Vapor–liquid Equilibrium
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thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
and
chemical engineering Chemical engineering is an engineering field which deals with the study of the operation and design of chemical plants as well as methods of improving production. Chemical engineers develop economical commercial processes to convert raw materials ...
, the vapor–liquid equilibrium (VLE) describes the distribution of a
chemical species Chemical species are a specific form of chemical substance or chemically identical molecular entities that have the same molecular energy level at a specified timescale. These entities are classified through bonding types and relative abundance of ...
between the vapor phase and a
liquid phase Liquid is a state of matter with a definite volume but no fixed shape. Liquids adapt to the shape of their container and are nearly incompressible, maintaining their volume even under pressure. The density of a liquid is usually close to that ...
. The concentration of a vapor in contact with its liquid, especially at equilibrium, is often expressed in terms of
vapor pressure Vapor pressure or equilibrium vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature in a closed system. The equilibrium vapor pressure is an indicat ...
, which will be a
partial pressure In a mixture of gases, each constituent gas has a partial pressure which is the notional pressure of that constituent gas as if it alone occupied the entire volume of the original mixture at the same temperature. The total pressure of an ideal g ...
(a part of the total gas pressure) if any other gas(es) are present with the vapor. The equilibrium vapor pressure of a liquid is in general strongly dependent on
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
. At vapor–liquid equilibrium, a liquid with individual components in certain concentrations will have an equilibrium vapor in which the concentrations or partial pressures of the vapor components have certain values depending on all of the liquid component concentrations and the temperature. The converse is also true: if a vapor with components at certain concentrations or partial pressures is in vapor–liquid equilibrium with its liquid, then the component concentrations in the liquid will be determined dependent on the vapor concentrations and on the temperature. The equilibrium concentration of each component in the liquid phase is often different from its concentration (or vapor pressure) in the vapor phase, but there is a relationship. The VLE concentration data can be determined experimentally or approximated with the help of theories such as
Raoult's law Raoult's law ( law) is a relation of physical chemistry, with implications in thermodynamics. Proposed by French chemist François-Marie Raoult in 1887, it states that the partial pressure of each component of an ideal mixture of ''liquids'' is ...
, Dalton's law, and Henry's law. Such vapor–liquid equilibrium information is useful in designing columns for
distillation Distillation, also classical distillation, is the process of separating the component substances of a liquid mixture of two or more chemically discrete substances; the separation process is realized by way of the selective boiling of the mixt ...
, especially
fractional distillation Fractional distillation is the separation of a mixture into its component parts, or fractions. Chemical compounds are separated by heating them to a temperature at which one or more fractions of the mixture will vaporize. It uses distillation ...
, which is a particular specialty of chemical engineers. Distillation is a process used to separate or partially separate components in a mixture by
boiling Boiling or ebullition is the rapid phase transition from liquid to gas or vapor, vapour; the reverse of boiling is condensation. Boiling occurs when a liquid is heated to its boiling point, so that the vapour pressure of the liquid is equal to ...
(vaporization) followed by
condensation Condensation is the change of the state of matter from the gas phase into the liquid phase, and is the reverse of vaporization. The word most often refers to the water cycle. It can also be defined as the change in the state of water vapor ...
. Distillation takes advantage of differences in concentrations of components in the liquid and vapor phases. In mixtures containing two or more components, the concentrations of each component are often expressed as
mole fraction In chemistry, the mole fraction or molar fraction, also called mole proportion or molar proportion, is a quantity defined as the ratio between the amount of a constituent substance, ''ni'' (expressed in unit of moles, symbol mol), and the to ...
s. The mole fraction of a given component of a mixture in a particular phase (either the vapor or the liquid phase) is the number of moles of that component in that phase divided by the total number of moles of all components in that phase. Binary mixtures are those having two components. Three-component mixtures are called ternary mixtures. There can be VLE data for mixtures with even more components, but such data is often hard to show graphically. VLE data is a function of the total pressure, such as 1  atm or at the pressure the process is conducted at. When a temperature is reached such that the sum of the equilibrium vapor pressures of the liquid components becomes equal to the total pressure of the system (it is otherwise smaller), then vapor bubbles generated from the liquid begin to displace the gas that was maintaining the overall pressure, and the mixture is said to boil. This temperature is called the ''
boiling point The boiling point of a substance is the temperature at which the vapor pressure of a liquid equals the pressure surrounding the liquid and the liquid changes into a vapor. The boiling point of a liquid varies depending upon the surrounding envi ...
'' of the liquid mixture at the given pressure. (It is assumed that the total pressure is held steady by adjusting the total volume of the system to accommodate the specific volume changes that accompany boiling.) The boiling point at an overall pressure of 1 atm is called the '' normal boiling point'' of the liquid mixture.


