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Chapter 12 Theory of separation methods
Chapter 12
Theory of separation methods C.H. Lochmu¨ller
12.1
INTRODUCTION
In the strictest meaning of the word, there is no theoretical basis for separation methods but rather models. Models are, in most cases, a mixture of empirical and chemical kinetic/chemical thermodynamics. There are many examples of attempts to relate known physical and chemical properties to observe and develop a ‘‘predictive theory’’. The goal is both an improvement in understanding of the underlying phenomena and an efficient approach to method development. This latter goal is of greater interest to the vast majority of those who make use of separation methods, as their interest is in either obtaining relatively pure material or in isolating components of a chemical mixture in time and space for quantitative measurement. If a separation method can also provide complimentary information leading to qualitative identification, so much the better. This chapter seeks to give the ‘‘user’’ of chemical separation methods the beginnings of a basis for understanding the methods described in this book and the ability to recognize ‘‘normal behavior’’ and to distinguish ‘‘anomalous behavior’’. Justice to all the important theory would require several volumes of substantial size and especially if historical justice were to be given to the development of current models. At times the chapter’s content will seem more conversational than hard scientific and the choice of style in any given instance, is deliberate. Stories are part of the history of separation methods, after all. 12.2
THE GREAT DIVIDE
Julius Caesar tells us ‘‘All Gaul is divided into three parts’’. Separation methods can be divided into two categories: equilibrium and Comprehensive Analytical Chemistry 47 S. Ahuja and N. Jespersen (Eds) Volume 47 ISSN: 0166-526X DOI: 10.1016/S0166-526X(06)47012-4 r 2006 Elsevier B.V. All rights reserved.
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non-equilibrium. The equilibria spoken of in this case are ‘‘chemical’’ rather than physical in the vast majority of cases. They are the kind of equilibria spoken of as ‘‘ionization in solution’’ or between two phases, such as gas and liquid in introductory chemistry courses. Finding the point between a knife tip and the end of its’ handle that when placed on a finger leaves the knife neither tipping toward one end or the other, is the location of the ‘‘center of gravity’’. Supported at that point the knife is said to be ‘‘in balance’’ and that balance is a physical equilibrium. It is based on simple mechanics and the concept of mass and length. Chemical equilibria arise from not just mass but the underlying properties of molecules and their interaction with each other. A sealed half-full bottle of gin contains a liquid which is 40% by volume ethyl alcohol and 60% water. If we call the liquid the condensed phase, the vapor phase is the gaseous region above the condensed phase. The composition of the vapor phase is found to be different than the condensed phase and richer in alcohol than water. At constant temperature, the mole fraction of alcohol in the vapor phase is higher than water. Why? Ethanol is more volatile than water. True and it is also true that the composition of ethanol in the vapor phase will reach a constant value in time. Separation theory attempts to understand why such a chemical equilibrium occurs between the two phases and why, if methanol is substituted for ethanol, the alcohol content in the vapor phase is higher. The fact that the equilibrium is established over time is a kinetic consideration, while the value of the equilibrium constant and its’ temperature dependence is a question of chemical thermodynamics. 12.2.1
Physical separation methods
Physical separation methods can be based on equilibrium considerations, but the majority are not. Ordinary filtration is an example of a non-equilibrium, physical method and so is ordinary centrifugation— e.g.—the separation of a precipitate from the suspending liquid using an artificial gravity field. There are separation methods, which are called ‘‘filtration’’ which are not such as gel filtration. Ultracentrifugation in a salt gradient is a physical equilibrium method. Separation of desirable components of a mixture is not a new desire on the part of humanity. Removal of water-borne particles including sand, silt, and bacteria is desirable for both sensory appeal and health. The use of sand as a filtration medium for such filtration is very old. The use of a pan to separate gold dust from silica sand by gold miners is a use of the large difference in density to wash the lighter silica away 398
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from the much denser gold, which resembles black sand. Not filtration but certainly a non-equilibrium, physical method. Molecules which exhibit optical activity are molecules which have a ‘‘handedness’’ in their structure. They are ‘‘chiral’’. Chemists often have reasons to obtain chemical pure aliquots of particular molecules. Since the chirality of molecules can influence biological effect in pharmaceuticals, the chiral purity of a drug substance can pose a challenge both in terms of obtaining the molecules and in assaying the chiral purity by instrumental methods. While diastereomers can have different physical properties including solubility, enantiomers have the same physical properties and the same chemical composition. How then to separate optically active molecules? The first example of the deliberate separation of optically active molecules is appropriate as an example of physical separation in the clearest sense of the term. The molecules are referred to as optically active because polarized light interacts differently with right- and lefthanded molecules. In the case of simple diastereomers the RR and SS forms are enantiomers while the RS and SR forms are not. The separation of the latter and former was first done under a microscope using crossed polarizers and the crystals which were seen were separated from those that caused little or no rotation of plane-polarized light by hand using tweezers. A truly physical separation of chemical species using a physical property of chemical origin! One could go on with examples such as the use of a shirt rather than sand reduce the silt content of drinking water or the use of a net to separate fish from their native waters. Rather than that perhaps we should rely on the definition of a chemical equilibrium and its’ presence or absencey. Chemical equilibria are dynamic with only the illusion of static state. Acetic acid dissociates in water to acetate-ion and hydrated hydrogen ion. At any instant, however, there is an acid molecule formed by recombination of acid anion and a proton cation while another acid molecule dissociates. The equilibrium constant is based on a dynamic process. Ordinary filtration is not an equilibrium process nor is it the case of crystals plucked from under a microscope into a waiting vial. 12.2.2
Chemical separation methods
Most, but not all, chemical separation methods are based on equilibrium in the sense of chemical equilibrium. Clearly, solubility is a chemical question but formation of a precipitate and filtration is a physical separation, which happens to use a favorable Ksp equilibrium 399
C.H. Lochmu ¨ ller TABLE 1 Possible combinations and typical separation methods using them in a notall-inclusive list Interface type
Separation method(s)
Gas–gas Gas–liquid Gas–solid Liquid–liquid Liquid–solid Solid–solid
Possible but no common examples Gas chromatography; distillation Gas chromatography; gas masks Extraction; liquid chromatography Liquid chromatography; solvent drying Possible but no common examples
value to begin the process. Distillation and extraction are indeed chemical separation methods based on equilibrium and differences in equilibrium constants. Chemical separations are often either a question of equilibrium established in two immiscible phases across the contact between the two phases. In the case of true distillation, the equilibrium is established in the reflux process where the condensed material returning to the pot is in contact with the vapor rising from the pot. It is a gas– liquid interface. In an extraction, the equilibrium is established by motion of the solute molecules across the interface between the immiscible layers. It is a liquid– liquid interface. If one adds a finely divided solid to a liquid phase and molecules are then distributed in equilibrium between the solid surface and the liquid, it is a liquid– solid interface (Table 1).
