P.
= viscosity of a solute = weight of adsorbent per unit
7,
= convenient temperature-inde-
7”
column volume wndent Parameter, EWation 8 = convenient parameters, Equations 20 and 21 ACKNOWLEDGMENT
G. J. Krige is indebted t o the Director General of the Atomic Energy Hoard for permission to participate in this project.
LITERATURE CITED
( 1 ) Berge, P. C. van, Pretorius, Lr., J . Gas chromalog. 2,235 ( 1964). ( 2 ) Boer. J . H. de. “Ilvnamical Character ’ o f Adsorption,” p. &, Clarendon Press, Oxford, 1953. (3) I M . , p. 55. (4) Krige, (;. J., Pretorius, V., ANAL. CHEM. 37. 1186 (1965). (5) Lange, ‘IV. A., Forker, G,., &I., eds., ‘[Handbook of Chemistry, 5th ed., pp. 1588-91, Handbook Publishers, Sandusky, Ohio, 1944. (6) McBain, J. W., “Sorption of Gases
and Vapours by Solids,” p. 65, Routledge and Sons, London, 1932. ( 7 ) Purnell, H., “Gas Chromatography,” Chap. .5, Wiley, New York, 1962. (8) Reid, R. C., Sherwood, T. K., “Properties of Gases and Liquids,” p. 200, RIcGraw-Hill, New York, 1958. ( 9 ) Young, D. M., Crowell, A. D., “Physical Adsorption of Gases,” p. 106, Butterworths, London, 1962. (10) Ibid., Chap. 8. ( 1 1 ) /bid., Chap. 11. (12) Ibid., p. 353. RECEIVED for review February 2, 1965. Accepted June 9, 1965.
Frontal Analysis Chromatography Theoretical Treatment of Stationary Front Widths Neglecting Radial Perrnea bility Fluctuations G. J. KRIGE’ and VICTOR PRETORIUS Department of Physical and Theoretical Chemistry, University of Pretoria, Pretoria, South Africa At sufficiently high sample inlet concentrations in frontal analysis, front broadening processes can be balanced by front sharpening processes, and a stationary front is then formed. Theoretical expressions are derived for the widths of stationary fronts taking sorption effects into account. The continuous injection of a single solute, as well as a binary solute mixture which yields well-resolved fronts at the column outlet, is considered. Stationary fronts can arise with linear, type I, and even type 111 distribution isotherms when sorption effects are present. Results obtained are more general than those existing in the literature, and are compared with expressions derived by other authors.
I
been shown theoretically (14), and verified experimentally (15), that sorption effects in frontal analysis at high sample inlet concentrations cause significant fluctuations in the flow velocity of the mobile phase in a chromatographic column, and complicate the expressions for the retention times and volumes of sharp fronts. I n the present paper a theoretical study is made of the effect of sorption mechanisms on the shapes and widths of the fronts observed when both a single solute and a binary solute mixture are continuously injected into a column7 i.e., it is no longer assumed that sharp fronts of zero width are formed. The treatment is simplified considerably a t T HAS
Present address, Department of Chemistry, University of British Columbia, Vancouver, Canada. Permanent, address, Atomic Energy Board, Pelindaba, Republic of South Africa. 1
this stage by stipulating that radial variations in the permeability of the column packing are negligibly small; the other basic assumptions pertaining to the study of preparative chromatography by frontal analysis as a whole have been stated in (14). Various authors (1, 3, 4 , 9 , 18, 19, 21) have deri\ ed expressions for the shapes and widths of the fronts formed when nonideality operates in conjunction with nonlinearity and/or sorption efferts. It has been shown that, at sufficiently high concentrations, the unchecked spreading of the front due to nonideality (front broadening processes) can be counteracted to a lesser or greater extent by nonlinear and sorption phenomena (front sharpening processes). After sufficient time has elapsed the two processes can balance one another, and the shape and width of the front no longer change as it moves down the column. Fronts of this type have been referred t o as stationary fronts (3, 4,18, 19,21). One of the most important differences between frontal analysis and elution development is that stationary fronts may be formed when the former technique is used. The front resolution function, R , defined as (12, 17)
is then proportional to the column length, L, and not to as is the case when the distribution isotherm is linear and flow rate changes due t o solute sorption are negligible. This suggests that by employing conditions under which stationary rather than nonstationary fronts are formed, the mini-
dc
mum column length required to yield a given separation could possibly be significantly reduced. Glueckauf and Coates (9) have derived expressions for the shape of a stationary front for a solute which follows a Freundlich or Langmuir type distribution isotherm, but have neglected sorption effects and nonideality due to processes other than lateral nonequilibrium. Schay and his associates (3, 4, 18, 19) have incorporated the effects due to fluctuations in the flow rate for different types of isotherms, but have made simplifying assumptions regarding nonideality. These are that the plate height is due to either longitudinal diffusion in the mobile phase or lateral nonequilibrium in the stationary phase; no attempt was made to consider these effects in conjunction with one another. Rosanquet and Morgan ( 1 , 2) have investigated the formation of stationary fronts when flow rate changes due to solute sorption play a n important role, but have considered only linear distribution isotherms. The phenomenon of stationary fronts is investigated in detail in this paper. Theoretical expressions are derived which are not subject t o the assumptions made in the treatments referred to above, and are compared with the existing results in the literature. GENERAL CONDITIONS UNDER W H I C H STATIONARY FRONTS ARE FORMED
For the purpose of the present discussion it may be assumed that the shapes and widths of solute fronts are determined by three factors in addition t o the column length : VOL 37, NO. 10, SEPTEMBER 1965
1 195
1. The nonideal effects, arising from lateral nonequilibrium, longitudinal diffusion, and eddy diffusion ( 5 ) . 2. The shape of the distribution isotherm of the solute between the mobile and stationary phases. 3. Changes in the flow rate due to continuous sorption of the solutes from the mobile phase.
