

Dalton's Law of Partial Pressures and Gay-Lussac's Pressure Law applied to gas-vapour mixtures

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