THE CONSERVATION EQUATION OF CHROMATOGRAPHY BY W. M. SMIT Received 18th July 1949 The conservation equation of chromatography thus far used does not account for the volume occupied by the adsorbed phase. When the equation of conservation of matter is applied to the process of adsorptive percolation and this volume is accounted for a general equation is obtained which may be more correct. This is shown by applying this equation to the equilibrium theory. Several effects occurring in chromato- graphy which cannot be explained by the de Vault equation may now be interpreted a t least qualitatively. Moreover the equation obtained reveals the possibility of a new method of adsorptive percolation. W. M. SMTT 39 In the development of the mathematical description of the process of adsorptive percolation two different methods of approach may be distin- guished.One may start with the assumption of instantaneous equilibrium between solution and adsorbent (equilibrium theory) or one may dismiss the idea of complete equilibrium and treat the problem as a kinetic one (kinetic theory). However in both cases the law of conservation of matter supplies the initial equation which in both cases must be fundamentally the same. The equations obtained thus far give a ready explanation of the phenomena occurring during development. But when applying these equations in explanation of other processes of adsorptive percolation like the continuous introduction method (frontal analysis) or the elution method (displacement method) difficulties are encountered.Moreover the occurrence of so-called adsorption azeotropes,l especially with closely related compounds needs further interpretation. Some experience gained with the adsorptive percolation of hydrocarbons suggested that the failures mentioned might be caused by the use of an approximate conservation equation. It also represents the weight of are different functions. Symbols The following symbols are used in this paper :- a interstitial volume of an adsorption column per unit of length (ml./cm.). b index referring to a sharp boundary. C concentration of solute expressed as units of weight of solute per unit of volume of solution (g./ml.). CA CB concentrations of substance A and substance B respectively. d density (g./ml.).f f l ( C ~ ) adsorption isotherm of component N. solute adsorbed per unit of weight of adsorbent a t the concentration CN (g./g.). &(CN) first derivative of the adsorption isotherm (dfn(CN) /dCN) m weight of adsorbent contained in the column per unit of length (g./cm.). f ( q ) volume occupied by the adsorbed phase (solute + solvent) per unit of q weight of solute adsorbed per unit of length of a column (g./cm.). length if a quantity q of solute is adsorbed (ml./cm.). S solvent. V volume of liquid forced into the column (ml.). v interstitial volume of dry adsorbent present in the column per unit of weight (ml*/g.) * x distance from the top of the column (cm.). 0 I (when combined with C) indices indicating zero and unit concentration respectively .I 2 3 (when combined with f) indices indicating that the functions fl fi and fs tion equation was given by Wilson 1. The Conservation Equations used thus far.-The first conserva- where C is the concentration of the solute x the distance from the top of the column q the weight of solute adsorbed per unit of length and V the volume of liquid poured into the adsorption column. This equation has been corrected by de Vault 3 for the interstitial volume a per unit of length of the adsorption column. De Vault arrives at the equation 1 Hirschler and Amon. Ind. Eng. Chenz 1947 39 1565. 2 Wilson J . Amer. Chem. SOC. 1940 62 1583. 3 De Vault J . Amer. Chem. SOC. 1943 65 532. CONSERVATION EQUATION OF CHROMATOGRAPHY 40 The same equation has been used implicitly or explicitly by other authors whether they adopted the equilibrium or the kinetic standpoint for the equilibrium or the kinetic conditions are introduced subsequently when the relation between C and q is inserted.Eqn. (2) is based on the assumption that the interstitial volume a is constant. This is the point where intro- duction of a correction might be useful. The adsorbed phase occupies a certain fraction of the original interstitial volume thus causing the liquid to flow at a greater linear rate than when the adsorbed phase is not present. As the quantity of material adsorbed is a function of the concentration the free interstitial volume that is the volume available to the jlowing liquid must be a function of the concentration.2. Derivation of the Corrected Conservation Equation.