12.3
CHEMICAL EQUILIBRIA AND THE PROPERTIES OF THE EQUILIBRIUM CONSTANT
Chemistry deals with molecules not atoms. True thermodynamics knows no molecules with much less properties of molecules derived from chemical effects. Its’ origin is in such concerns as heat flow and the heat equivalent of mechanical work. Most of us have heard in physical chemistry about how it was the drilling of cannon barrels that created the connection between work and heat energy. One can take entire semester course on thermodynamics in physics and in engineering and never deal with the solution thermodynamics, which often dominates chemistry courses. To the extent that thermodynamics has been used in developing a theory for separation methods, it is almost entirely chemical thermodynamics. 400
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In talking about thermodynamics and the properties of chemical equilibrium constants it is very difficult for chemists to avoid attempts to include the influence of forces between molecules on the magnitude of the equilibrium constant and differences between observed equilibrium constants. For the purpose of this chapter, it is convenient to talk first about the equilibrium constant and the macroscopic properties of matter which affect it first. Next, the reader will be introduced to the concept of forces between molecules, their relative magnitude and influence is separations.
12.3.1
Thermodynamics and kinetics in separations
If all that is achieved, in an attempted separation, is the transfer of the components of the mixture (atoms and/or molecules) from one place to another, no separation is achieved. The result is just mass transport. An analogy would be carrying a bag of mixed groceries from the store to a car. The tomatoes, onions, garlic and meat are still in the bag. What is needed is a means to have the components move to the end of an experiment but at different apparent rates of speed. For most equilibrium separation methods, the answer lies in controlling the thermodynamics of the system. Molecules or atoms are made to move in one of the two immiscible phases while undergoing transfer to the other, stationary phase and quickly back to the mobile phase. If the net effects in the molecules or atoms move at different rates depending on their chemical character, some will move faster and others more slowly. The goal would be to have molecules of type A moving faster than type B and both faster than type C and so on. Ideally, it would be best if they remained in tight ‘‘packets’’ as the forward motion of the mobile phase caused them to migrate in space and the difference in net rate moved the packets further and further apart as time passed. The simplest rules of thermodynamics suggest that energy must be expended to do work—‘‘You cannot get something for nothing,’’ and that even if work is done some energy is forever lost to useful work— ‘‘You cannot even get what you paid for’’. And that this entropy effect is such that the entropy of the universe is forever driving toward a maximum—‘‘Nature spontaneously falls into a mess!’’ Humor aside, the consequence is that any narrow packet as described above will spread over space in an attempt to make the local and universal mole fraction of A, B or C y the same everywhere. 401
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Put in ordinary terms, the more successful we are in causing a separation, the more propensities there are for a re-mixing of the components. There are many ways this can occur but there are a fewer number of important routes to mixing. It seems reasonable that we examine these before we consider all the possible ways in which thermodynamics can be controlled in general terms. In almost all equilibrium separation systems, the separation can occur either in a packed bed of particles or fibers or in an open channel or tube. The stationary phase is either coated on the walls of the channel or on the particles/fibers of the packed bed. If there were no mixing mechanisms an infinitely narrow packet containing the components would become a series of infinitely narrow packets of pure components moving at different velocities toward the end of the packed bed or tube.
12.3.2
Mechanisms for band broadening
How to best describe this broadening we expect to occur? One way is by analogy to random error in measurements. We know or assume there is a truly correct answer to any measurement of quantity present and attempt to determine that number. In real measurements there are real, if random, sources of error. It is convenient to talk about standard deviation of the measurement but actually the error in measurement accumulates as the sum of the square of each error process or variance producing mechanism or total variance ¼ sum of the individual variances. If we ignore effects outside the actual separation process (e.g. injection/spot size, connecting tubing, detector volume), this sum can be factored into three main influences: 1. 2. 3.
The random path of molecules taken in a packed bed (called Random Path Term). Ordinary diffusion along the length of the path taken by a packet of molecules (called Longitudinal Molecular Diffusion). Departure from true equilibrium of the molecules moving over the stationary phase and moving in and out of the stationary phase (called Resistance to Mass Transfer).
If we give variance then s2 ½total ¼ s2 ½RPT þ s2 ½LMD þ s2 ½RTMT
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Multipath processes
If flow occurred in an open channel with no particles or fibers and if there were no other mixing mechanisms, all particles would transit the same distance from beginning to end. In a packed bed, each time a molecule or atom encounters a particle or fiber it must go around it to continue on. It is analogous to encounter a tree in a field—one either walks around it to the right or the left and that is equivalent to flipping a coin. Some molecules will encounter more particles than others as illustrated in the following scheme where each encounter causes a chance in direction and the path of a hypothetical molecule is traced by a line (Scheme 1). Since the paths are random and the number of collisions large, the effect is to make the paths random and distributed as a Gaussian distribution. If you are familiar with the Random Walk model, this can be considered as analogous Random Path Model. The actual situation is a bit more complicated but for practical purposes the intuitive insights that the degree of broadening will be influenced by particle size dp (and, thus, distance between particles or fibers) and the geometry of the packing. The variance produced per unit length l of the bed traversed (s2/l) can be given the symbol H: H ¼ 2ldp
Scheme 1. Traces of the path of molecules in a packed bed.
