1.6: Multicomponent Phase Diagrams

Introduction

As might be expected multicomponent phase diagrams are more complex than phase diagrams for pure substances. These multicomponent phase diagrams are useful for understanding how distillation works and the behavior of mixtures. We will start with quantifying the number of possible phases that can coexist under particular constraints in a system and then discuss phase separation and distillation.

Gibbs Phase Rule

The Gibbs phase rule describes the number of compositional and phase variables that can be varied freely for a system at equilibrium. We will call these the degrees of freedom (F). Given a system of C components that can form P different phases and two state variables to vary (pressure and temperature), if there were no constraints we would have F = CP + 2 degrees of freedom.

The first constraint is that for each phase present in a system, the mole fraction of all but one component can be varied independently. This is because, the relationship

\[\sum_i \chi_i =1 \nonumber \]

places a constraint on the last mole fraction. Thus, we lose one degree of freedom for each phase that is present. This leads to: F = 2 + CP - P.

The second constraint comes from the fact that the chemical potential of all the phases in equilibrium must be the same. There are P-1 equations of the form \(\mu_A = \mu_B\), where A and B represent different phases, specifying this constraint. Thus for each component we lose another P-1 degrees of freedom. This leads to: F = 2 + CP - P -C(P-1).

Simplification leads to the common form of the Gibbs phase rule.

\[\begin{align} F &= 2+ PC-P - C(P-1) \nonumber \\[4pt] &= 2 + PC - P -PC +C \nonumber \\[4pt] &= 2+C-P \label{Phase} \end{align} \]

Equation \ref{Phase} is the Gibbs phase rule.

Example \(\PageIndex{1}\):

Show that the maximum number of phases that can co-exist at equilibrium for a single component system is \(P = 3\).

Solution

The maximum number of components will occur when the number of degrees of freedom is zero.

\[ \begin{align*} 0 &= 2+1 -P \\[4pt] P&=3 \end{align*} \]

Note: This shows that there can never be a “quadruple point” for a single component system!

Because a single component system at its triple point has no degrees of freedom, the triple point makes a very convenient physical condition at which to define a temperature. For example, the International Practical Temperature Scale of 1990 (IPT-90) uses the triple points of hydrogen, neon, oxygen, argon, mercury, and water to define several low temperatures. (The calibration of a platinum resistance thermometer at the triple point of argon, for example, is described by Strouse (Strouse, 2008)). The advantage to using a triple point is that the compound sets both the temperature and pressure, rather than forcing the researcher to set a pressure and then measure the temperature of a phase change, introducing an extra parameter than can introduce uncertainty into the measurement.

As suggested by the Gibbs Phase Rule, the most important variables describing a mixture are pressure, temperature and composition. In the case of single component systems, composition is not important so only pressure and temperature are typically depicted on a phase diagram. However, for mixtures with two components, the composition is of vital import, so there is generally a choice that must be made as to whether the other variable to be depicted is temperature or pressure.

Temperature-composition diagrams are very useful in the description of binary systems, many of which will form two-phase compositions at a variety of temperatures and compositions.

Partially Miscible Liquids

A pair of liquids is considered partially miscible if there is a set of compositions over which the liquids will form a two-phase liquid system. This is a common situation and is the general case for a pair of liquids where one is polar and the other non-polar (such as water and vegetable oil.) Another case that is commonly used in the organic chemistry laboratory is the combination of diethyl ether and water. In this case, the differential solubility in the immiscible solvents allows the two-phase liquid system to be used to separate solutes using a separatory funnel method.

Figure \(\PageIndex{1}\) :

As is the case for most solutes, their solubility is dependent on temperature. For many binary mixtures of immiscible liquids, miscibility increases with increasing temperature. And then at some temperature (known as the upper critical temperature), the liquids become miscible in all compositions. An example of a phase diagram that demonstrates this behavior is shown in Figure \(\PageIndex{1}\). An example of a binary combination that shows this kind of behavior is that of methyl acetate and carbon disufide, for which the critical temperature is approximately 230 K at one atmosphere (Ferloni & Spinolo, 1974). Similar behavior is seen for hexane/nitrobenzene mixtures, for which the critical temperature is 293 K.

Figure \(\PageIndex{2}\) :

Another condition that can occur is for the two immiscible liquids to become completely miscible below a certain temperature, or to have a lower critical temperature. An example of a pair of compounds that show this behavior is water and trimethylamine. A typical phase diagram for such a mixture is shown in Figure \(\PageIndex{2}\). Some combinations of substances show both an upper and lower critical temperature, forming two-phase liquid systems at temperatures between these two temperatures. An example of a combination of substances that demonstrate the behavior is nicotine and water.

The Lever Rule

The composition and amount of material in each phase of a two phase liquid can be determined using the lever rule. This rule can be explained using the following diagram.

