Gas-Liquid Systems: One Condensable Component & Humidity Calculations

Gas-Liquid System: One Condensable Component

Introduction to Gas-Liquid Systems

Gas-liquid systems with one condensable component are fundamental in chemical engineering and physical chemistry. These systems typically involve a non-condensable gas (e.g., air) and a condensable vapor (e.g., water vapor or benzene vapor). Understanding the equilibrium and saturation properties of such systems is essential for calculations involving humidity, dew point, and vapor-liquid equilibrium.

• Condensable Component: A vapor that can condense into a liquid under system conditions (e.g., water vapor).

• Non-condensable Gas: A gas that does not condense under the same conditions (e.g., air).

Phases in Gas-Liquid Systems

• Liquid Phase: The condensed form of the condensable component.

• Vapor Phase: The gaseous form of the condensable component.

• Gas Phase: The non-condensable gas present in the system.

Saturation and Equilibrium

• Saturated Vapor: The vapor phase is in equilibrium with its liquid at a given temperature and pressure.

• No Net Exchange: At equilibrium, there is no net exchange of molecules between the liquid and vapor phases.

• Saturation Condition: The gas phase contains the maximum amount of vapor possible at the system temperature and pressure.

• Raoult's Law (Single Condensable Species):

$P_{H_2O} = y_{H_2O} P$

At saturation, $P_{H_2O} = P^o$ (the vapor pressure of the pure condensable component at the system temperature).

Dew Point

• Dew Point: The temperature at which the vapor in a mixture of vapor and non-condensable gas just begins to condense.

Partial Saturation / Humidity

1. Relative Saturation (Relative Humidity)

• Definition: The ratio of the partial pressure of the vapor to the equilibrium vapor pressure at the same temperature, expressed as a percentage.

$\%RS (\%RH) = \frac{P_{vapor}}{P^o} \times 100$

2. Molal Saturation (Molal Humidity)

• Definition: The ratio of the moles of vapor to the moles of dry gas.

$MS(MH) = \frac{n_{vapor}}{n_{dry\ gas}} = \frac{P_{vapor}}{P - P_{vapor}} \frac{MW_{vapor}}{MW_{dry\ gas}}$

3. Absolute Saturation (Absolute Humidity or Humidity)

• Definition: The mass of vapor per mass of dry gas.

$AS(H) = \frac{mass_{vapor}}{mass_{dry\ gas}} = \frac{n_{vapor}(MW_{vapor})}{n_{dry\ gas}(MW_{dry\ gas})}$

$= \frac{P_{vapor}(MW_{vapor})}{(P - P_{vapor})(MW_{dry\ gas})}$

4. Percent Saturation (Percent Humidity)

• Definition: The ratio of the actual humidity to the humidity at saturation, expressed as a percentage.

$\%S(\%H) = \frac{P_{vapor}}{P^o} \frac{P - P^o}{P - P_{vapor}} \times 100$

$= \%RS \times \frac{P - P^o}{P - P_{vapor}}$

Sample Problems

Sample Problem #9

Problem Statement: A mixture of benzene and air at 36°C and 100 kPa is found to have a dew point of 16°C. Calculate its:

• A. Composition by volume

• B. kg-mol benzene/kg-mol air

• C. wt benzene/wt air

Approach: Use vapor pressure data for benzene at the dew point and system temperature, apply Dalton's and Raoult's laws, and convert between mole and mass ratios as needed.

Sample Problem #10

Problem Statement: 250 grams of steam is injected into a large enclosed container containing 55 kg air. If the temperature is at 25°C and pressure 120 kPa, calculate:

• A. % relative humidity

• B. Molal humidity

• C. Humidity

• D. % humidity

• E. Dew point

Approach: Calculate the partial pressure of water vapor, determine the relevant humidity values using the formulas above, and use vapor pressure tables to find the dew point.

Vapor Pressure Data Table

The provided tables list vapor pressures of various inorganic and organic liquids as a function of temperature, which are essential for solving the sample problems above. The general form for vapor pressure calculation is:

$\log_{10} P = C_1 + \frac{C_2}{T} + C_3 T + C_4 \log_{10} T + C_5 P^n$

Application: Use the appropriate constants for the substance of interest (e.g., benzene or water) to determine vapor pressures at given temperatures.

Summary Table: Humidity Definitions and Formulas

Term	Definition	Formula
Relative Humidity (%RH)	Partial pressure of vapor / Saturation vapor pressure × 100	$\frac{P_{vapor}}{P^o} \times 100$
Molal Humidity (MH)	Moles vapor / Moles dry gas	$\frac{P_{vapor}}{P - P_{vapor}} \frac{MW_{vapor}}{MW_{dry\ gas}}$
Absolute Humidity (H)	Mass vapor / Mass dry gas	$\frac{P_{vapor}(MW_{vapor})}{(P - P_{vapor})(MW_{dry\ gas})}$
Percent Humidity (%H)	Actual humidity / Saturation humidity × 100	$\frac{P_{vapor}}{P^o} \frac{P - P^o}{P - P_{vapor}} \times 100$

Additional info: These concepts are foundational for chemical engineering and physical chemistry, especially in the study of vapor-liquid equilibrium, air conditioning, and environmental engineering. While not directly part of the standard Organic Chemistry curriculum, they are relevant for students studying physical properties of organic vapors and their interactions with air.