Thermodynamic description of vapor–liquid equilibrium

The field of
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
describes when vapor–liquid equilibrium is possible, and its properties. Much of the analysis depends on whether the vapor and liquid consist of a single component, or if they are mixtures.


Pure (single-component) systems

If the liquid and vapor are pure, in that they consist of only one molecular component and no impurities, then the equilibrium state between the two phases is described by the following equations: :P^\text = P^\text\,; :T^\text = T^\text; and :\tilde^\text = \tilde^\text where P^\text and P^\text are the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
s within the liquid and vapor, T^\text and T^\text are the
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
s within the liquid and vapor, and \tilde^\text and \tilde^\text are the molar Gibbs free energies (units of energy per
amount of substance In chemistry, the amount of substance (symbol ) in a given sample of matter is defined as a ratio () between the particle number, number of elementary entities () and the Avogadro constant (). The unit of amount of substance in the International ...
) within the liquid and vapor, respectively.Balzhiser et al. (1972), ''Chemical Engineering Thermodynamics''. In other words, the temperature, pressure and molar Gibbs free energy are the same between the two phases when they are at equilibrium. An equivalent, more common way to express the vapor–liquid equilibrium condition in a pure system is by using the concept of
fugacity In thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of chemical equilibrium. It is equal to the pressure of an ideal gas which has the same tempe ...
. Under this view, equilibrium is described by the following equation: :f^\text(T_s,P_s) = f^\text(T_s,P_s) where f^\text(T_s,P_s)and f^\text(T_s,P_s)are the fugacities of the liquid and vapor, respectively, at the system temperature and pressure . It is often convenient to use the quantity \phi=f/P, the dimensionless ''fugacity coefficient'', which is 1 for an
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is ...
.


Multicomponent systems

In a multicomponent system, where the vapor and liquid consist of more than one type of compounds, describing the equilibrium state is more complicated. For all components in the system, the equilibrium state between the two phases is described by the following equations: :P^\text = P^\text; :T^\text = T^\text; and :\bar_i^\text = \bar_i^\text where and are the temperature and pressure for each phase, and \bar_i^\text and \bar_i^\textare the partial molar Gibbs free energy also called
chemical potential In thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potent ...
(units of energy per
amount of substance In chemistry, the amount of substance (symbol ) in a given sample of matter is defined as a ratio () between the particle number, number of elementary entities () and the Avogadro constant (). The unit of amount of substance in the International ...
) within the liquid and vapor, respectively, for each phase. The partial molar Gibbs free energy is defined by: :\bar_i \ \stackrel\ \frac where is the ( extensive) Gibbs free energy, and is the
amount of substance In chemistry, the amount of substance (symbol ) in a given sample of matter is defined as a ratio () between the particle number, number of elementary entities () and the Avogadro constant (). The unit of amount of substance in the International ...
of component .