Figure \(\PageIndex{3}\) :

Suppose that the temperature and composition of the mixture is given by point b in the above diagram. The horizontal line segment that passes through point b, is terminated at points a and c, which indicate the compositions of the two liquid phases. Point a indicates the mole faction of compound B (\(\chi_B^A\)) in the layer that is predominantly A, whereas the point c indicates the composition (\(\chi_B^B\) )of the layer that is predominantly compound B. The relative amounts of material in the two layers is then inversely proportional to the length of the tie-lines a-b and b-c, which are given by \(l_A\) and \(l_B\) respectively. In terms of mole fractions,

\[ l_A = \chi_B - \chi_B^A \nonumber \]

and

\[ l_B = \chi_B^B - \chi_B \nonumber \]

The number of moles of material in the A layer (\(n_A\)) and the number of moles in the B layer (\(n_B\)) are inversely proportional to the lengths of the two lines \(l_A\) and \(l_B\).

\[ n_A l_A = n_B l_B \nonumber \]

Or, substituting the above definitions of the lengths \(l_A\) and \(l_B\), the ratio of these two lengths gives the ratio of moles in the two phases.

\[ \dfrac{n_A}{n_B} = \dfrac{l_B}{l_A} = \dfrac{ \chi_B^B - \chi_B}{\chi_B - \chi_B^A} \nonumber \]

 

Distillation

Distillation can be understood in terms of the lever rule, which provides a simple way of determining the relative quantities (not just the compositions) of two phases in equilibrium. The plot below shows the boiling point diagram of a simple binary mixture of composition . At the temperature corresponding to the tie line, the composition of the liquid corresponds to (more A than B) and that of the vapor to (more B than A or "enriched" in B).

Figure \(\PageIndex{4}\):

The relative quantities of the liquid and the vapor we identified above are given by the lengths of the tie-line segments labeled a and b. Thus in this particular example, in which b is about four times longer than a, we can say that the mole ratio of vapor (of composition ) to liquid (composition ) is 4.

 

Fractional Distillation

Now if we collect and condense the vapor, we will have a new liquid with composition enriched in B. If we re-vaporize it at the lower boiling temperature of this new liquid we create a new vapor that is even more enriched in B. Repeating this process is called fractional distillation. To make this more concrete, suppose you want to separate a liquid mixture composed of 20 mole-% B and 80 mole-% A, with A being the more volatile.

Figure \(\PageIndex{5}\): Steps in Fractional Distillation

As we heat the mixture whose overall composition is indicated by , the first vapor is formed at T₀ and has the composition y₀, found by extending the horizontal dashed line until it meets the vapor curve. This vapor is clearly enriched in B; if it is condensed, the resulting liquid will have a mole fraction x_B > 0.7.

As the liquid continues to boil, the boiling temperature rises. When it reaches T₁, we will have boiled away half of the liquid. At this point, the "system" composition (liquid plus vapor) is still the same (), but is now equally divided between the liquid, which we call "residue" R₁, and the condensed vapor, the distillate D₁.

How do we know it is equally divided? We have picked T₁ so that the tie line is centered on the system concentration, so by the lever rule, R₁ and D₁ contain equal numbers of moles.

We now take the condensed liquid D₁ having the composition , and distill half of it, obtaining distillate of composition D₂.

.. and then carry out yet another distillation, this time using D₃ as our feedstock.

Our four-stage fractionation has enriched the more volatile solute from 20 to slightly over 80 mole-percent in D₄. The less volatile component A is most concentrated in R₁. R₂ through R₄ are thrown away (but not down the sink, please!)

Figure \(\PageIndex{6}\): Boiling point diagrams illustrating fractional distillation"

This may be sufficient for some purposes, but we might wish to do much better, using perhaps 1000 stages instead of just 4. What could be more tedious?

Fractionation with reflux

Not to worry! The multiple successive distillations can be carried out "virtually" by inserting a fractionating column between the boiling flask and the condenser. The key is that the boiling point of each fraction enriched in the more volatile component is lower than the previous fraction. So as we go up the column to lower temperatures the condensation - vaporization cycle repeats.

Figure \(\PageIndex{7}\):

These columns are made with indentations or are filled with materials that provide a large surface area extending through the vertical temperature gradient (higher temperature near the bottom, lower temperature at the top.) The idea is that hot vapors condense at various levels in the column and the resulting liquid drips down (refluxes) to a lower level where it is vaporized, which corresponds roughly to a re-distillation. Vigreux columns having multiple indentations are widely used. Simple columns can be made by filling a glass tube with beads, short glass tubes, or even stainless steel kitchen-type scouring pads. More elaborate ones have spinning steel ribbons.

