# Introduction to Chemical Engineering Processes/Vapor-Liquid equilibrium

## Phase Equlibrium

Many processes in chemical engineering do not only involve a single phase but a combination of two immiscible liquids, or a stream containing both gas and liquid. It is very important to recognize and be able to calculate when these phases are in equilibrium with each other, and how much is in each phase. This knowledge will be especially useful when you study separation processes, for many of these processes work by somehow distorting the equilibrium so that one phase is especially rich in one component, and the other is rich in the other component.

More specifically, there are three important criteria for different phases to be in equilibrium with each other:

1. The temperature of the two phases is the same at equilibrium.
2. The partial pressure of every component in the two phases is the same at equilibrium.
3. The Gibbs free energy of every component in the two phases is the same at equilibrium.

The third criteria will be explored in more depth in another course; it is a consequence of the first two criteria and the second law of thermodynamics.

## Single-Component Phase Equilibrium

If there is only a single component in a mixture, there is only a single possible temperature (at a given pressure) for which phase equilibrium is possible. For example, water at standard pressure (1 atm) can only remain in equilibrium at 100°C. Below this temperature, all of the water condenses, and above it, all of the water vaporizes into steam.

At a given temperature, the unique atmospheric pressure at which a pure liquid boils is called its vapor pressure. Students may benefit from conceptualizing vapor pressure as the minimum pressure required to keep the fluid in the liquid phase. If the atmospheric pressure is higher than the vapor pressure, the liquid will not boil. Vapor pressure is strongly temperature-dependent. Water at 100°C has a vapor pressure of 1 atmosphere, which explains why water on Earth (which has an atmosphere of about 1 atm) boils at 100°C. Water at a temperature of 20°C(a typical room temperature) will only boil at pressures under 0.023 atm, which is its vapor pressure at that temperature.

## Multiple-Component Phase Equilibrium: Phase Diagrams

In general, chemical engineers are not dealing with single components; instead they deal with equilibrium of mixtures. When a mixture begins to boil, the vapor does not, in general, have the same composition as the liquid. Instead, the substance with the lower boiling temperature (or higher vapor pressure) will have a vapor concentration higher than that with the higher boiling temperature, though both will be present in the vapor. A similar argument applies when a vapor mixture condenses.

The concentrations of the vapor and liquid when the overall concentration and one of the temperature or pressure are fixed can easily be read off of a phase diagram. In order to read and understand a phase diagram, it is necessary to understand the concepts of bubble point and dew point for a mixture.

### Bubble Point and Dew Point

In order to be able to predict the phase behavior of a mixture, scientists and engineers examine the limits of phase changes, and then utilize the laws of thermodynamics to determine what happens in between those limits. The limits in the case of gas-liquid phase changes are called the bubble point and the dew point.

The names imply which one is which:

1. The bubble point is the point at which the first drop of a liquid mixture begins to vaporize.
2. The dew point is the point at which the first drop of a gaseous mixture begins to condense.

If you are able to plot both the bubble and the dew points on the same graph, you come up with what is called a Pxy or a Txy diagram, depending on whether it is graphed at constant temperature or constant pressure. The "xy" implies that the curve is able to provide information on both liquid and vapor compositions, as we will see when we examine the thermodynamics in more detail.

#### Txy and Pxy diagrams

The easier of the two diagrams to calculate (but sometimes harder to grasp intuitively) is the Pxy diagram, which is shown below for an idealized Benzene-Toluene system:

In order to avoid getting confused about what you're looking at, think: what causes a liquid to vaporize? Two things should come to mind:

• Increasing the temperature
• Decreasing the pressure

Therefore, the region with the higher pressure is the liquid region, and that of lower pressure is vapor, as labeled. The region in between the curves is called the two-phase region.

 Note: You may be tempted to try and memorize something like the dew point line is on the bottom in a Pxy diagram and on the top in a Txy diagram. This is, however, strongly discouraged, as you will very likely become confused if you depend on this type of memorizing. Instead think: which half of the graph will contain liquid and which half will be vapor? Then use the definitions of "dew" and "bubble" points to determine which line is which.