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The symbol l in this case was first introduced by Van Deemter to account for packing inhomogeneity. 12.3.4
Broadening by diffusion
If a band of molecules or atoms are placed in a container such as a tube at the center of its length, the molecules will diffuse in the direction of lower concentration. Clearly this occurs in all directions but in a tube the walls are a limit and so our concern is along the axis of movement caused by the mobile phase and this is referred to as longitudinal molecular diffusion. The Einstein equation for diffusion along a single axis states that the process is Gaussian and the variance is time dependent: s2 ¼ 2Dt where D is a characteristic diffusion coefficient. The time a sample spends in the moving phase on its transit from the beginning to the end of the bed is proportional not to the flow rate but the flow velocity of the mobile phase, which we can call u. If the time to transit a distance l is l/u then for a unit distance t ¼ 1/u. It can be seen then that the variance contribution per unit length in an open tube is s2/l ¼ 2Dm/u where Dm is the diffusion coefficient for the molecules of interest in the moving or mobile phase. Van Deemter added a factor g which has the value of 1 in open space and less than 1 in a packed bed of particles or fibers. The final equation becomes H ¼ 2gDm =u when we use the symbol H as before. How important is diffusion in band broadening? Clearly it depends on the magnitude of Dm. In gases the value of D is relatively large compared to liquids with nitrogen in air being 1 cm2 s1. The value for a small molecule like benzene in ethanol liquid is ~104 cm2 s1. Longitudinal molecular diffusion is clearly more of concern in gaseous mobile phase systems but as will be seen later, diffusion is important in condensed or liquid systems as well. 12.3.5
Resistance to mass transfer (RTMT)
A molecule carried along in the mobile phase must contact the stationary phase if the system is an equilibrium separation method. That means that a molecule moving between particles must sense the chemical potential in the direction of the stationary phase and move toward it. 404
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The diving force is the Gibbs free energy change and the mechanism is diffusion. This is, by analogy, akin to dropping a bowling ball from a moving airplane in that the approach of the molecule will take a parabolic path to the stationary phase moving forward because of the flow of the mobile phase and ‘‘down’’ by diffusion rather than gravity. The Van Deemter expression for this broadening in terms of variance due to mobile phase resistance to mass transfer to the stationary phase is s2RM ¼ f ðkÞd2p =Dm u where f(k) is a function related to that fraction of molecules in the moving phase because of the equilibrium. A solute entering onto or into a stationary phase must re-enter the moving phase when the molecules in the moving phase move on. For molecules that are adsorbed on a surface, this leaving can be a rapid process with little variation in how fast any given molecule returns. When the stationary phase is a condensed liquid film, there can be significant differences depending on how near to the surface interface the molecule is at the time when reentry into the mobile phase is required to maintain equilibrium. Here the orders of magnitude show a smaller value of Ds—the diffusion coefficient of the molecule in the liquid stationary phase—compared to the gas phase is again important and an expression analogous to the moving phase RTMT can be written. An expression can be written which lumps together RTMT in both mobile and stationary phase and sums the variance contributions which can be written as H ¼ A þ B=u þ Cu Here, A is the random path, B the longitudinal molecular diffusion and C the RTMT contributions with the velocity of the mobile phase u shown separately. H is referred to as the ‘‘Height Equivalent to a Theoretical Plate’’ and is terminology borrowed from distillation. While the distillation HETP is not truly applicable, the terminology has persisted. It can be shown that the H in this expression is the equivalent to variance/unit length. This is the expression introduced by Van Deemter and co-workers in 1956 in a discussion of band broadening. 12.3.6
Practical consequences
The Van Deemter equation contains a linear term A influenced only by particle size and geometry of packing of the particles. It is a constant and sets a lower limit for variance production. The diffusion term B is 405
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inversely proportional to velocity and will become asymptotic to the value of A at large u suggesting that high velocities are best. It is unfortunate for practical analysts and analysis time that the C term is directly dependent on velocity. This means attempts to speed up analysis by increasing flow velocity will become a trade off in band broadening vs. analysis time. Smaller particle size will reduce the A term and improve the mobile phase contribution to C but forcing a gas or liquid through a bed of tinier particles will lead to a need for higher pressures at the inlet, compression of a gas and wear on pumps for liquid delivery. Better, more uniform packing will help as would particles that are spherical and of quite narrow particle size distribution. Modern packings are spherical and of narrow particle size distribution leading to more permeable and lower variance contribution. Fibrous beds such as paper or the fiber-webs of polymer gels are difficult to produce in high uniformity of fiber-gap and fiber-diameter at present.
12.4
A CHEMIST’S VIEW OF EQUILIBRIUM CONSTANTS
A plug of molecules of the same kind moving under the flow of the mobile phase and in equilibrium with a stationary phase will require a volume of mobile phase Vr to be washed from one end of a separator to the other. A packed tube (or column) full of particles has an open total volume occupied by mobile phase. If there is no equilibrium—if the equilibrium constant is K ¼ 0—it is void volume that is required for an infinitely narrow plug through the system. It is more common to define that volume V0 as the volume here half the plug has eluted. If K is finite and non-zero, the net effect is for the plug to move more slowly and for it to take a volume larger than V0 to achieve elution. The retention of the band or ‘‘peak’’ beyond what V0 predicts depends on the magnitude of the equilibrium constant and logically on the volume Vs or area As of the stationary phase. The equation of importance is Vr ¼ V0+KVs and the net retention Vr0 ¼ KVs. Two main factors influence the value of the equilibrium constant and these are the chemical nature of the mobile and stationary phases. Chemistry is molecules and while true thermodynamics knows no molecules or forces between molecules, chemists think in terms of molecular properties. Among those properties, there is a consideration of the kinds of forces that exist between molecules. Granted that thermodynamics are energy not force considerations but it is useful to understand the main forces involved in the interaction between molecules. Put another way, 406
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it is the chemist who asks why the boiling point of water (a small molecule) is 1001C but if we methylate water— H2O2H ) CH3 2O2H ) CH3 2O2CH3 —the boiling point changes to 301C on the first methylation and even more so in the dimethyl case. One way to think about that observation is in terms of forces between molecules. The concept of forces between molecules was an area of intense interest in the early third of the 20th century. The focus at that time was not on explaining chemical separation concepts. The focus, rather, was on explaining why liquids exist, why solids exist above absolute zero, why members of homologous series have incremental changes in their boiling points depending on—e.g.—carbon number. Even more challenging was the question of why helium can be liquefied! Among the scientists who pondered this question are Nobel laureates—e.g. Debye—and near Nobel laureates—e.g.—London. We are all familiar with forces between bodies on a macroscopic scale. If one sits at their desk on earth pondering the view and inadvertently lets go of their lunch sandwich it falls until it lands on your manuscript, your lap or the floor—right? Better yet, a cannonball and a softball fall at the same rate. It is the definition of ‘‘gravity’’. Long after the Tower of Pisa experiments, we discovered why the moon is in seemingly stable orbit around the earth and the earth around the sun is related to forces between bodies. We are also familiar with electrostatic forces and their effects. Dust from the air in the room is attracted to the screen of everyone’s TV because of the electrostatic charge it develops. It would seem reasonable that attraction occurs on a smaller scale and even on a molecular scale that does not involve energies on the order of true chemical bonds. 12.4.1
Categories of forces
It is possible to make a list of forces between molecules in the order of weakest to strongest:
Dispersion forces. These are weak attractions caused by instantaneous fluctuation of the electron distribution in molecules and even atoms. They were first posed by Fritz London whose focus was on helium liquefaction. Such London forces fall off with the sixth power of the distance of separation. Any individual fluctuation creates a +/ local charge and that instantaneous dipole can interact with other such instantaneous dipoles nearby. The important 407
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408
insight is that while weak individually the sum of all these interactions occurring at any moment is a significant component of the internal energy of the system. In a homologous series like pentane, hexane, heptane, octane each additional—CH2- group adds one more possibility of a local fluctuation, which on average increases the overall chance of attraction between molecules. The result should be an increment in the boiling point of these liquids that is monotonic and this is what is observed. (The reader is invited to make a plot of boiling point vs. carbon number for the alkanes beginning with propane and ending with dodecane.) Dipole– dipole forces. If there is some substituent on a molecule, which is sufficiently electronegative to cause a permanent polarization of a chemical bond, this creates a permanent dipole. The results are referred to as Debye forces. There are untold examples but a simple one to consider is a comparison of the properties of benzene and chlorobenzene. Benzene is non-polar in the sense we use the word here. Chlorobenzene is polar. The reason is that chlorine is more electronegative than carbon and the electron distribution in chlorobenzene is determined by the electron-withdrawing effect of the chlorine substituent. The result is a permanent molecular dipole, which can react with other permanent dipoles and cause larger changes in the observed melting and boiling point that the difference in molecular weight between benzene and chlorobenzene would predict. What about dichlorobenzenes? Substitution of another hydrogen by chlorine creates another local dipole and a more polar molecule. Certainly 1,2 and 1,3-dichlorobenzene are more polar but 1,4-dichlorobenzene poses a quandary. It is what chemist call a polar molecule but the opposing chlorines result in a net molecular dipole of zero. Right? Perhaps it is best for chemists to think in terms of ‘‘bond-dipoles’’ rather than molecular dipoles and in many cases this is the case in chemical separation discussions. (Reader should look up the properties of the dichlorobenzenes, such as melting and boiling point and liquid density.) Dipole-induced dipole forces. A molecule with a strong molecular or bond dipole can induce a dipole in a molecule nearby that is polarizable. These Keesom forces have the same inverse 6th power dependence with distance. An example could be the interaction of chlorobenzene with naphthalene. Hydrogen bond forces. The hydrogen bond is a special case of dipole–dipole interaction but is often spoken of separately because most hydrogen bonds are more energetic than other dipole interactions.