-To intro- duce the correction for the volume of the adsorbed phase the conservation equation will be derived once more for a solution containing a single solute. The following assumptions which do not differ from those usually applied have been made. Within an adsorption band the concentration is a con- tinuous function of x. Within a cross-section of infinitesimal length & (3C/3x)v is considered constant. The same applies to (3q/3V) when an infinitesimal volume of liquid dV is introduced into the column. When an infinitesimal volume dV is introduced into the column an equal volume has to pass through a cross-section at x where x is a point within the adsorption band. As the concentration of the solution entering the cross-section at x and the concentration of the solution leaving the cross- section at (x + dx) differ by (3Cj3X)V h the amount of solute present in this section of the column is increased by (ac/3x)V dx.dv.This amount is distributed between the liquid phase and the adsorbed phase present in the section. The volume of the liquid phase present in the section is equal to (a -f(d)dx wheref(q) means the volume of the adsorbed phase per unit of length within the section considered. Thus the amount of solute present in the liquid phase is equal t o (a - f(4))Cdx. The amount of solute present in the adsorbed phase is equal to qdx where 4. = mfn(C,). Thus the total amount of solute present in the section is equal t o If an infinitesimal volume dV is forced through this section the increase of solute may therefore also be represented by (a - f W + 4 ) h .According to the law of conservation of matter the amount of solute intro- duced by the volume dV should be equal to the increase of the amount of solute present in the section. Thus 'Martin and Synge Biocheun. J. 1941 35 1941 Weiss J . Chew. sot. 1943 '45. 297. Thomas Ann. N.Y. Acad. Sci. 1948 49 161. W. M. SMIT 41 This is the corrected conservation equation of chromatography in its general form for a solution containing a single solute. As to.f(q) it should be borne in mind that it is not correct to put by f l ( C A ) . At equilibrium f ( d = 4 M where d means the density of the adsorbed solute.For besides the solute the solvent is adsorbed too. As t o this point it might be useful to stress that adsorption isotherms of pure gases and so-called isotherms of solutions containing a single solute usually have a different meaning. The adsorption isotherm of a pure gas shows the relation between the concentration !(pressure) of the gas and the weight adsorbed. However the so-called adsorption isotherm of a solute shows the relation between the concentration and the weight of solute adsorbed preferentially. ‘This relation might be better called a preferential adsorption isotherm. ‘To avoid confusion it may be stated here that in this paper the adsorption isotherm of a solute means the relation between the total amount of solute .adsorbed per unit of weight of adsorbent and the concentration of the :solution with which it is in equilibrium.3. Application of Eqn. (3) to the Equilibrium Theory.-Attempts will now be made to derive a fundamental equation of chromatography ,on the basis of the equilibrium theory for a liquid mixture of two substances A and B. Which of the two is considered as solvent does not matter. For the sake of convenience it is assumed that A and B have equal densities and no volume change occurs on adsorption or mixing. If the densities of A and B are represented by d and the concentration is expressed as g./ml. we may put C.4 + CB = d . Suppose the adsorption column considered contains m g. adsorbent per unit of length. The adsorption isotherm of A when diluted by B is represented y = mfi (CAI (for non-equilibrium another relation between q and C should be introduced but this will not be discussed).If f2(C4 represents the adsorption isotherm of B when diluted by A we may put and or f ( 4 ) = {fi(cA) + f d C B > Imld* If ZI is the interstitial volume of the dry adsorbent per unit of weight Substitution in eqn. (3) gives a = mu. (3cA/3x)V + m[(v -fi(cA)/a -f2(cB)/d>(acA/av)z - { (afl(cA)/3v)x + (3f2(CB)/3v)z)CA/d + (af,(cA)/av)%l = 0. On further substitution viz. ( ~ C A / ~ X ) V = - (3V/ax)cA (~CA/~V)X (afi (C A) /a v) = ( J f l ( CA) / ~ C A ) (JCA/J v) = f i c ~ ( ~CA/J V)z (3f,cB/3v)x = (afz(cB)/acB) (acB/av) = - fkcB(aCA/aV)~ the following equation is obtained - (dlr/Jx)~,(aC~/’~)% + mCv - f l ( c ~ ) / d - j 2 ( C ~ ) / d - {fi(c~) - fS(c~) ) C A / ~ fi(CA)I(acA/av)~ = 0 - PviWc + [vd - f l ( c ~ ) - f z ( C ~ ) + CB~~(CA) + C,fh(C~)]m/d = 0.B* 42 CONSERVATION EQUATION OF CHROMATOGRAPHY Thus (bX/bV)C = - _ _ _ ~ m[vd - f i ( c A ) -fi(cB) -k CBf;(CA) + c A f L ( c B ) l * a T concentrations thus if b( (&v/bV)CA}/aCA is positive it means that the highest concentration of A possible will move at the greatest rate. Consequently it will overtake all lower concentrations of A which might be in front of it. Thus a sharp front boundary will be finally formed and the lowest concentration possible will be trailing behind forming a