Now that we have this curve, what can we do with it? There are several critical pieces of information we can gather from this graph by simple techniques, which have complete analogies in the Txy diagram. First, note that the two lines intersect at ${\displaystyle x_{Benzene}=0}$ and at ${\displaystyle x_{Benzene}=1}$. These intersections are the pure-component vapor pressures at ${\displaystyle T=20^{o}C}$, since a pure component boils at its vapor pressure.

We can determine, given the mole fraction of one component and a pressure, whether the system is gas, liquid, or two-phase, which is critical information from a design standpoint. For example, if the Benzene composition in the Benzene-Toluene system is 40% and the pressure is 25 mmHg, the entire mixture will be vapor, whereas if the pressure is raised to 50 mmHg it will all condense. The design of a flash evaporator at 20oC would require a pressure between about 30 and 40 mmHg (the 2-phase region).

We can also determine the composition of each component in a 2-phase mixture, if we know the overall composition and the vapor pressure. First, start on the x-axis at the overall composition and go up to the pressure you want to know about. Then from this point, go left until you reach the bubble-point curve to find the liquid composition, and go to the right until you reach the dew-point curve to find the vapor composition. See the below diagram.

This method "works" because the pressure must be constant between each phase while the two phases are in equilibrium. The bubble and dew compositions are the only liquid and vapor compositions that are stable at a given pressure and temperature, so the system will tend toward those values.

Another useful rule is the lever rule which can be used to calculate the percentage of all the material that is in a given phase (as opposed to the composition of the vapor). The lever rule equation is [1]:

 ${\displaystyle \%Liquid={\frac {D2}{D1+D2}}}$

- Lever Rule

and therefore,

${\displaystyle \%Vapor={\frac {D1}{D1+D2}}}$.

The phase whose percent you're calculating is simply the one which you are going away from for the line segment in the numerator; for example, D2 is going from the point of interest to the vapor phase, so if D2 is in the numerator then you're calculating percent of liquid.

Txy diagrams have entirely analogous rules, but just be aware that the graph is "reversed" somewhat in shape. It's somewhat harder to calculate even in an ideal case, requiring an iterative solution, but is more useful for isobaric (constant-pressure) systems and is worth the effort. The extreme ends of the txy diagram are the boiling temperatures of pure toluene (xb = 0) and benzene (xb = 1) at 760 mmHg.

#### VLE phase diagram summary

To summarize, here's the information you can directly garner from a phase diagram. Many of these can be used for all types of phase diagrams, not just VLE.

1. You can use it to tell you what phase(s) you are in at a given composition, temperature, and/or pressure.
2. You can use it to tell you what the composition of each phase will be, if you're in a multiphase region.
3. You can use it to tell you how much of the original solution is in each phase, if you're in a multiphase region.
4. You can use it to gather some properties of the pure materials from the endpoints (though these are usually the best-known of all the mixture properties).

This data is invaluable in systems design which is why you'll be drilled with it before you graduate.

## Raoult's Law: the Simplest Case

The simplest case (by far) to analyze occurs when an ideal solution is in equilibrium with an ideal gas. This is potentially a good approximation when two very similar liquids (the archetypal example is benzene and toluene) are dissolved in each other. It is also a good approximation for solvent properties (NOT solute properties; there is another law for that) when a very small amount of a solute is dissolved.

In an ideal liquid, the pressure exerted by a certain component on the gas is proportional to the vapor pressure of the pure liquid. The only thing that may prevent the liquid from exerting this much pressure is the fact that another component is present. Therefore, the partial pressure of the liquid component on the gas component is:

${\displaystyle P_{A}=x_{A}*P_{A}^{*}}$

where ${\displaystyle P_{A}^{*}}$ is the vapor pressure of pure component A.

 Note: For all VLE equations, ${\displaystyle x_{A}}$ denote a liquid mole fraction of component A, and ${\displaystyle y_{A}}$ is the vapor mole fraction [/itex] of component A.

