Question

At a given temperature, the humid volume is

• A. Linear function of humidity
• B. Inverse function of humidity
• C. Square function of humidity
• D. No specific function of humidity

Hint:

Humid volume \( v_H \) increases linearly with absolute humidity \( Y \), reflecting the added volume of water vapor in the mixture.

Answer 1

The Correct Option is A

Step 1: Define humid volume and humidity.
Humid volume (\( v_H \)) is the volume of a humid gas (e.g., air-water vapor mixture) per unit mass of dry gas (m³/kg dry gas). It accounts for the volume occupied by both the dry gas and the water vapor at a given temperature and pressure.
Humidity (\( Y \)) is the absolute humidity, defined as the mass of water vapor per unit mass of dry gas (kg water/kg dry air). Step 2: Derive the relationship between humid volume and humidity.
For a humid gas (e.g., air-water vapor mixture) at a given temperature \( T \) and total pressure \( P \), the humid volume is the specific volume of the mixture. Assume ideal gas behavior for both the dry air and water vapor:
Let \( m_d \): Mass of dry air (kg),
\( m_v \): Mass of water vapor (kg),
Absolute humidity: \( Y = \frac{m_v}{m_d} \).
The total number of moles of the mixture is the sum of moles of dry air and water vapor: Moles of dry air: \( n_d = \frac{m_d}{M_d} \), where \( M_d \) is the molecular weight of dry air (≈ 29 kg/kmol),
Moles of water vapor: \( n_v = \frac{m_v}{M_v} = \frac{Y m_d}{M_v} \), where \( M_v \) is the molecular weight of water (≈ 18 kg/kmol).
Total moles per kg of dry air: \[ n_{\text{total}} = \frac{m_d}{M_d} + \frac{Y m_d}{M_v} = m_d \left( \frac{1}{M_d} + \frac{Y}{M_v} \right). \] Per kg of dry air (\( m_d = 1 \)): \[ n_{\text{total}} = \frac{1}{M_d} + \frac{Y}{M_v}. \] Using the ideal gas law (\( V = \frac{nRT}{P} \)), the humid volume \( v_H \) (volume per kg of dry air) is: \[ v_H = \frac{n_{\text{total}} R T}{P} = \left( \frac{1}{M_d} + \frac{Y}{M_v} \right) \frac{R T}{P}. \] \[ v_H = \frac{R T}{P} \left( \frac{1}{M_d} + \frac{Y}{M_v} \right). \] At a given temperature \( T \) and pressure \( P \), \( \frac{R T}{P} \), \( M_d \), and \( M_v \) are constants. Let \( a = \frac{R T}{P M_d} \), \( b = \frac{R T}{P M_v} \), so: \[ v_H = a + b Y. \] This is a linear function of humidity \( Y \). Step 3: Evaluate the options.
(1) Linear function of humidity: Correct, as \( v_H = a + b Y \), which is linear in \( Y \). Correct.
(2) Inverse function of humidity: Incorrect, as the relationship is not \( v_H \propto \frac{1}{Y} \). Incorrect.
(3) Square function of humidity: Incorrect, as the relationship is not \( v_H \propto Y^2 \). Incorrect.
(4) No specific function of humidity: Incorrect, as humid volume is a linear function of humidity. Incorrect.
Step 4: Select the correct answer.
At a given temperature, the humid volume is a linear function of humidity, matching option (1).