diffuse boundary. If however b{ (bX/bV)C,}/bcA is negative the inverse will be obtained.A further calculation which will not be extended here shows that if adsorption occurs according to the Langmuir theory of adsorption a leading front boundary of A will be formed if ~ I ( ~ A > / ~ A > f2(cB>/cB holds good at all concentrations. This simple condition only means that the concentration of A in the liquid phase decreases if adsorbent is added to any mixture of A and B. If the adsorption isotherms of A and B are of the Freundlich type and fl(c~>/c~ > fi(c~)/C~ a leading front boundary will also be formed at least at low or high con- centrations. At medium concentrations no general conclusion can be drawn. Thus it may be assumed if f i ( c A > / c A > f2(cB)/cB a sharp front boundary of A will generally be formed.As has already been stated eqn. (4) only holds good within an adsorption band. When a sharp boundary occurs whether in front or in the rear another conservation equation has to be applied to the boundary. For- either in front or at the rear of such a boundary no trace of the substance considered should be present. Therefore the conservation equation of a sharp boundary is CdV = (a - f ( 4 ) ) C + q l h where C applies to the concentration in the boundary. If eqn. (5) is applied to the mixture of A and B and equilibrium is instantaneously established the following equation holds good for a sharp boundary where the subscript b indicates that the concentration refers to that in the sharp boundary. (4) De Vault,3 whose paper has been very useful to the author assumed the interstitial volume to be constant and obtained According to the initial assumptions eqn.(4) holds good only within the adsorption band. In the same way as de Vault did it can be shown that at a boundary of an adsorption band eqn. (4) may lead to physical impos- sibilities thus indicating the occurrence of a discontinuity (sharp boundary). If 'v is increased at a constant rate (3~/bV)c may be called the rate of transport of the concentration C A . If (bx/bV)cA increases with increasing (5) -~ W. M. SMIT and are obtained. 43 For the diffuse boundary (zero concentration) the following equation is obtained where the subscripts o and I indicate zero and unit concentration d respec- tively.Applying de Vault’s theory 4. Discussion.-Eqn. (4) represents the rate of transport of a certain concentration of A within the adsorption band. But the corresponding concentration of B should attain the same value. Thus Consequently (a@ Y) c = (a@ V> c, if CB = d - CF,. Eqn. (4) meets this condition whereas eqn. (4a) does not f:(CA) =fi(CB) unless which as a rule cannot be true. As has already been shown a sharp leading boundary of A will be formed if f i ( c A ) / c A > fi(CB)/CB- (ax/3VC* zvj I (eqn. (6)). This means that the rate of transport of any boundary is always smaller than the rate of transport of the liquid front which of course is equal to Both rates become equal if Cg = 0 i.e. CA = d. If however r/mvd. fi(cA)/cA < f i ( C B ) / c B the front boundary will be diffuse and a sharp boundary will be present at the rear.In this case the rate of transport of the boundaries is always greater than the rate of transport of the liquid front unless C A = d. Thus it may be stated if f i ( C d / C ~ > f & d / C ~ substance A will always lose on the liquid front whereas B has a tendency to overtake the liquid front until its concentration d has become equal to unity. This may serve as a suitable interpretation of the phenomena which obtain when a solution containing a single solute is introduced into the dry column (continuous introduction method). If a mixture of A and B is introduced into a dry column A will move at a slower rate than the liquid front. Therefore pure B will appear in front of the adsorption band and its quantity will increase if introduction is continued.Since at the top of the column the concentration of A is equal to the initial concentration and this is the highest Concentration of A possible under these conditions it will overtake all lower concentrations of A which might be in front of it. Thus within the adsorption band the concentration of A is equal to the initial concentration and the adsorption band will move at a constant rate. Elution (displacement) may now be explained too. Suppose two sub- stances A and B are present in an adsorption column and a new solvent S is added at the top of the column. Further .f3(cS>/cS >fi(cA)/c.I > f i ( c B > / c B . CONSERVATION EQUATION OF CHROMATOGRAPHY 44 According to the above deductions any A or B which happens to become diluted by S will move at a greater rate than the liquid front whereas the rate of S should be smaller than (or when undiluted equal to) the rate of the liquid front.Therefore S will remain undiluted and act like a piston