Recall that the partial pressure of an ideal gas in a mixture is given by:

${\displaystyle P_{A}=y_{A}*P}$

Therefore, since the partial pressures must be equal at equilibrium, we have the Raoult's Law equation for each component:

 Raoult's Law for component A ${\displaystyle y_{A}*P=x_{A}*P_{A}^{*}}$

### Vapor Pressure Correlations

Unfortunately, life isn't that simple even when everything is ideal. Vapor pressure is not by any stretch of the imagination a constant. In fact, it has a very strong dependence on temperature. Therefore, people have spent a good deal of time and energy developing correlations with which to predict the vapor pressure of a given substance at any reasonable temperature.

One of the most successful correlations is called the Antoine Equation, which uses three coefficients, A, B, and C, which depend on the substance being analyzed. The Antoine Equation is as follows.

 Antoine Equation ${\displaystyle log(P_{A}^{*})=A-{\frac {B}{T+C}}}$
 Note: In the external link provided in the appendix, the logarithm is to base 10, T is in degrees Celsius, and P* is in mmHg. Other sources use different forms (for example, natural log or P* in bars) so be wary.

### Bubble Point and Dew Point with Raoult's Law

#### Key concept

When calculating either a bubble point or a dew point, one quantity is key, and this is the overall composition, denoted with the letter z. This is to distinguish it from the single-phase composition in either the liquid or the gas phase. It is necessary to distinguish between them because the composition of the two phases will almost always be different at equilibrium.

It is important to remember that the dew and bubble points of a multi-component mixture are limits. The bubble point is the point at which a very small amount of the liquid has evaporated - so small, in fact, that in essence, the liquid phase composition remains the same as the overall composition. Making this assumption, it is possible to calculate the composition of that single bubble of vapor that has formed.

Similarly, the dew point is the point at which a very small amount of the vapor has condensed, so that the gas phase composition remains the same as the overall composition, and thus it is possible to calculate the composition of the single bubble of liquid.

#### Bubble Point

Recall that the dew point of a solution is the set of conditions (either a temperature at constant pressure or a pressure at constant temperature) at which the first drops of a vapor mixture begin to condense.

Let us first consider how to calculate the bubble point (at a constant temperature) of a mixture of 2 components A and B, assuming that the mixture follows Raoult's Law under all conditions. To begin, write Raoult's Law for each component in the mixture.

${\displaystyle y_{A}*P=x_{A}*P_{A}^{*}}$

${\displaystyle y_{B}*P=x_{B}*P_{B}^{*}}$

We can add these two equations together to yield:

${\displaystyle P(y_{A}+y_{B})=P_{A}^{*}x_{A}+P_{B}^{*}x_{B}}$

Now since ${\displaystyle y_{A}}$ and ${\displaystyle y_{B}}$ are mole fractions and A and B are the only components of the mixture, ${\displaystyle y_{A}+y_{B}=1}$. In addition, recall that since we are considering the bubble point, the liquid composition is essentially equal to the overall composition. Therefore, ${\displaystyle x_{A}=z_{A}}$ and ${\displaystyle x_{B}=z_{B}}$.

 Note: This is only true at the bubble point, not in general.

Hence we have the following equation valid at the bubble point for an ideal equilibrium:

 Bubble point equation for two components under Raoult's Law ${\displaystyle P=z_{A}*P_{A}^{*}+z_{B}*P_{B}^{*}}$

Therefore, if the temperature and overall composition are known, the bubble pressure can be determined directly.

If the pressure is held constant and the bubble point temperature is required, it is necessary to calculate the temperature by an iterative method. The temperature dependence is contained in the Antoine equation for vapor pressure of each component. One method to solve for the temperature is to:

1. Guess a temperature
2. Use the guess and the Antoine equation to calculate the vapor pressure of each component in the mixture.
3. Calculate an equilibrium pressure using the bubble point pressure equation.
4. If the calculated pressure does not equal the known pressure, it is necessary to change the temperature and try again.

This process is ideally suited to spreadsheet functions such as Excel's "goalseek" routine. An example calculation will be shown in the next section.