Boiling-point diagrams

Binary mixture VLE data at a certain overall pressure, such as 1 atm, showing mole fraction vapor and liquid concentrations when boiling at various temperatures can be shown as a two-dimensional
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
called a boiling-point diagram. The mole fraction of component 1 in the mixture can be represented by the symbol . The mole fraction of component 2, represented by , is related to in a binary mixture as follows: : In multi-component mixtures in general with n components, this becomes: : The preceding equilibrium equations are typically applied for each phase (liquid or vapor) individually, but the result can be plotted in a single diagram. In a binary boiling-point diagram, temperature () (or sometimes pressure) is graphed vs. . At any given temperature (or pressure) where both phases are present, vapor with a certain mole fraction is in equilibrium with liquid with a certain mole fraction. The two mole fractions often differ. These vapor and liquid mole fractions are represented by two points on the same horizontal isotherm (constant ) line. When an entire range of temperatures vs. vapor and liquid mole fractions is graphed, two (usually curved) lines result. The lower one, representing the mole fraction of the boiling liquid at various temperatures, is called the '' bubble point curve''. The upper one, representing the mole fraction of the vapor at various temperatures, is called the ''dew point curve''. These two curves necessarily meet where the mixture becomes purely one component, namely where (and , pure component 2) or (and , pure component 1). The temperatures at those two points correspond to the boiling points of each of the two pure components. For certain pairs of substances, the two curves also coincide at some point strictly between and . When they meet, they meet tangently; the dew-point temperature always lies above the boiling-point temperature for a given composition when they are not equal. The meeting point is called an
azeotrope An azeotrope () or a constant heating point mixture is a mixture of two or more liquids whose proportions cannot be changed by simple distillation.Moore, Walter J. ''Physical Chemistry'', 3rd e Prentice-Hall 1962, pp. 140–142 This happens beca ...
for that particular pair of substances. It is characterized by an azeotrope temperature and an azeotropic composition, often expressed as a mole fraction. There can be maximum-boiling azeotropes, where the azeotrope temperature is at a maximum in the boiling curves, or minimum-boiling azeotropes, where the azeotrope temperature is at a minimum in the boiling curves. If one wants to represent a VLE data for a three-component mixture as a boiling point "diagram", a three-dimensional graph can be used. Two of the dimensions would be used to represent the composition mole fractions, and the third dimension would be the temperature. Using two dimensions, the composition can be represented as an equilateral triangle in which each corner represents one of the pure components. The edges of the triangle represent a mixture of the two components at each end of the edge. Any point inside the triangle represents the composition of a mixture of all three components. The mole fraction of each component would correspond to where a point lies along a line starting at that component's corner and perpendicular to the opposite edge. The bubble point and
dew point The dew point is the temperature the air needs to be cooled to (at constant pressure) in order to produce a relative humidity of 100%. This temperature depends on the pressure and water content of the air. When the air at a temperature above the ...
data would become curved surfaces inside a triangular prism, which connect the three boiling points on the vertical temperature "axes". Each face of this triangular prism would represent a two-dimensional boiling-point diagram for the corresponding binary mixture. Due to their three-dimensional complexity, such boiling-point diagrams are rarely seen. Alternatively, the three-dimensional curved surfaces can be represented on a two-dimensional graph by the use of curved isotherm lines at graduated intervals, similar to iso-altitude lines on a map. Two sets of such isotherm lines are needed on such a two-dimensional graph: one set for the bubble point surface and another set for the dew point surface.


''K'' values and relative volatility values

The tendency of a given chemical species to partition itself preferentially between liquid and vapor phases is the Henry's law constant. There can be VLE data for mixtures of four or more components, but such a boiling-point diagram is hard to show in either tabular or graphical form. For such multi-component mixtures, as well as binary mixtures, the vapor–liquid equilibrium data are represented in terms of values ( vapor–liquid distribution ratios) defined by :K_i=\frac where and are the mole fractions of component  in the phases and respectively. For
Raoult's law Raoult's law ( law) is a relation of physical chemistry, with implications in thermodynamics. Proposed by French chemist François-Marie Raoult in 1887, it states that the partial pressure of each component of an ideal mixture of ''liquids'' is ...
:K_i=\frac P For modified
Raoult's law Raoult's law ( law) is a relation of physical chemistry, with implications in thermodynamics. Proposed by French chemist François-Marie Raoult in 1887, it states that the partial pressure of each component of an ideal mixture of ''liquids'' is ...
:K_i=\frac P where \gamma_i is the activity coefficient, is the
partial pressure In a mixture of gases, each constituent gas has a partial pressure which is the notional pressure of that constituent gas as if it alone occupied the entire volume of the original mixture at the same temperature. The total pressure of an ideal g ...
and is the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
. The values of the ratio are correlated empirically or theoretically in terms of temperature, pressure and phase compositions in the form of equations, tables or graph such as the DePriester charts.DePriester, C.L., ''Chem. Eng. Prog. Symposium Series'', 7, 49, pages 1–43 For binary mixtures, the ratio of the values for the two components is called the
relative volatility Relative volatility is a measure comparing the vapor pressures of the components in a liquid mixture of chemicals. This quantity is widely used in designing large industrial distillation processes. In effect, it indicates the ease or difficulty of ...
denoted by :\alpha=\frac =\frac which is a measure of the relative ease or difficulty of separating the two components. Large-scale industrial distillation is rarely undertaken if the relative volatility is less than 1.05 with the volatile component being and the less volatile component being . values are widely used in the design calculations of
continuous distillation Continuous distillation, a form of distillation, is an ongoing separation in which a mixture is continuously (without interruption) fed into the process and separated fractions are removed continuously as output streams. Distillation is the se ...
columns for distilling multicomponent mixtures.