forcing A and B in front of it. The same applies to A as compared to B. Therefore the final result of elution will be three adjacent adsorption bands containing S A and B at unit concentrations As to the developer method little further explanation seems to be necessary. If more than one solute is present and separation of the solutes has been obtained already the rates of transport of the different solutes are established by eqn. (6) and (7) and thus the distance between the adsorption bands will usually increase on further development and at the same time the bands will be broadened.The mere process of separation of a mixture of solutes is more com- plicated because of mutual alteration of the adsorption isotherms. Never- theless it may be accepted that the separation itself occurs qualitatively along the same lines. As a rule when the developer method is applied a sharp front boundary occurs. Experimentally asymmetric adsorption bands are usually found the highest concentration being near the front. Eqn. (6) and (7) reveal another possibility which may not as yet have been recognized. An adsorption band of A when present will move at a greater rate than the liquid front But if the whole column has previously been wetted with B the substance A cannot reach the liquid front before leaving the column.If a mixture of A and B is introduced into a column previously wetted with B the lowest concentration of A will move at the greatest rate. Thus an adsorption band with a diffuse front and a sharp rear boundary will be formed immediately. This band is transported through the column if there- upon pure B is added to the column at the top. The band is broadened during this procedure but its rate is conditioned by eqn. (6) and (7). If still another solute say D is present which is also adsorbed less strongly than B it will behave in the same way but it may be that both limits of the rate of transport of the adsorption band of D are smaller than those of B.Thus this method which is supposed to be called development with an eluent may cause separation. According to former conceptions this method would be impossible because B the eluent should not allow A or D to become adsorbed. However a very simple experiment showed the present conception to be correct. A mixture of cetane and cetene could be partly separated by silica gel previously wetted with benzene using benzene as developing liquid. I t is well known that benzene may act as an eluent (displacer) for cetane and cetene when adsorbed on silica gel. The advantage of this method is that the total volume of " developer " necessary to collect all the solutes at the bottom of the column is smaller than the interstitial volume of the column.This means a saving of time. However the separating efficiency of this method may be in many cases low. The occurrence of rates of transport greater than the rate of the liquid front is also demonstrated in an earlier paper,5 though not mentioned explicitly. Rates of transport up to about 1-3 times the rate of the liquid front have been found with chloroform and dodecylbenzene. A few words may be added as to the occurrence of so-called adsorption azeotropes as described by Hirschler and Am0n.l When an adsorption azeotrope occurs fi (c A ) / c A - 5 Smit Anal. chim. A d a 1948. 2 671. w. M. SMIT At the azeotropic concentration ~ ~ ( C A ) / C A -~~(CB)/CB = 0. 45 changes sign at a certain concentration of say A. At concentrations lower than the azeotropic concentration the sign may be positive and when the mixture is introduced into the column pure B appears in front of the adsorption band containing A.Beyond the azeotropic concentration pure A will appear in front and B is contained within the adsorption band. The rate of transport of A and B both become equal to the rate of the liquid front and no separation occurs. So the conclusion derived from eqn. (6) and (7) agrees with experiment. At the same time it is clear that the occurrence of adsorption azeotropes is limited to substances having comparable adsorption affinities. No attempt has been made to solve the differential equations nor to develop formulze for more than two substances. As de Vault already stated this becomes very complicated.Moreover as real adsorption isotherms are used in our equation the quantitative solution is of no use as the real adsorption isotherms are not available. The main purpose of this paper has been to arrive at a rather simple formula which permits a qualitative explanation of the different methods of percolation but on the other hand it might show that the quantitative deductions made thus far have to be handled with care. Acknowledgment is due to the Management of the Bataafsche Petroleum Maatschappij for their permission to publish this paper. KoninklijkelShell-Laboratorium Amsterdam. THE