If there is more than one component, a similar derivation yields the following:

 Bubble point equation for multiple components under Raoult's Law ${\displaystyle P=\Sigma (z_{i}*P_{i}^{*})}$ (summed over all components i)

#### Dew Point

The Dew Point calculation is similar, although the equation that results from the derivation is somewhat more complex. The starting point is the same: assume that Raoult's Law applies to each component.

${\displaystyle y_{A}*P=x_{A}*P_{A}^{*}}$

${\displaystyle y_{B}*P=x_{B}*P_{B}^{*}}$

Now we want to eliminate the liquid compositions in a similar manner to how we eliminated the vapor compositions in the previous derivation. To do this we need to divide by the vapor pressures:

${\displaystyle {\frac {y_{A}*P}{P_{A}^{*}}}=x_{A}}$

${\displaystyle {\frac {y_{B}*P}{P_{B}^{*}}}=x_{B}}$

Adding the equations and recalling that ${\displaystyle x_{A}+x_{B}=1}$, we have:

${\displaystyle {\frac {y_{A}*P}{P_{A}^{*}}}+{\frac {y_{B}*P}{P_{B}^{*}}}=1}$.

Since this is the dew point, the gas-phase composition is essentially the overall composition, and therefore we have the following dew point equation:

 Dew Point Equation for Two Components under Raoult's Law ${\displaystyle {\frac {1}{P}}={\frac {z_{A}}{P_{A}^{*}}}+{\frac {z_{B}}{P_{B}^{*}}}}$
 Note: This is only valid at the dew point, just as the other equation was only valid at the bubble point.

For multiple components, the equation is similarly:

 Dew Point Equation for Multiple Components under Raoult's Law ${\displaystyle {\frac {1}{P}}=\Sigma ({\frac {z_{A}}{P_{A}^{*}}})}$

### Phase Diagrams Resulting from Raoult's Law

By holding one variable constant, varying a second, and calculating the other two, it is possible to calculate a phase diagram from Raoult's Law. Typical Pxy and Txy diagrams derived from Raoult's Law were shown in the previous section for the benzene-toluene system.

Diagrams for systems that follow Raoult's Law are relatively "nice"; it can be shown that they will never have azeotropes, which would be indicated by intersection of the bubble and dew point lines. In addition, since only one parameter in the equation depends on the temperature (the vapor pressure) and the pressure dependence is explicit, the dew and bubble point lines are relatively easy to calculate.

## Non-ideal VLE

Deviations from Raoult's Law occur because not all solutions are ideal, nor are all gas mixtures. Therefore, methods have been developed in order to take these nonidealities into account.

### Henry's Law

The third non-ideal method, Henry's Law, is especially useful for dilute solutions, and states that at very low concentrations, the partial pressure of the dilute component over a liquid mixture is proportional to the concentration:

 Henry's Law For a dilute component A, ${\displaystyle P_{A}=H_{A}*x_{A}}$ where ${\displaystyle H_{A}}$ is a constant and ${\displaystyle x_{A}}$ is the liquid-phase mole fraction of A

This law is very similar to Raoult's Law, except that the proportionality constant is not the pure-component vapor pressure but is empirically determined from VLE data. Like the pure-component vapor pressure, the Henry's constant is dependent on temperature and the nature of component A. Unlike the pure-component vapor pressure, it also depends on the solvent, so when utilizing tables of Henry's constants, make sure that the solvents match.

 Note: If Henry's Law applies to one component of a two-component mixture, the other component is often concentrated enough for Raoult's Law to apply to a reasonable approximation. Therefore, for a mixture of components A and B, where A is dilute and B is concentrated, a system similar to the following is common: ${\displaystyle y_{A}*P=x_{A}*H_{A}}$ ${\displaystyle y_{B}*P=x_{B}*P_{B}^{*}}$

Henry's Law constants are generally very small, and are most useful when the concentration is less than 10% (depending on how accurate you want it, the concentration may need to be less than this).

### Excess Gibbs Energy

The other two commonly-used correction parameters, the activity coefficient and the fugacity coefficient, are based on how non-ideal a given phase is. For a gas, the degree of non-ideality present is called the residual Gibbs energy, while for a liquid it is called excess Gibbs energy. The distinction is made because the Gibbs energy of an ideal gas and the Gibbs energy of an ideal solution are very different, as are the natures of how real solutions and real gasses deviate from ideality.

