Vapor-Liquid/Solid System

Q: What is a gas-liquid system:

A: As a general rule, any liquid that doesn't entirely fill a container involves a gas-liquid system in that container. When you take your glass of lemonade and smell it, you are smelling the vapour of the lemonade (the liquid). (Under some conditions, though, a liquid can have a negligible vapour above it. For instance, water under the right conditions, can have completely dry air above it.)

Usually a liquid has some vapour above above it (and the vapour of course has physical properties associated with it, such as pressure).

It is also true that the pressure exerted by a liquid to the air applies to solids too. That is, just above a solid, there is an equilibrium between the solid and its vapor.

Now that we understand that liquids and solids have (just above their surface) molecules in the gas phase due to their presence, we are ready for a relationship that tells us the pressure of this vapor.

Before we lay out the rules, though. Two asides: 1) vapour (the European spelling) and vapor have both been used in this page (for fun?). 2) Perhaps, for a condensible gas phase species, it shouldn't be referring to as a gas, but rather as a vapor, thus we are talking about vapor-liquid/solid systems, oops.

It turns out that vapour pressure is a direct function of temperature. That is, if given a vapour pressure for a substance, you can get temperature without any other info, and vise versa! You can do this in a variety of ways in this course. We'll find that the following tables and graphs offer the relationships we need to move between vapour pressures and temperatures:

Note: There are 2 major challenges to being able to use these interrelationships effectively:

Recognition: In a mass balance problem, if you're given temperature, know that you can get the vapor pressure of the gas in a gas-liquid system (which will lead to other properties, eg. see the section on Raoult's Law).

Tables/Graphs: Learn to use all the tables and graphs, and the advantages of each. For example, Appendix B4 means having to use Antoine's Equation which means an algebra exercise. For more information on Antoine's equation,

Example 1

A propane tank is at a temperature of 100 degrees Fahrenheit. Determine its vapour pressure in lb-force/in.², using both the Cox chart and Antoine's equation. Which method was more convenient? Can you think of a scenario when the less convenient method could become more convenient?

Example 2

Liquid water is heated from 10 degrees Celcius to 50 degrees Celcius. Find the vapour pressure of the water at 10, 20, 30, 40, and 50 degrees Celcius in mmHg.

How do I recognize if a gas-liquid system has only a single condensible species?

The direct answer to this question is to compare the boiling points. Any boiling points that are far below the temperatures involved imply that those species are considered uncondesible.

A cup of water: the gas above the water is composed of water and air (nitrogen and oxygen). Thus there are three gases present above the cup of water (water, nitrogen, and oxygen). However, only the water will condense upon a cooling from 70°C to 50°C (bp's: H₂O: 100°C, Nitrogen: -195°C, Oxygen: -218°C). So when we cool the system , some of the H₂O gas molecules will condense and join the liquid water in the glass, but none of the nitrogen or oxygen will condense because we'd have to cool them down much, much further to get them to condense at all. This is an example where we have a system with only a single condensible species.

A mixture of an alcohol (say, ethanol) and water: the gas above the liquid is composed of H₂O, EtOH, and air. Here, upon a cooling from 70°C to 50°C, both the alcohol (bp 78°C) and water (bp 100°C) will condense. Thus we have a system where there are two condensible species.

So you now know how to identify if a gas-liquid system has a single condesible species. But, what are the interrelationships that will help us?

There are four saturation formulas that apply to gas-liquid systems that have only a single condensible species, and they are listed here:

4 Saturation Formulas

Relative Saturation
Molal Saturation
Absolute Saturation
Percentage Saturation

P is the Pressure of the total gas mixture
p_i is the Partial Pressure of component i
p_i^* is the Vapor Pressure of component i
MW_i is the molecular weight of component i
MW_dry is the molecular weight of total gas mixture

There is special notation and terminology that we use for air-water systems since they are so common. However, the term saturation in each of these equations is designated for any generic gas-vapor system: when the gas is air and the vapor is water vapor, then the word saturation is replaced with humidity for air-water systems.
Take the first equation for example:
The name "relative saturation" changes to "relative humidity". When we look at s_r, it changes to h_r, and both p_i and p_i^* specifically refer to water. This means the pressure could be written as p_H2O and p_H2O^*.
The formulas stay the same whether we are talking about any gas-vapor system or specifically an air-water system; there is just a slight change in terminology and notation. The most important thing to remember is that if a problem statement gives or asks for one of these quantities, you now know where to look them up and how to use them.

Example 1

A gas consists of 10% styrene and 90% air in a tank at 100 degrees Celcius and 1000 mmHg. Calculate the relative saturation, molal saturation, and percentage saturation.

Example 2

Water enters a boiler at 80 degrees Celcius under atmospheric pressure. If its relative humidity is 60%, what is the vapor pressure, partial pressure, and mole fraction of water in the air.

Raoult's Law relates a component's partial pressure, vapor pressure, and mole fraction in the liquid phase in order to find the mole fraction in the gas phase. However, if we know any three of the variables, we can solve for the fourth one! We use this relationship for nearly pure liquids, that is, where x_A is close to one. Here's the relation:

If we recall Dalton's law of partial pressures for a gas (p_A = y_A * P) we can combine this with Raoult's law to get the following important set of interrelations:

If we have a completely pure liquid, x_A = 1, and Raoult's Law reduces to the following:

For dilute components (x_A is about 0), we use Henry's Law, where we have to look up Henry's law constant for a component, which is a function temperature.

These interrelations relate the fraction of a component in a liquid-phawse versus the fraction of a component in the gas phase.
To exemplify the possible utility of this interrelation, suppose we are doing a balance on a piece of chemical equipment and we have a system of air and a liquid that is mostly methanol. Early on in the problem solution, we may want to know how much methanol is in the gas phase(y_MeOH), or even how much is in the liquid phase (x_MeOH). With the information given, we can use Raoult's Law on the methanol (liquid is mostly pure methanol), and Henry's Law on the air (its mole fraction in the liquid is close to one) to determine the mole fractions of methanol in the gas and liquid phases.

A reaction energy balance problem that I did for chapter 8 involved a balance on a condenser that required the implementation of these concepts. I challenge you to solve it!

Example 1

An equimolar liquid mixture of benzene (B) and toluene (T) is in equilibrium with its vapor at 30.0 degrees Celcius. What is the system pressure and the composition of the vapor?

Example 2

A gas stream containing 40.0 mole % hydrogen, 35.0% carbon monoxide, 20.0 % carbon dioxide, and 5.0% methane is cooled from 1000 degrees Celcius to 10 degrees Celcius at a constant absolute pressure of 35.0 atm. Gas enters the cooler at 120 m³/min and, upon leaving the cooler, is fed to an absorber where it is contacted with refrigerated liquid methanol. The methanol is fed to the absorber at a molar flow rate 1.2 times that of the inlet gas and absorbs essentially all of the CO₂, 98% of the methane, and none of the other components of the feed gas. The gas leaving the absorber, which is saturated with methanol at -12 degrees Celcius, is fed to a cross-country pipeline. Calculate the volumetric flow rate of methanol entering the absober in m₃/min and the molar flow rate of methanol in the gas leaving the absorber assuming ideal gas behavior.

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