CONSERVATION EQUATION OF CHROMATOGRAPHY BY W. M. SMIT Received 18th July 1949 The conservation equation of chromatography thus far used does not account for the volume occupied by the adsorbed phase.When the equation of conservation of matter is applied to the process of adsorptive percolation and this volume is accounted for a general equation is obtained which may be more correct. This is shown by applying this equation to the equilibrium theory. Several effects occurring in chromato-graphy which cannot be explained by the de Vault equation may now be interpreted, a t least qualitatively. Moreover the equation obtained reveals the possibility of a new method of adsorptive percolation W. M. SMTT 39 In the development of the mathematical description of the process of adsorptive percolation two different methods of approach may be distin-guished. One may start with the assumption of instantaneous equilibrium between solution and adsorbent (equilibrium theory) or one may dismiss the idea of complete equilibrium and treat the problem as a kinetic one (kinetic theory).However in both cases the law of conservation of matter supplies the initial equation which in both cases must be fundamentally the same. The equations obtained thus far give a ready explanation of the phenomena occurring during development. But when applying these equations in explanation of other processes of adsorptive percolation like the continuous introduction method (frontal analysis) or the elution method (displacement method) difficulties are encountered. Moreover the occurrence of so-called adsorption azeotropes,l especially with closely related compounds needs further interpretation.Some experience gained with the adsorptive percolation of hydrocarbons suggested that the failures mentioned might be caused by the use of an approximate conservation equation. Symbols The following symbols are used in this paper :-a interstitial volume of an adsorption column per unit of length (ml./cm.). b index referring to a sharp boundary. C concentration of solute expressed as units of weight of solute per unit of CA CB concentrations of substance A and substance B respectively. d density (g./ml.). f f l ( C ~ ) adsorption isotherm of component N. &(CN) first derivative of the adsorption isotherm (dfn(CN) /dCN), m weight of adsorbent contained in the column per unit of length (g./cm.). q weight of solute adsorbed per unit of length of a column (g./cm.).f ( q ) volume occupied by the adsorbed phase (solute + solvent) per unit of S solvent. V volume of liquid forced into the column (ml.). v interstitial volume of dry adsorbent present in the column per unit of weight x distance from the top of the column (cm.). 0 I (when combined with C) indices indicating zero and unit concentration I 2 3 (when combined with f) indices indicating that the functions fl fi and fs volume of solution (g./ml.). It also represents the weight of solute adsorbed per unit of weight of adsorbent a t the concentration CN (g./g.). length if a quantity q of solute is adsorbed (ml./cm.). (ml*/g.) * respectively . are different functions. 1. The Conservation Equations used thus far.-The first conserva-tion equation was given by Wilson : where C is the concentration of the solute x the distance from the top of the column q the weight of solute adsorbed per unit of length and V the volume of liquid poured into the adsorption column.This equation has been corrected by de Vault 3 for the interstitial volume a per unit of length of the adsorption column. De Vault arrives at the equation : 1 Hirschler and Amon. Ind. Eng. Chenz 1947 39 1565. 2 Wilson J . Amer. Chem. SOC. 1940 62 1583. 3 De Vault J . Amer. Chem. SOC. 1943 65 532 40 CONSERVATION EQUATION OF CHROMATOGRAPHY The same equation has been used implicitly or explicitly by other authors whether they adopted the equilibrium or the kinetic standpoint for the equilibrium or the kinetic conditions are introduced subsequently when the relation between C and q is inserted.Eqn. (2) is based on the assumption that the interstitial volume a is constant. This is the point where intro-duction of a correction might be useful. The adsorbed phase occupies a certain fraction of the original interstitial volume thus causing the liquid to flow at a greater linear rate than when the adsorbed phase is not present. As the quantity of material adsorbed is a function of the concentration the free interstitial volume that is the volume available to the jlowing liquid must be a function of the concentration. 2. Derivation of the Corrected Conservation Equation.