The residual Gibbs energy is based on the ideal gas and is defined as follows:

 Residual Gibbs Energy definition (for a gas phase) ${\displaystyle G^{R}=G_{real}-G_{\mbox{ideal gas}}}$

The excess Gibbs energy of a liquid phase is based on an ideal solution and is defined as:

 Excess Gibbs Energy definition (for a liquid phase) ${\displaystyle G^{E}=G_{real}-G_{\mbox{ideal solution}}}$

### Activity Coefficients

The activity coefficient takes into account variation from Raoult's Law due to liquid excess Gibbs energy. It may be defined as:

 ${\displaystyle ln\gamma _{i}={\frac {G_{i}^{E}}{RT}}}$

where ${\displaystyle \gamma }$ is a composition-dependent value which is also different for each component. It therefore is a measure of the effect of each component in contributing to the nonideality of the mixture.

Raoult's Law can be extended using activity coefficients as follows:

 Extended Raoult's Law ${\displaystyle y_{A}*P=x_{A}*P_{A}^{*}*\gamma _{i}}$

The extended Raoult's law may be used to calculate activity coefficients: the vapor pressure and equilibrium composition are measured at a low pressure (to avoid gaseous nonideality). Then, since the activity coefficient is only weakly dependent on pressure (liquid properties often change very little with pressure), the same values of the activity coefficient may be used at higher pressures to aid in determining the change in equilibrium properties.

Once activity coefficients are determined at a wide variety of concentrations, it is often desired to condense the information into one equation. See this publication for an interesting read on this topic, though it will probably make more sense after you take thermodynamics, it offers a good description of what you will see.

 Note: The definition of the activity coefficient implies that an ideal solution will have an activity coefficient equal to 1 (since its excess Gibbs energy is 0). Thus for an ideal solution the equilibrium equation reduces back to Raoult's law.

### Fugacity Coefficients

The fugacity coefficient of a gas is defined in a similar way to the activity coefficient, but it is based on the residual Gibbs energy:

 ${\displaystyle ln\phi _{i}={\frac {G_{i}^{R}}{RT}}}$

The fugacity coefficient of a gas depends on temperature, as can be seen clearly from the definition. It also depends heavily on the pressure. Indeed, if you have data available that relates the compressibility of a pure gas, Z, as a function of pressure at constant temperature, the fugacity can be computed using calculus or estimated (roughly) using the following equation if the change in pressure between each set of points is constant:

${\displaystyle ln\phi _{i}=\Delta P\Sigma ({\frac {Z_{k}-1}{P_{k}}})}$

where "k" referring to a data point

 Note: Since the compressibility (${\displaystyle Z_{k}}$) of an ideal gas is 1 regardless of what the pressure is, the fugacity coefficient of an ideal gas is 1 as well. Therefore, like the activity coefficient, the fugacity coefficient provides us with a measure of how nonideal a given gas or mixture of gasses is.

To do this calculation, it is necessary to extrapolate so that the first data point is taken at P = 0..

Example:

Given the following data:

P (atm)      Z
0.1          0.98
0.2          0.96
0.3          0.95


Calculate the fugacity coefficient at 0.2 atm and 0.3 atm.

Solution: It is necessary to first extrapolate to zero pressure:

${\displaystyle P=0\rightarrow Z~1.0}$

Then insert the data into the formula:

${\displaystyle ln\phi _{i}=(0.2-0.1)*({\frac {0.98-1}{0.1}}+{\frac {0.96-1}{0.2}})}$

 To do: Finish this example

Raoult's law can be modified to account for nonideal gasses in a similar way to its modification for nonideal liquids:

 VLE Equation for nonideal gasses and nonideal liquids ${\displaystyle \phi _{i}*y_{i}*P=\gamma _{i}*x_{i}*P_{i}^{*}}$

This equation is true except at very high pressures, a case covered in this paper briefly.

 To do: revamp to include saturation fugacity (which I forgot to include with liquid)