Theory of separation methods
Hydrogen bonds involve a hydrogen donor and a hydrogen bond receptor—e.g.—ethanol and acetone. 12.4.2
Use of forces in manipulation of separations
There are many examples of the deliberate use of chemical intuition and understanding of forces between molecules in getting selective interactions that enhance separation. An entire book could be written just cataloging examples. It is more appropriate here to give a few stationary phase examples. Aromatic molecules with no polar substituent include benzene derivatives or other, more polyaromatic molecules, such as naphthalene, phenanthrene, and anthracene. These are polarizable. Paraffins are not polarizable by comparison. In gas–liquid systems, aromatic molecules will show stronger interactions with polar stationary phases that paraffins of comparable boiling point and, thus, polar stationary phases can aid in improving separation of substituted aromatics. A more familiar example of deliberate design of a stationary phase is familiar to organic chemists. It is called ‘‘argentation’’ and is done by treating a packing material with silver nitrate and using a mobile phase in which the silver salt is not soluble. Silver forms adducts with olefins and the adducts formed with cis geometric isomers is more stable than the corresponding trans isomer. The result is that cis-olefins show larger equilibrium constants for stationary phases that contain silver ions and are thus retained longer. 12.4.3
Properties of equilibrium constants
It is a great deal of work to actually determine a true equilibrium constant and most chemical separation methods speak in terms of values which are proportional to the actual equilibrium constant. At constant flow, the time that a given type of molecule is retained is related to the time for the void volume to pass after the sample is placed in a column or on a plate with the addition of the time for the net retention volume. If the flow remains constant, the temperature of the separation remains constant and no stationary phase is gained or lost, one can attempt qualitative identification using retention times. It is more reasonable to calculate the ratio of net retention volume to the void volume and call the result partition factor or capacity factor, k0 . k0 ¼ ðtr t0 Þ=0 409
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Since k0 is proportional to K, mathematical treatments appropriate for equilibrium constants apply to the capacity factor k0 . Plots of log k0 vs. 1/T where T is in degrees Kelvin for any given molecule should yield a straight line over reasonable ranges of temperature. Log k'
I/T [degrees Kelvin]
Since members of a homologous series have incremental boiling point differences and if the amount of any homolog in the moving gas phase is related to vapor pressure at the temperature of the experiment, plots of log k0 vs. carbon number should also be a straight line. (The enthalpy of vaporization increases monotonically with carbon number.) This in fact is observed in gas–liquid equilibrium separation systems. It is the basis of ‘‘retention index’’ systems pioneered by Kovats for qualitative identification. 12.4.4
Choosing mobile and stationary phases
Mobile phases can be gases or liquids and stationary phases can be solids or liquids. If as suggested K and thus k0 is proportional to vapor pressure in gas–liquid and gas–solid systems then manipulation of the apparent volatility is dependent on non-ideal behavior in the gas mobile phase and in the liquid phase in gas–liquid systems (the most common). Put in other terms k0 is proportional to 1/gp where g is the activity coefficient of the sample molecules in the liquid and p the vapor pressure at the temperature of the experiment. Except at very high density, the contribution of the gas phase to non-ideality is small and fugacity coefficients are near 1. Changing mobile phase from helium, to hydrogen, to nitrogen to carbon dioxide changes k0 by a few percent. The power is in the choice of stationary phase and, in part, is why there are 410
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so many stationary phases for gas–liquid separations. (Truthfully, many of the thousands of reported stationary phases are of equal value.) In liquid–solid and liquid–liquid systems, manipulation of mobile phase properties is the dominant approach. Mixtures of poor and good solvents for the types of molecules being separated are prepared and retention observed. There are fewer stationary phases for liquid mobile phase systems for this reason and at least one other. It is desirable that the equilibrium constant for a ‘‘solute’’ be not zero or very large lest there be no net retention or near infinite retention. The ‘‘catch’’ comes in the fact that liquids, which are relatively good solvents for a given type of molecule are also solvents for each other. This means the risk involved is by washing off the stationary phase with the mobile phase. Yet liquid–liquid methods offer much promise for relatively nonvolatile but soluble molecules and their separation of one from the other. The discovery of liquid–liquid chromatography earned Martin and Synge the Nobel Prize when they applied it to amino acids with water mobile phases and organic liquid stationary phases. The solution to stationary phase solubility is solved for most cases by covalently attaching the molecules to the solid chromatography packing. The chemistry used for the vast majority of such phases in current use is an adaptation of the sylilation chemistry used to modify window glass surfaces. The surface of silica gel contains Si-OH functions and these can be either esterified by cooking the silica with alkanols or by reaction with chloro- or alkoxysilanes. The most common attached molecules are propyl, octyl, and octadecyl hydrocarbons. This can either be thought of as converting the hydrophilic silica surface to a carbonlike one or as a very thin film of hydrocarbon. These materials are stable to moderate pH water-based mobile phases. They are the basis of ‘‘reversed-phase’’ method where the moving phase is water or watermethanol, water acetonitrile, and water tetrahydrofuran mixtures (polar) and a hydrocarbon-like stationary phase.
Further Reading R.P.W. Scott, Liquid Chromatography Column Theory, Wiley, UK, ISBN 0 471 93305 8, 1992. A.J.P. Martin and R.L.M. Synge, Biochem. J., 35 (1941) 358. Roger M. Smith, Retention and Selectivity in Chromatography, Elsevier, Amsterdam, ISBN 0-444-81539-2, 1995. 411
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REVIEW QUESTIONS 1.
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If gas–liquid and gas–solid separations are dependent on the ‘‘saturation vapor pressure’’ of the chemical component undergoing equilibration: (a) What is the expected effect when the temperature of the system is raised? (b) If the system is a gas-liquid system sketch what a plot of log Vr0 vs. 1/T would look like including when the T is below the freezing point of the stationary phase. (c) Why might it be better to sample the vapor phase above a solution as a sample to determine trace materials in the solution? Does solubility in the mobile phase influence net retention? Polyethylene glycols are more soluble in cold solvents than warm. What is the noted effect if the temperature of the liquid chromatography separation of poly ethers is raised? The addition of a detergent such as sodium sulfate can cause a protein to completely unfold. What would be the effect of adding sodium dodecylsulfate to the mobile phase of a protein size separation? Explain why? Separation scientists speak of a ‘‘general elution problem’’ when asked to develop a universal separation method using chromatography. What is the problem? (Hint: you may need to consult works by Snyder or Heftmann in the library.)