-To intro-duce the correction for the volume of the adsorbed phase the conservation equation will be derived once more for a solution containing a single solute.The following assumptions which do not differ from those usually applied, have been made. Within an adsorption band the concentration is a con-tinuous function of x. Within a cross-section of infinitesimal length &, (3C/3x)v is considered constant. The same applies to (3q/3V) when an infinitesimal volume of liquid dV is introduced into the column. When an infinitesimal volume dV is introduced into the column an equal volume has to pass through a cross-section at x where x is a point within the adsorption band. As the concentration of the solution entering the cross-section at x and the concentration of the solution leaving the cross-section at (x + dx) differ by the amount of solute present in this section of the column is increased by This amount is distributed between the liquid phase and the adsorbed phase present in the section.The volume of the liquid phase present in the section is equal to wheref(q) means the volume of the adsorbed phase per unit of length within the section considered. Thus the amount of solute present in the liquid phase is equal t o The amount of solute present in the adsorbed phase is equal to qdx where 4. = mfn(C,). Thus the total amount of solute present in the section is equal t o If an infinitesimal volume dV is forced through this section the increase of solute may therefore also be represented by (3Cj3X)V h, (ac/3x)V dx.dv. (a -f(d)dx, (a - f(4))Cdx. (a - f W + 4 ) h . According to the law of conservation of matter the amount of solute intro-duced by the volume dV should be equal to the increase of the amount of solute present in the section.Thus 'Martin and Synge Biocheun. J. 1941 35 1941 Weiss J . Chew. sot. 1943 '45. 297. Thomas Ann. N.Y. Acad. Sci. 1948 49 161 W. M. SMIT 41 This is the corrected conservation equation of chromatography in its As to.f(q) it should be borne in mind that it is not correct to put general form for a solution containing a single solute. f ( d = 4 M where d means the density of the adsorbed solute. For besides the solute the solvent is adsorbed too. As t o this point it might be useful to stress that adsorption isotherms of pure gases and so-called isotherms of solutions containing a single solute usually have a different meaning. The adsorption isotherm of a pure gas shows the relation between the concentration !(pressure) of the gas and the weight adsorbed.However the so-called adsorption isotherm of a solute shows the relation between the concentration and the weight of solute adsorbed preferentially. ‘This relation might be better called a preferential adsorption isotherm. ‘To avoid confusion it may be stated here that in this paper the adsorption isotherm of a solute means the relation between the total amount of solute .adsorbed per unit of weight of adsorbent and the concentration of the :solution with which it is in equilibrium. 3. Application of Eqn. (3) to the Equilibrium Theory.-Attempts will now be made to derive a fundamental equation of chromatography ,on the basis of the equilibrium theory for a liquid mixture of two substances A and B.Which of the two is considered as solvent does not matter. For the sake of convenience it is assumed that A and B have equal densities and no volume change occurs on adsorption or mixing. If the densities of A and B are represented by d and the concentration is expressed as g./ml., we may put Suppose the adsorption column considered contains m g. adsorbent per unit of length. The adsorption isotherm of A when diluted by B is represented by f l ( C A ) . At equilibrium, (for non-equilibrium another relation between q and C should be introduced but this will not be discussed). If f2(C4 represents the adsorption isotherm of B when diluted by A we may put C.4 + CB = d . y = mfi (CAI f ( 4 ) = {fi(cA) + f d C B > Imld* If ZI is the interstitial volume of the dry adsorbent per unit of weight, a = mu.Substitution in eqn. (3) gives (3cA/3x)V + m[(v -fi(cA)/a -f2(cB)/d>(acA/av)z - { (afl(cA)/3v)x + (3f2(CB)/3v)z)CA/d + (af,(cA)/av)%l = 0. On further substitution viz., ( ~ C A / ~ X ) V = - (3V/ax)cA (~CA/~V)X, (afi (C A) /a v) = ( J f l ( CA) / ~ C A ) (JCA/J v) = f i c ~ ( ~CA/J V)z and (3f,cB/3v)x = (afz(cB)/acB) (acB/av) = - fkcB(aCA/aV)~, the following equation is obtained : - (dlr/Jx)~,(aC~/’~)% + mCv - f l ( c ~ ) / d - j 2 ( C ~ ) / d - {fi(c~) - fS(c~) ) C A / ~ or fi(CA)I(acA/av)~ = 0, - PviWc + [vd - f l ( c ~ ) - f z ( C ~ ) + CB~~(CA) + C,fh(C~)]m/d = 0. B 42 CONSERVATION EQUATION OF CHROMATOGRAPHY * (4) Thus a (bX/bV)C = - _ _ _ ~ m[vd - f i ( c A ) -fi(cB) -k CBf;(CA) + c A f L ( c B ) l De Vault,3 whose paper has been very useful to the author assumed the interstitial volume to be constant and obtained T According to the initial assumptions eqn.