If a sample of unsaturated air with a relative humidity of 40\% at 30^\circ\text{C} is com… — JAMB UTME

If a sample of unsaturated air with a relative humidity of $40\%$ at $30^\circ\text{C}$ is compressed isothermally to one-third of its original volume, the final relative humidity of the air will be $100\%$ .

Answer: Answer

Answer

True. The relative humidity cannot exceed 100% because water vapour condenses into liquid once its partial pressure reaches the saturated vapour pressure at that temperature.

The statement is true because relative humidity is the ratio of actual partial vapour pressure to saturated vapour pressure at a specific temperature. Although isothermal compression to one-third volume triples the partial pressure of water vapour (giving a theoretical 120%), partial vapour pressure is bounded by the saturated vapour pressure. When the relative humidity reaches 100%, condensation occurs, keeping the relative humidity at exactly 100%.

Step-by-Step Solution

Apply Boyle's law to calculate the theoretical partial vapour pressure after isothermal compression.

Since volume is reduced to

\frac{1}{3}V_1

at constant temperature, the partial pressure of water vapour would increase by a factor of 3 (

p_2 = 3p_1

For an unsaturated vapour at constant temperature, partial pressure is inversely proportional to volume.

Determine the theoretical relative humidity from the pressure ratio.

\text{Theoretical R.H.} = 3 \times 40\% = 120\%.

Relative humidity is defined as

\text{R.H.} = \frac{p}{p_s} \times 100\%

, where

p

is the partial pressure and

p_s

is the saturated vapour pressure.

Apply the physical limit imposed by saturated vapour pressure.

The actual relative humidity cannot exceed

100\%

; excess vapour condenses.

Saturated vapour pressure

p_s

represents the maximum possible partial pressure of water vapour in air at a given temperature.

Key Concept

Saturated Vapour Pressure Limit and Relative Humidity under Isothermal Compression

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