Theory of separation methods C.H. Lochmu¨ller
12.1
INTRODUCTION
In the strictest meaning of the word, there is no theoretical basis for separation methods but rather models. Models are, in most cases, a mixture of empirical and chemical kinetic/chemical thermodynamics. There are many examples of attempts to relate known physical and chemical properties to observe and develop a ‘‘predictive theory’’. The goal is both an improvement in understanding of the underlying phenomena and an efficient approach to method development. This latter goal is of greater interest to the vast majority of those who make use of separation methods, as their interest is in either obtaining relatively pure material or in isolating components of a chemical mixture in time and space for quantitative measurement. If a separation method can also provide complimentary information leading to qualitative identification, so much the better. This chapter seeks to give the ‘‘user’’ of chemical separation methods the beginnings of a basis for understanding the methods described in this book and the ability to recognize ‘‘normal behavior’’ and to distinguish ‘‘anomalous behavior’’. Justice to all the important theory would require several volumes of substantial size and especially if historical justice were to be given to the development of current models. At times the chapter’s content will seem more conversational than hard scientific and the choice of style in any given instance, is deliberate. Stories are part of the history of separation methods, after all. 12.2
THE GREAT DIVIDE
Julius Caesar tells us ‘‘All Gaul is divided into three parts’’. Separation methods can be divided into two categories: equilibrium and Comprehensive Analytical Chemistry 47 S. Ahuja and N. Jespersen (Eds) Volume 47 ISSN: 0166-526X DOI: 10.1016/S0166-526X(06)47012-4 r 2006 Elsevier B.V. All rights reserved.
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non-equilibrium. The equilibria spoken of in this case are ‘‘chemical’’ rather than physical in the vast majority of cases. They are the kind of equilibria spoken of as ‘‘ionization in solution’’ or between two phases, such as gas and liquid in introductory chemistry courses. Finding the point between a knife tip and the end of its’ handle that when placed on a finger leaves the knife neither tipping toward one end or the other, is the location of the ‘‘center of gravity’’. Supported at that point the knife is said to be ‘‘in balance’’ and that balance is a physical equilibrium. It is based on simple mechanics and the concept of mass and length. Chemical equilibria arise from not just mass but the underlying properties of molecules and their interaction with each other. A sealed half-full bottle of gin contains a liquid which is 40% by volume ethyl alcohol and 60% water. If we call the liquid the condensed phase, the vapor phase is the gaseous region above the condensed phase. The composition of the vapor phase is found to be different than the condensed phase and richer in alcohol than water. At constant temperature, the mole fraction of alcohol in the vapor phase is higher than water. Why? Ethanol is more volatile than water. True and it is also true that the composition of ethanol in the vapor phase will reach a constant value in time. Separation theory attempts to understand why such a chemical equilibrium occurs between the two phases and why, if methanol is substituted for ethanol, the alcohol content in the vapor phase is higher. The fact that the equilibrium is established over time is a kinetic consideration, while the value of the equilibrium constant and its’ temperature dependence is a question of chemical thermodynamics. 12.2.1
Physical separation methods
Physical separation methods can be based on equilibrium considerations, but the majority are not. Ordinary filtration is an example of a non-equilibrium, physical method and so is ordinary centrifugation— e.g.—the separation of a precipitate from the suspending liquid using an artificial gravity field. There are separation methods, which are called ‘‘filtration’’ which are not such as gel filtration. Ultracentrifugation in a salt gradient is a physical equilibrium method. Separation of desirable components of a mixture is not a new desire on the part of humanity. Removal of water-borne particles including sand, silt, and bacteria is desirable for both sensory appeal and health. The use of sand as a filtration medium for such filtration is very old. The use of a pan to separate gold dust from silica sand by gold miners is a use of the large difference in density to wash the lighter silica away 398
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from the much denser gold, which resembles black sand. Not filtration but certainly a non-equilibrium, physical method. Molecules which exhibit optical activity are molecules which have a ‘‘handedness’’ in their structure. They are ‘‘chiral’’. Chemists often have reasons to obtain chemical pure aliquots of particular molecules. Since the chirality of molecules can influence biological effect in pharmaceuticals, the chiral purity of a drug substance can pose a challenge both in terms of obtaining the molecules and in assaying the chiral purity by instrumental methods. While diastereomers can have different physical properties including solubility, enantiomers have the same physical properties and the same chemical composition. How then to separate optically active molecules? The first example of the deliberate separation of optically active molecules is appropriate as an example of physical separation in the clearest sense of the term. The molecules are referred to as optically active because polarized light interacts differently with right- and lefthanded molecules. In the case of simple diastereomers the RR and SS forms are enantiomers while the RS and SR forms are not. The separation of the latter and former was first done under a microscope using crossed polarizers and the crystals which were seen were separated from those that caused little or no rotation of plane-polarized light by hand using tweezers. A truly physical separation of chemical species using a physical property of chemical origin! One could go on with examples such as the use of a shirt rather than sand reduce the silt content of drinking water or the use of a net to separate fish from their native waters. Rather than that perhaps we should rely on the definition of a chemical equilibrium and its’ presence or absencey. Chemical equilibria are dynamic with only the illusion of static state. Acetic acid dissociates in water to acetate-ion and hydrated hydrogen ion. At any instant, however, there is an acid molecule formed by recombination of acid anion and a proton cation while another acid molecule dissociates. The equilibrium constant is based on a dynamic process. Ordinary filtration is not an equilibrium process nor is it the case of crystals plucked from under a microscope into a waiting vial. 12.2.2
Chemical separation methods
Most, but not all, chemical separation methods are based on equilibrium in the sense of chemical equilibrium. Clearly, solubility is a chemical question but formation of a precipitate and filtration is a physical separation, which happens to use a favorable Ksp equilibrium 399
C.H. Lochmu ¨ ller TABLE 1 Possible combinations and typical separation methods using them in a notall-inclusive list Interface type
Separation method(s)
Gas–gas Gas–liquid Gas–solid Liquid–liquid Liquid–solid Solid–solid
Possible but no common examples Gas chromatography; distillation Gas chromatography; gas masks Extraction; liquid chromatography Liquid chromatography; solvent drying Possible but no common examples
value to begin the process. Distillation and extraction are indeed chemical separation methods based on equilibrium and differences in equilibrium constants. Chemical separations are often either a question of equilibrium established in two immiscible phases across the contact between the two phases. In the case of true distillation, the equilibrium is established in the reflux process where the condensed material returning to the pot is in contact with the vapor rising from the pot. It is a gas– liquid interface. In an extraction, the equilibrium is established by motion of the solute molecules across the interface between the immiscible layers. It is a liquid– liquid interface. If one adds a finely divided solid to a liquid phase and molecules are then distributed in equilibrium between the solid surface and the liquid, it is a liquid– solid interface (Table 1).