Vapor–liquid equilibrium diagrams

For each component in a binary mixture, one could make a vapor–liquid equilibrium diagram. Such a diagram would graph liquid mole fraction on a horizontal axis and vapor mole fraction on a vertical axis. In such VLE diagrams, liquid mole fractions for components 1 and 2 can be represented as and respectively, and vapor mole fractions of the corresponding components are commonly represented as and . Similarly for binary mixtures in these VLE diagrams: \begin x_1 + x_2 &= 1 \\ y_1 + y_2 &= 1 \end Such VLE diagrams are square with a diagonal line running from the () corner to the () corner for reference. These types of VLE diagrams are used in the McCabe–Thiele method to determine the number of equilibrium stages (or
theoretical plate A theoretical plate in many separation processes is a hypothetical zone or stage in which two phases, such as the liquid and vapor phases of a substance, establish an equilibrium with each other. Such equilibrium stages may also be referred to as ...
s) needed to distill a given composition binary feed mixture into one distillate fraction and one bottoms fraction. Corrections can also be made to take into account the incomplete efficiency of each tray in a distillation column when compared to a theoretical plate.


Raoult's law

At boiling and higher temperatures the sum of the individual component partial pressures becomes equal to the overall pressure, which can symbolized as . Under such conditions, Dalton's law would be in effect as follows: P_\text = P_1 + P_2 + \cdots Then for each component in the vapor phase: y_1 = \frac, \quad y_2 = \frac, \quad \cdots where = partial pressure of component 1, = partial pressure of component 2, etc.
Raoult's law Raoult's law ( law) is a relation of physical chemistry, with implications in thermodynamics. Proposed by French chemist François-Marie Raoult in 1887, it states that the partial pressure of each component of an ideal mixture of ''liquids'' is ...
is approximately valid for mixtures of components between which there is very little interaction other than the effect of dilution by the other components. Examples of such mixtures includes mixtures of
alkane In organic chemistry, an alkane, or paraffin (a historical trivial name that also has other meanings), is an acyclic saturated hydrocarbon. In other words, an alkane consists of hydrogen and carbon atoms arranged in a tree structure in whi ...
s, which are non- polar, relatively inert compounds in many ways, so there is little attraction or repulsion between the molecules. Raoult's law states that for components 1, 2, etc. in a mixture: P_1 = x_1 P^\circ_1, \quad P_2 = x_2 P^\circ_2, \quad \cdots where , , etc. are the vapor pressures of components 1, 2, etc. when they are pure, and , , etc. are mole fractions of the corresponding component in the liquid. Recall from the first section that vapor pressures of liquids are very dependent on temperature. Thus the pure vapor pressures for each component are a function of temperature (): For example, commonly for a pure liquid component, the
Clausius–Clapeyron relation The Clausius–Clapeyron relation, in chemical thermodynamics, specifies the temperature dependence of pressure, most importantly vapor pressure, at a discontinuous phase transition between two phases of matter of a single constituent. It is nam ...
may be used to approximate how the vapor pressure varies as a function of temperature. This makes each of the partial pressures dependent on temperature also regardless of whether Raoult's law applies or not. When Raoult's law is valid these expressions become: P_1 T = x_1 P^\circ_1 T, \quad P_2 T = x_2 P^\circ_2 T, \quad \cdots At boiling temperatures if Raoult's law applies, the total pressure becomes: P_\text = x_1 P^\circ_1 T + x_2 P^\circ_2 T + \cdots At a given such as 1 atm and a given liquid composition, can be solved for to give the liquid mixture's boiling point or bubble point, although the solution for may not be mathematically analytical (i.e., may require a numerical solution or approximation). For a binary mixture at a given , the bubble point  can become a function of (or ) and this function can be shown on a two-dimensional graph like a binary boiling point diagram. At boiling temperatures if Raoult's law applies, a number of the preceding equations in this section can be combined to give the following expressions for vapor mole fractions as a function of liquid mole fractions and temperature: y_1 = x_1 \frac, \quad y_2 = x_2 \frac, \quad \cdots Once the bubble point 's as a function of liquid composition in terms of mole fractions have been determined, these values can be inserted into the above equations to obtain corresponding vapor compositions in terms of mole fractions. When this is finished over a complete range of liquid mole fractions and their corresponding temperatures, one effectively obtains a temperature  function of vapor composition mole fractions. This function effectively acts as the dew point function of vapor composition. In the case of a binary mixture, and the above equations can be expressed as: \begin y_1 &= x_1 \frac \\ y_2 &= (1 - x_1) \frac \end For many kinds of mixtures, particularly where there is interaction between components beyond simply the effects of dilution, Raoult's law does not work well for determining the shapes of the curves in the boiling point or VLE diagrams. Even in such mixtures, there are usually still differences in the vapor and liquid equilibrium concentrations at most points, and distillation is often still useful for separating components at least partially. For such mixtures, empirical data is typically used in determining such boiling point and VLE diagrams.
Chemical engineer A chemical engineer is a professional equipped with the knowledge of chemistry and other basic sciences who works principally in the chemical industry to convert basic raw materials into a variety of Product (chemistry), products and deals with ...
s have done a significant amount of research trying to develop equations for correlating and/or predicting VLE data for various kinds of mixtures which do not obey Raoult's law well.