(4) holds good only within the adsorption band. In the same way as de Vault did it can be shown that at a boundary of an adsorption band eqn. (4) may lead to physical impos-sibilities thus indicating the occurrence of a discontinuity (sharp boundary). If 'v is increased at a constant rate (3~/bV)c may be called the rate of transport of the concentration C A . If (bx/bV)cA increases with increasing concentrations thus if b( (&v/bV)CA}/aCA is positive, it means that the highest concentration of A possible will move at the greatest rate. Consequently it will overtake all lower concentrations of A which might be in front of it.Thus a sharp front boundary will be finally formed and the lowest concentration possible will be trailing behind, forming a diffuse boundary. b{ (bX/bV)C,}/bcA is negative, the inverse will be obtained. A further calculation which will not be extended here shows that if adsorption occurs according to the Langmuir theory of adsorption a leading, front boundary of A will be formed if ~ I ( ~ A > / ~ A > f2(cB>/cB holds good at all concentrations. This simple condition only means that the concentration of A in the liquid phase decreases if adsorbent is added to any mixture of A and B. If the adsorption isotherms of A and B are of the Freundlich type and fl(c~>/c~ > fi(c~)/C~, a leading front boundary will also be formed at least at low or high con-centrations.At medium concentrations no general conclusion can be drawn. Thus it may be assumed if a sharp front boundary of A will generally be formed. As has already been stated eqn. (4) only holds good within an adsorption band. When a sharp boundary occurs whether in front or in the rear, another conservation equation has to be applied to the boundary. For-either in front or at the rear of such a boundary no trace of the substance considered should be present. If however, f i ( c A > / c A > f2(cB)/cB, Therefore the conservation equation of a sharp boundary is CdV = (a - f ( 4 ) ) C + q l h (5) where C applies to the concentration in the boundary. If eqn. (5) is applied to the mixture of A and B and equilibrium is instantaneously established the following equation holds good for a sharp boundary : -~ where the subscript b indicates that the concentration refers to that in the sharp boundary W.M. SMIT 43 For the diffuse boundary (zero concentration) the following equation is obtained : where the subscripts o and I indicate zero and unit concentration d respec-tively. Applying de Vault’s theory, and are obtained. 4. Discussion.-Eqn. (4) represents the rate of transport of a certain concentration of A within the adsorption band. But the corresponding concentration of B should attain the same value. Thus if CB = d - CF,. Eqn. (4) meets this condition whereas eqn. (4a) does not, which as a rule cannot be true. (a@ Y) c = (a@ V> c,, unless f:(CA) =fi(CB), As has already been shown a sharp leading boundary of A will be formed if f i ( c A ) / c A > fi(CB)/CB-Consequently I (ax/3VC* zvj (eqn.(6)). This means that the rate of transport of any boundary is always smaller than the rate of transport of the liquid front which of course is equal to r/mvd. If however, fi(cA)/cA < f i ( C B ) / c B , the front boundary will be diffuse and a sharp boundary will be present at the rear. In this case the rate of transport of the boundaries is always greater than the rate of transport of the liquid front unless C A = d. Thus f i ( C d / C ~ > f & d / C ~ it may be stated if substance A will always lose on the liquid front whereas B has a tendency to overtake the liquid front until its concentration d has become equal to unity.This may serve as a suitable interpretation of the phenomena which obtain when a solution containing a single solute is introduced into the dry column (continuous introduction method). If a mixture of A and B is introduced into a dry column A will move at a slower rate than the liquid front. Therefore pure B will appear in front of the adsorption band and its quantity will increase if introduction is continued. Since at the top of the column the concentration of A is equal to the initial concentration and this is the highest Concentration of A possible under these conditions it will overtake all lower concentrations of A which might be in front of it. Thus within the adsorption band the concentration of A is equal to the initial concentration and the adsorption band will move at a constant rate.Suppose two sub-stances A and B are present in an adsorption column and a new solvent S is added at the top of the column. Both rates become equal if Cg = 0 i.e. CA = d. Elution (displacement) may now be explained too. Further .f3(cS>/cS >fi(cA)/c.I > f i ( c B > / c B 44 CONSERVATION EQUATION OF CHROMATOGRAPHY According to the above deductions any A or B which happens to become diluted by S will move at a greater rate than the liquid front whereas the rate of S should be smaller than (or when undiluted equal to) the rate of the liquid front. Therefore S will remain undiluted and act like