12.3
CHEMICAL EQUILIBRIA AND THE PROPERTIES OF THE EQUILIBRIUM CONSTANT
Chemistry deals with molecules not atoms. True thermodynamics knows no molecules with much less properties of molecules derived from chemical effects. Its’ origin is in such concerns as heat flow and the heat equivalent of mechanical work. Most of us have heard in physical chemistry about how it was the drilling of cannon barrels that created the connection between work and heat energy. One can take entire semester course on thermodynamics in physics and in engineering and never deal with the solution thermodynamics, which often dominates chemistry courses. To the extent that thermodynamics has been used in developing a theory for separation methods, it is almost entirely chemical thermodynamics. 400
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In talking about thermodynamics and the properties of chemical equilibrium constants it is very difficult for chemists to avoid attempts to include the influence of forces between molecules on the magnitude of the equilibrium constant and differences between observed equilibrium constants. For the purpose of this chapter, it is convenient to talk first about the equilibrium constant and the macroscopic properties of matter which affect it first. Next, the reader will be introduced to the concept of forces between molecules, their relative magnitude and influence is separations.
12.3.1
Thermodynamics and kinetics in separations
If all that is achieved, in an attempted separation, is the transfer of the components of the mixture (atoms and/or molecules) from one place to another, no separation is achieved. The result is just mass transport. An analogy would be carrying a bag of mixed groceries from the store to a car. The tomatoes, onions, garlic and meat are still in the bag. What is needed is a means to have the components move to the end of an experiment but at different apparent rates of speed. For most equilibrium separation methods, the answer lies in controlling the thermodynamics of the system. Molecules or atoms are made to move in one of the two immiscible phases while undergoing transfer to the other, stationary phase and quickly back to the mobile phase. If the net effects in the molecules or atoms move at different rates depending on their chemical character, some will move faster and others more slowly. The goal would be to have molecules of type A moving faster than type B and both faster than type C and so on. Ideally, it would be best if they remained in tight ‘‘packets’’ as the forward motion of the mobile phase caused them to migrate in space and the difference in net rate moved the packets further and further apart as time passed. The simplest rules of thermodynamics suggest that energy must be expended to do work—‘‘You cannot get something for nothing,’’ and that even if work is done some energy is forever lost to useful work— ‘‘You cannot even get what you paid for’’. And that this entropy effect is such that the entropy of the universe is forever driving toward a maximum—‘‘Nature spontaneously falls into a mess!’’ Humor aside, the consequence is that any narrow packet as described above will spread over space in an attempt to make the local and universal mole fraction of A, B or C y the same everywhere. 401
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Put in ordinary terms, the more successful we are in causing a separation, the more propensities there are for a re-mixing of the components. There are many ways this can occur but there are a fewer number of important routes to mixing. It seems reasonable that we examine these before we consider all the possible ways in which thermodynamics can be controlled in general terms. In almost all equilibrium separation systems, the separation can occur either in a packed bed of particles or fibers or in an open channel or tube. The stationary phase is either coated on the walls of the channel or on the particles/fibers of the packed bed. If there were no mixing mechanisms an infinitely narrow packet containing the components would become a series of infinitely narrow packets of pure components moving at different velocities toward the end of the packed bed or tube.
12.3.2
Mechanisms for band broadening
How to best describe this broadening we expect to occur? One way is by analogy to random error in measurements. We know or assume there is a truly correct answer to any measurement of quantity present and attempt to determine that number. In real measurements there are real, if random, sources of error. It is convenient to talk about standard deviation of the measurement but actually the error in measurement accumulates as the sum of the square of each error process or variance producing mechanism or total variance ¼ sum of the individual variances. If we ignore effects outside the actual separation process (e.g. injection/spot size, connecting tubing, detector volume), this sum can be factored into three main influences: 1. 2. 3.
The random path of molecules taken in a packed bed (called Random Path Term). Ordinary diffusion along the length of the path taken by a packet of molecules (called Longitudinal Molecular Diffusion). Departure from true equilibrium of the molecules moving over the stationary phase and moving in and out of the stationary phase (called Resistance to Mass Transfer).
If we give variance then s2 ½total ¼ s2 ½RPT þ s2 ½LMD þ s2 ½RTMT
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Multipath processes
If flow occurred in an open channel with no particles or fibers and if there were no other mixing mechanisms, all particles would transit the same distance from beginning to end. In a packed bed, each time a molecule or atom encounters a particle or fiber it must go around it to continue on. It is analogous to encounter a tree in a field—one either walks around it to the right or the left and that is equivalent to flipping a coin. Some molecules will encounter more particles than others as illustrated in the following scheme where each encounter causes a chance in direction and the path of a hypothetical molecule is traced by a line (Scheme 1). Since the paths are random and the number of collisions large, the effect is to make the paths random and distributed as a Gaussian distribution. If you are familiar with the Random Walk model, this can be considered as analogous Random Path Model. The actual situation is a bit more complicated but for practical purposes the intuitive insights that the degree of broadening will be influenced by particle size dp (and, thus, distance between particles or fibers) and the geometry of the packing. The variance produced per unit length l of the bed traversed (s2/l) can be given the symbol H: H ¼ 2ldp
Scheme 1. Traces of the path of molecules in a packed bed.
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The symbol l in this case was first introduced by Van Deemter to account for packing inhomogeneity. 12.3.4
Broadening by diffusion
If a band of molecules or atoms are placed in a container such as a tube at the center of its length, the molecules will diffuse in the direction of lower concentration. Clearly this occurs in all directions but in a tube the walls are a limit and so our concern is along the axis of movement caused by the mobile phase and this is referred to as longitudinal molecular diffusion. The Einstein equation for diffusion along a single axis states that the process is Gaussian and the variance is time dependent: s2 ¼ 2Dt where D is a characteristic diffusion coefficient. The time a sample spends in the moving phase on its transit from the beginning to the end of the bed is proportional not to the flow rate but the flow velocity of the mobile phase, which we can call u. If the time to transit a distance l is l/u then for a unit distance t ¼ 1/u. It can be seen then that the variance contribution per unit length in an open tube is s2/l ¼ 2Dm/u where Dm is the diffusion coefficient for the molecules of interest in the moving or mobile phase. Van Deemter added a factor g which has the value of 1 in open space and less than 1 in a packed bed of particles or fibers. The final equation becomes H ¼ 2gDm =u when we use the symbol H as before. How important is diffusion in band broadening? Clearly it depends on the magnitude of Dm. In gases the value of D is relatively large compared to liquids with nitrogen in air being 1 cm2 s1. The value for a small molecule like benzene in ethanol liquid is ~104 cm2 s1. Longitudinal molecular diffusion is clearly more of concern in gaseous mobile phase systems but as will be seen later, diffusion is important in condensed or liquid systems as well. 12.3.5
Resistance to mass transfer (RTMT)
A molecule carried along in the mobile phase must contact the stationary phase if the system is an equilibrium separation method. That means that a molecule moving between particles must sense the chemical potential in the direction of the stationary phase and move toward it. 404