See also

*
Continuous distillation Continuous distillation, a form of distillation, is an ongoing separation in which a mixture is continuously (without interruption) fed into the process and separated fractions are removed continuously as output streams. Distillation is the se ...
* Dortmund Data Bank (includes a collection of VLE data) * Fenske equation * Flash evaporation * DECHEMA model * Hand boiler * Van Laar equation * Margules activity model *
Pervaporation Pervaporation (or pervaporative separation) is a processing method for the separation of mixtures of liquids by partial vaporization through a non-porous or porous membrane. Theory The term ''pervaporation'' is a portmanteau of the two steps of ...
*
Supercooling Supercooling, also known as undercooling, is the process of lowering the temperature of a liquid below its freezing point without it becoming a solid. Per the established international definition, supercooling means ''‘cooling a substance be ...
*
Superheated steam Superheated steam is steam at a temperature higher than its vaporization point at the absolute pressure where the temperature is measured. Superheated steam can therefore cool (lose internal energy) by some amount, resulting in a lowering of its ...


External links


Distillation Principals
by Ming T. Tham,
University of Newcastle upon Tyne Newcastle University (legally the University of Newcastle upon Tyne) is a public university, public research university based in Newcastle upon Tyne, England. It has overseas campuses in Singapore and Malaysia. The university is a red brick un ...
(scroll down to Relative Volatility)
Introduction to Distillation: Vapor Liquid Equilibria


(Chemical Engineering Dept., Prof. Richard Rowley, Brigham Young University)
NIST Standard Reference Database 103b
(Describes the extensive VLE database available from
NIST The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical s ...
)
Some VLE data sets and diagrams for mixtures of 30 common components
a small subset of the Dortmund Data Bank
Where can I get the vapor-liquid phase equilibrium data?
Reference to the various phase equilibrium data sources
Can. J. Chem. Eng. ternary and multicomponent systems from binary ones
* George Schlowsky, Alan Erickson, and Thomas A. Schafer, Modular Process Systems, Inc.
Operations & Maintenance - Generating your own VLE Data
Chemical Engineering, March 1995, McGraw-Hill, Inc.


References

{{DEFAULTSORT:Vapor-Liquid Equilibrium Chemical engineering thermodynamics Equilibrium chemistry Distillation Phases of matter