Separation efficiency: theoretical plates

The operation of fractionating columns can best be understood by reference to a bubble-cap column. The one shown here consists of four sections, or "plates" through which hot vapors rise and bubble up through pools of condensate that collect on each plate. The intimate contact between vapor and liquid promotes equilibration and re-distillation at successively higher temperatures at each higher plate in the column. Unlike the case of the step-wise fractional distillation we discussed above, none of the intermediate residues is thrown away; they simply drip down back into the pot where their fractionation journey begins again, always leading to a further concentration of the less-volatile component in the remaining liquid. At the same time, the vapor emerging from the top plate (5) provides a continuing flow of volatile-enriched condensate, although in diminishing quantities as it is depleted in the boiling pot.

Figure \(\PageIndex{8}\):

If complete equilibrium is attained between the liquid and vapor at each stage, then we can describe the system illustrated above as providing "five theoretical plates" of separation (remember that the pot represents the first theoretical plate.) Equilibrium at each stage requires a steady-state condition in which the quantity of vapor moving upward at each stage is equal to the quantity of liquid draining downward — in other words, the column should be operating in total reflux, with no net removal of distillate. So any real distillation process will be operated at a reflux ratio that provides optimum separation in a reasonable period of time.

Figure \(\PageIndex{9}\): ​​​​​​ Reflux ratio can be adjusted by placing a special control head on top of the fractionation column. The laboratory-type device shown here illustrates the general concept. Reflux control is mainly of importance in large-scale industrial operations.

Some of the more advanced laboratory-type devices (such as some spinning-steel band columns) are said to offer up to around 200 theoretical plates of separating power.

Azeotropes: the Limits of Distillation

The boiling point diagrams presented in the foregoing section apply to solutions that behave in a reasonably ideal manner — that is, to solutions that do not deviate too far from Raoult's law. As we explained above, mixtures of liquids whose intermolecular interactions are widely different do not behave ideally, and may be impossible to separate by ordinary distillation. The reason for this is that under certain conditions, the compositions of the liquid and of the vapor in equilibrium with it become identical, precluding any further separation. These cross-over points appear as "kinks" in the boiling point diagrams.

High- and low-boiling azeotropes

Thus in this boiling point diagram for a mixture exhibiting a positive deviation from Raoult's law, successive fractionations of mixtures correspond to either or bring the distillation closer to the azeotropic composition indicated by the dashed vertical line. Once this point is reached, further distillation simply yields more of the same "high-boiling" azeotrope.

Figure \(\PageIndex{10}\): High Boiling Azeotrope

Distillation of a mixture having a negative deviation from Raoult's law leads to a similar stalemate, in this case yielding a "low-boiling" azeotrope. High- and low-boiling azeotropes are commonly referred to as constant-boiling mixtures, and they are more common than most people think.

Figure \(\PageIndex{11}\): Low Boiling Azeotrope

An example of a common low boiling azeotrope is water + ethanol, which yields ~96% by weight ethanol as the limit of purification by simple fractional distillation.

 Phase diagrams including multiphase liquids and solids

It is possible that when the temperature is lowered enough below the boiling points of the solutions the liquid could undergo phase separation as in figures \(\PageIndex{1}\) and \(\PageIndex{3}\). Thus you could have a more complex diagram that looks something like the figure below. Exactly how the various boundaries relate depends on the particular system. For example the two phase liquid region boundary could touch the two phase vapor boundaries at the low-boiling azeotrope point in the figure.

Figure \(\PageIndex{12}\): Phase diagram for a binary solution with the boiling point of a low boiling azeotrope that is higher then when components are miscible (single phase).

Solids mixtures can also exhibit different phase compositions. A phase diagram for two immiscible solids and the liquid phase (which is miscible in all proportions) is shown in the figure below. The point labeled “e₂” is the eutectic point, meaning the composition for which the mixture of the two solids has the lowest melting point. The four main regions can be described as below:

• Two-phase solid
• (Region I) Solid (mostly A) and liquid (A and B)
• (Region II) Solid (mostly B) and liquid (A and B)
• Single phase liquid (A and B)

Figure \(\PageIndex{13}\): Phase diagram of a two-component system that exhibits an eutectic point.

The unlabeled regions on the sides of the diagram indicate regions where one solid is so miscible in the other, that only a single phase solid forms. This is different than the “two-phase solid” region where there are two distinct phases, meaning there are regions (crystals perhaps) that are distinctly mostly A or B, even though they are intermixed within on another. Region I contains two phases: a solid phase that is mostly compound A, and a liquid phase which contains both A and B. A sample in region II (such as the temperature/composition combination depicted by point b) will consist of two phases: 1 is a liquid mixture of A and B with a composition given by that at point a, and the other is a single phase solid that is mostly pure compound B, but with traces of A entrained within it. As always, the lever rule applies in determining the relative amounts of material in the two phases.

Many other additional complexities exist in the phase diagrams for binary mixtures of solids, liquids or gases. Even more complexity is found as expected from the Gibb's phase rule in diagrams of ternary and higher mixtures.

Keep in mind that the diagrams we have been looking at are slices of multi-dimensional spaces where surfaces (1 dimension less than the number of degrees of freedom) separate the different phase regions.