a piston forcing A and B in front of it. The same applies to A as compared to B.Therefore the final result of elution will be three adjacent adsorption bands containing S A and B at unit concentrations As to the developer method, little further explanation seems to be necessary. If more than one solute is present and separation of the solutes has been obtained already the rates of transport of the different solutes are established by eqn. (6) and (7) and thus the distance between the adsorption bands will usually increase on further development and at the same time the bands will be broadened. The mere process of separation of a mixture of solutes is more com-plicated because of mutual alteration of the adsorption isotherms. Never-theless it may be accepted that the separation itself occurs qualitatively along the same lines.As a rule when the developer method is applied a sharp front boundary occurs. Experimentally asymmetric adsorption bands are usually found the highest concentration being near the front. Eqn. (6) and (7) reveal another possibility which may not as yet have been recognized. An adsorption band of A when present will move at a greater rate than the liquid front But if the whole column has previously been wetted with B, the substance A cannot reach the liquid front before leaving the column. If a mixture of A and B is introduced into a column previously wetted with B the lowest concentration of A will move at the greatest rate. Thus an adsorption band with a diffuse front and a sharp rear boundary will be formed immediately. This band is transported through the column if there-upon pure B is added to the column at the top.The band is broadened during this procedure but its rate is conditioned by eqn. (6) and (7). If still another solute say D is present which is also adsorbed less strongly than B it will behave in the same way but it may be that both limits of the rate of transport of the adsorption band of D are smaller than those of B. Thus this method which is supposed to be called development with an eluent may cause separation. According to former conceptions this method would be impossible because B the eluent should not allow A or D to become adsorbed. However, a very simple experiment showed the present conception to be correct. A mixture of cetane and cetene could be partly separated by silica gel previously wetted with benzene using benzene as developing liquid.I t is well known that benzene may act as an eluent (displacer) for cetane and cetene when adsorbed on silica gel. The advantage of this method is that the total volume of " developer " necessary to collect all the solutes at the bottom of the column is smaller than the interstitial volume of the column. This means a saving of time. However the separating efficiency of this method may be in many cases low. The occurrence of rates of transport greater than the rate of the liquid front is also demonstrated in an earlier paper,5 though not mentioned explicitly. Rates of transport up to about 1-3 times the rate of the liquid front have been found with chloroform and dodecylbenzene. A few words may be added as to the occurrence of so-called adsorption azeotropes as described by Hirschler and Am0n.l When an adsorption azeotrope occurs fi (c A ) / c A -5 Smit Anal.chim. A d a 1948. 2 671 w. M. SMIT 45 changes sign at a certain concentration of say A. At concentrations lower than the azeotropic concentration the sign may be positive and when the mixture is introduced into the column pure B appears in front of the adsorption band containing A. Beyond the azeotropic concentration pure A will appear in front and B is contained within the adsorption band. At the azeotropic concentration ~ ~ ( C A ) / C A -~~(CB)/CB = 0. The rate of transport of A and B both become equal to the rate of the liquid front and no separation occurs. So the conclusion derived from eqn. (6) and (7) agrees with experiment. At the same time it is clear that the occurrence of adsorption azeotropes is limited to substances having comparable adsorption affinities. No attempt has been made to solve the differential equations nor to develop formulze for more than two substances. As de Vault already stated this becomes very complicated. Moreover as real adsorption isotherms are used in our equation the quantitative solution is of no use as the real adsorption isotherms are not available. The main purpose of this paper has been to arrive at a rather simple formula which permits a qualitative explanation of the different methods of percolation but on the other hand it might show that the quantitative deductions made thus far have to be handled with care. Acknowledgment is due to the Management of the Bataafsche Petroleum Maatschappij for their permission to publish this paper. KoninklijkelShell-Laboratorium, Amsterdam