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The diving force is the Gibbs free energy change and the mechanism is diffusion. This is, by analogy, akin to dropping a bowling ball from a moving airplane in that the approach of the molecule will take a parabolic path to the stationary phase moving forward because of the flow of the mobile phase and ‘‘down’’ by diffusion rather than gravity. The Van Deemter expression for this broadening in terms of variance due to mobile phase resistance to mass transfer to the stationary phase is s2RM ¼ f ðkÞd2p =Dm u where f(k) is a function related to that fraction of molecules in the moving phase because of the equilibrium. A solute entering onto or into a stationary phase must re-enter the moving phase when the molecules in the moving phase move on. For molecules that are adsorbed on a surface, this leaving can be a rapid process with little variation in how fast any given molecule returns. When the stationary phase is a condensed liquid film, there can be significant differences depending on how near to the surface interface the molecule is at the time when reentry into the mobile phase is required to maintain equilibrium. Here the orders of magnitude show a smaller value of Ds—the diffusion coefficient of the molecule in the liquid stationary phase—compared to the gas phase is again important and an expression analogous to the moving phase RTMT can be written. An expression can be written which lumps together RTMT in both mobile and stationary phase and sums the variance contributions which can be written as H ¼ A þ B=u þ Cu Here, A is the random path, B the longitudinal molecular diffusion and C the RTMT contributions with the velocity of the mobile phase u shown separately. H is referred to as the ‘‘Height Equivalent to a Theoretical Plate’’ and is terminology borrowed from distillation. While the distillation HETP is not truly applicable, the terminology has persisted. It can be shown that the H in this expression is the equivalent to variance/unit length. This is the expression introduced by Van Deemter and co-workers in 1956 in a discussion of band broadening. 12.3.6
Practical consequences
The Van Deemter equation contains a linear term A influenced only by particle size and geometry of packing of the particles. It is a constant and sets a lower limit for variance production. The diffusion term B is 405
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inversely proportional to velocity and will become asymptotic to the value of A at large u suggesting that high velocities are best. It is unfortunate for practical analysts and analysis time that the C term is directly dependent on velocity. This means attempts to speed up analysis by increasing flow velocity will become a trade off in band broadening vs. analysis time. Smaller particle size will reduce the A term and improve the mobile phase contribution to C but forcing a gas or liquid through a bed of tinier particles will lead to a need for higher pressures at the inlet, compression of a gas and wear on pumps for liquid delivery. Better, more uniform packing will help as would particles that are spherical and of quite narrow particle size distribution. Modern packings are spherical and of narrow particle size distribution leading to more permeable and lower variance contribution. Fibrous beds such as paper or the fiber-webs of polymer gels are difficult to produce in high uniformity of fiber-gap and fiber-diameter at present.
12.4
A CHEMIST’S VIEW OF EQUILIBRIUM CONSTANTS
A plug of molecules of the same kind moving under the flow of the mobile phase and in equilibrium with a stationary phase will require a volume of mobile phase Vr to be washed from one end of a separator to the other. A packed tube (or column) full of particles has an open total volume occupied by mobile phase. If there is no equilibrium—if the equilibrium constant is K ¼ 0—it is void volume that is required for an infinitely narrow plug through the system. It is more common to define that volume V0 as the volume here half the plug has eluted. If K is finite and non-zero, the net effect is for the plug to move more slowly and for it to take a volume larger than V0 to achieve elution. The retention of the band or ‘‘peak’’ beyond what V0 predicts depends on the magnitude of the equilibrium constant and logically on the volume Vs or area As of the stationary phase. The equation of importance is Vr ¼ V0+KVs and the net retention Vr0 ¼ KVs. Two main factors influence the value of the equilibrium constant and these are the chemical nature of the mobile and stationary phases. Chemistry is molecules and while true thermodynamics knows no molecules or forces between molecules, chemists think in terms of molecular properties. Among those properties, there is a consideration of the kinds of forces that exist between molecules. Granted that thermodynamics are energy not force considerations but it is useful to understand the main forces involved in the interaction between molecules. Put another way, 406
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it is the chemist who asks why the boiling point of water (a small molecule) is 1001C but if we methylate water— H2O2H ) CH3 2O2H ) CH3 2O2CH3 —the boiling point changes to 301C on the first methylation and even more so in the dimethyl case. One way to think about that observation is in terms of forces between molecules. The concept of forces between molecules was an area of intense interest in the early third of the 20th century. The focus at that time was not on explaining chemical separation concepts. The focus, rather, was on explaining why liquids exist, why solids exist above absolute zero, why members of homologous series have incremental changes in their boiling points depending on—e.g.—carbon number. Even more challenging was the question of why helium can be liquefied! Among the scientists who pondered this question are Nobel laureates—e.g. Debye—and near Nobel laureates—e.g.—London. We are all familiar with forces between bodies on a macroscopic scale. If one sits at their desk on earth pondering the view and inadvertently lets go of their lunch sandwich it falls until it lands on your manuscript, your lap or the floor—right? Better yet, a cannonball and a softball fall at the same rate. It is the definition of ‘‘gravity’’. Long after the Tower of Pisa experiments, we discovered why the moon is in seemingly stable orbit around the earth and the earth around the sun is related to forces between bodies. We are also familiar with electrostatic forces and their effects. Dust from the air in the room is attracted to the screen of everyone’s TV because of the electrostatic charge it develops. It would seem reasonable that attraction occurs on a smaller scale and even on a molecular scale that does not involve energies on the order of true chemical bonds. 12.4.1
Categories of forces
It is possible to make a list of forces between molecules in the order of weakest to strongest:
Dispersion forces. These are weak attractions caused by instantaneous fluctuation of the electron distribution in molecules and even atoms. They were first posed by Fritz London whose focus was on helium liquefaction. Such London forces fall off with the sixth power of the distance of separation. Any individual fluctuation creates a +/ local charge and that instantaneous dipole can interact with other such instantaneous dipoles nearby. The important 407
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insight is that while weak individually the sum of all these interactions occurring at any moment is a significant component of the internal energy of the system. In a homologous series like pentane, hexane, heptane, octane each additional—CH2- group adds one more possibility of a local fluctuation, which on average increases the overall chance of attraction between molecules. The result should be an increment in the boiling point of these liquids that is monotonic and this is what is observed. (The reader is invited to make a plot of boiling point vs. carbon number for the alkanes beginning with propane and ending with dodecane.) Dipole– dipole forces. If there is some substituent on a molecule, which is sufficiently electronegative to cause a permanent polarization of a chemical bond, this creates a permanent dipole. The results are referred to as Debye forces. There are untold examples but a simple one to consider is a comparison of the properties of benzene and chlorobenzene. Benzene is non-polar in the sense we use the word here. Chlorobenzene is polar. The reason is that chlorine is more electronegative than carbon and the electron distribution in chlorobenzene is determined by the electron-withdrawing effect of the chlorine substituent. The result is a permanent molecular dipole, which can react with other permanent dipoles and cause larger changes in the observed melting and boiling point that the difference in molecular weight between benzene and chlorobenzene would predict. What about dichlorobenzenes? Substitution of another hydrogen by chlorine creates another local dipole and a more polar molecule. Certainly 1,2 and 1,3-dichlorobenzene are more polar but 1,4-dichlorobenzene poses a quandary. It is what chemist call a polar molecule but the opposing chlorines result in a net molecular dipole of zero. Right? Perhaps it is best for chemists to think in terms of ‘‘bond-dipoles’’ rather than molecular dipoles and in many cases this is the case in chemical separation discussions. (Reader should look up the properties of the dichlorobenzenes, such as melting and boiling point and liquid density.) Dipole-induced dipole forces. A molecule with a strong molecular or bond dipole can induce a dipole in a molecule nearby that is polarizable. These Keesom forces have the same inverse 6th power dependence with distance. An example could be the interaction of chlorobenzene with naphthalene. Hydrogen bond forces. The hydrogen bond is a special case of dipole–dipole interaction but is often spoken of separately because most hydrogen bonds are more energetic than other dipole interactions.
Theory of separation methods
Hydrogen bonds involve a hydrogen donor and a hydrogen bond receptor—e.g.—ethanol and acetone. 12.4.2
Use of forces in manipulation of separations
There are many examples of the deliberate use of chemical intuition and understanding of forces between molecules in getting selective interactions that enhance separation. An entire book could be written just cataloging examples. It is more appropriate here to give a few stationary phase examples. Aromatic molecules with no polar substituent include benzene derivatives or other, more polyaromatic molecules, such as naphthalene, phenanthrene, and anthracene. These are polarizable. Paraffins are not polarizable by comparison. In gas–liquid systems, aromatic molecules will show stronger interactions with polar stationary phases that paraffins of comparable boiling point and, thus, polar stationary phases can aid in improving separation of substituted aromatics. A more familiar example of deliberate design of a stationary phase is familiar to organic chemists. It is called ‘‘argentation’’ and is done by treating a packing material with silver nitrate and using a mobile phase in which the silver salt is not soluble. Silver forms adducts with olefins and the adducts formed with cis geometric isomers is more stable than the corresponding trans isomer. The result is that cis-olefins show larger equilibrium constants for stationary phases that contain silver ions and are thus retained longer. 12.4.3
Properties of equilibrium constants
It is a great deal of work to actually determine a true equilibrium constant and most chemical separation methods speak in terms of values which are proportional to the actual equilibrium constant. At constant flow, the time that a given type of molecule is retained is related to the time for the void volume to pass after the sample is placed in a column or on a plate with the addition of the time for the net retention volume. If the flow remains constant, the temperature of the separation remains constant and no stationary phase is gained or lost, one can attempt qualitative identification using retention times. It is more reasonable to calculate the ratio of net retention volume to the void volume and call the result partition factor or capacity factor, k0 . k0 ¼ ðtr t0 Þ=0 409
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Since k0 is proportional to K, mathematical treatments appropriate for equilibrium constants apply to the capacity factor k0 . Plots of log k0 vs. 1/T where T is in degrees Kelvin for any given molecule should yield a straight line over reasonable ranges of temperature. Log k'
I/T [degrees Kelvin]
Since members of a homologous series have incremental boiling point differences and if the amount of any homolog in the moving gas phase is related to vapor pressure at the temperature of the experiment, plots of log k0 vs. carbon number should also be a straight line. (The enthalpy of vaporization increases monotonically with carbon number.) This in fact is observed in gas–liquid equilibrium separation systems. It is the basis of ‘‘retention index’’ systems pioneered by Kovats for qualitative identification. 12.4.4
Choosing mobile and stationary phases
Mobile phases can be gases or liquids and stationary phases can be solids or liquids. If as suggested K and thus k0 is proportional to vapor pressure in gas–liquid and gas–solid systems then manipulation of the apparent volatility is dependent on non-ideal behavior in the gas mobile phase and in the liquid phase in gas–liquid systems (the most common). Put in other terms k0 is proportional to 1/gp where g is the activity coefficient of the sample molecules in the liquid and p the vapor pressure at the temperature of the experiment. Except at very high density, the contribution of the gas phase to non-ideality is small and fugacity coefficients are near 1. Changing mobile phase from helium, to hydrogen, to nitrogen to carbon dioxide changes k0 by a few percent. The power is in the choice of stationary phase and, in part, is why there are 410
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so many stationary phases for gas–liquid separations. (Truthfully, many of the thousands of reported stationary phases are of equal value.) In liquid–solid and liquid–liquid systems, manipulation of mobile phase properties is the dominant approach. Mixtures of poor and good solvents for the types of molecules being separated are prepared and retention observed. There are fewer stationary phases for liquid mobile phase systems for this reason and at least one other. It is desirable that the equilibrium constant for a ‘‘solute’’ be not zero or very large lest there be no net retention or near infinite retention. The ‘‘catch’’ comes in the fact that liquids, which are relatively good solvents for a given type of molecule are also solvents for each other. This means the risk involved is by washing off the stationary phase with the mobile phase. Yet liquid–liquid methods offer much promise for relatively nonvolatile but soluble molecules and their separation of one from the other. The discovery of liquid–liquid chromatography earned Martin and Synge the Nobel Prize when they applied it to amino acids with water mobile phases and organic liquid stationary phases. The solution to stationary phase solubility is solved for most cases by covalently attaching the molecules to the solid chromatography packing. The chemistry used for the vast majority of such phases in current use is an adaptation of the sylilation chemistry used to modify window glass surfaces. The surface of silica gel contains Si-OH functions and these can be either esterified by cooking the silica with alkanols or by reaction with chloro- or alkoxysilanes. The most common attached molecules are propyl, octyl, and octadecyl hydrocarbons. This can either be thought of as converting the hydrophilic silica surface to a carbonlike one or as a very thin film of hydrocarbon. These materials are stable to moderate pH water-based mobile phases. They are the basis of ‘‘reversed-phase’’ method where the moving phase is water or watermethanol, water acetonitrile, and water tetrahydrofuran mixtures (polar) and a hydrocarbon-like stationary phase.
Further Reading R.P.W. Scott, Liquid Chromatography Column Theory, Wiley, UK, ISBN 0 471 93305 8, 1992. A.J.P. Martin and R.L.M. Synge, Biochem. J., 35 (1941) 358. Roger M. Smith, Retention and Selectivity in Chromatography, Elsevier, Amsterdam, ISBN 0-444-81539-2, 1995. 411
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REVIEW QUESTIONS 1.
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If gas–liquid and gas–solid separations are dependent on the ‘‘saturation vapor pressure’’ of the chemical component undergoing equilibration: (a) What is the expected effect when the temperature of the system is raised? (b) If the system is a gas-liquid system sketch what a plot of log Vr0 vs. 1/T would look like including when the T is below the freezing point of the stationary phase. (c) Why might it be better to sample the vapor phase above a solution as a sample to determine trace materials in the solution? Does solubility in the mobile phase influence net retention? Polyethylene glycols are more soluble in cold solvents than warm. What is the noted effect if the temperature of the liquid chromatography separation of poly ethers is raised? The addition of a detergent such as sodium sulfate can cause a protein to completely unfold. What would be the effect of adding sodium dodecylsulfate to the mobile phase of a protein size separation? Explain why? Separation scientists speak of a ‘‘general elution problem’’ when asked to develop a universal separation method using chromatography. What is the problem? (Hint: you may need to consult works by Snyder or Heftmann in the library.)