When the solutes are highly diluted with carrier, and only nonideal effects are important, the fronts broaden steadily as they move down the column, and their shape a t the column outlet is described by the first integral of the probability function (16,I7). However, as the sample inlet concentration is increased, the shapes, and widths, of the fronts are affected by nonlinearity and sorption effects in addition to the nonideal mechanisms. The formation of stationary fronts under these conditions may best be demonstrated by first considering the continuous introduction of a single solute, which may be diluted with a carrier, into a chromatographic column containing the carrier and a suitable sorbent. The general mass balance equations applicable to this problem are ( I t ? ) , for the carrier,
a
- (1 at
X)
+a D
[u(l
- X)l -
(1
- X) = 0
(2)
and for the solute of mole fraction X,
The effective diffusion coefficient of the solute in the mobile phase, D, is regarded as constant in the treatment. The average linear flow velocity of the mobile phase, u, is in general a function of x and t . Y/X is equal to the ratio of the mass of solute in the stationary phase to that in the mobile phase; a t equilibrium, Y is equal to kXX,where kx denotes the value of the mass distribution coefficient a t the mole fraction X of the solute. Equations 2 and 3 may be used to investigate the role played by factors (2) and (3) in the formation of stationary fronts. The discussion is simplified by considering each factor in the absence of the other, and by neglecting nonideality. Ideal, Nonlinear Chromatography by Frontal Analysis at Negligible Solute Concentrations. Under these conditions D = 0, Y = k X X , and X> L,, and that a stationary front is formed. Equations 2 and 3, the general mass balance equations applicable to the problem, will be solved by assuming that the front progression velocity, v f , is not a function of time. This is not strictly true when the inlet and outlet pressures are kept constant (14). However, if L >> L,, the front will be able to adapt itself to flow rate changes while they occur, and the front width a t a given instant may be found by neglecting the variation of vf with time. Under these conditions a given mole fraction will be observed a t a point xII in the column (xII - X I ) / V J seconds after it is observed a t a point X I . The variation of X with x and t may thus be accounted for by introducing a single compound variable, w, defined as (18)
The variation of the average flow velocity in the mobile phase, u, with x and t is similar to that of X. The following transformations of operators are obtained from Equation 20 :
It is therefore found from Equations 2 and 21 that
-dX+ - -1 -du- - 1 d (UX) du v / dv Vf du D dZX Vf2 dv2
uX _ _ D_dX_ Vf
v,2
dv
-- 0 (25)
Similarly, Equation 23 becomes Y--
U
Of
+ [l
+ k'(1 - X . ) ] = 0
(26) When u is eliminated from Equations 25 and 26 it is found that dX -_D _
dw
Y(l
- X)
- k'X(1 - 3) (27)
Y is eliminated from Equation 27 by assuming that b
- (kxX) = p ( k X X at
- Y)
(28)
where p is a function of variables such as the mass distribution coefficient, the diffusion coefficient in the mobile phase, and the particle diameter, but is not explicitly dependent on X , x, or t. Relations similar to the above are usually employed in studies of chromatographic processes (6, 8, 1I), and are dependent on the assumption that the deviation from equilibrium is slight. The validity of Equation 28 is discussed in detail below and in the appendix. From Equations 21,27, and 28 it may easily be shown that dX - --
dw
- 0 (22)
By applying Equation 21 to Equation 3, and by subtracting the result from Equation 22, a second relation is obtained:
k X X ( 1 - X) - ktX(1 - X * )
{ - X) L d (kXX) +}: -.
(29)
dX Vf Integration of Equation 29 yields an expression for the front shape in terms of (u VI), where U I is an integration I.(
+
VOL. 37, NO. 10, SEPTEMBER 1965
1197
constant. When the distribution isotherm is linear-Le., k x = k' = k-it is found that {log P
x + (XS - 1) log ( X t - X ) 1 -
D log
(x+x)
=
Xi(w
+
VI)
(30)
when solutes are highly diluted with carrier (1'7). Equations 29 and 34 yield and Equation 41 becomes
WtS =
)
(SX
X = I / P X '~
kv1
1
+
+ bXp '
ax ax ( l + k ) z + U - = bX
- Xi) 2 ki(1 - xl) kh (a - X i )
;bt2 + D a2x
2 ku
-
where kh denotes the value of k X a t X = l/zX$. Alternatively, from Equations 33 and 34, w'st --
2k
+ k ) 2 H , - (1 + k)2C7
(32)
where C, is the coefficient of u in the expression for H , and is independent of the flow velocity. In order to apply the above equation under nonlinear conditions k is simply replaced by k X and C, evaluated a t a mole fraction X. From Equations 29 and 32 it follows that
ax _-
(kXX)
Xal/2X'
(36)
wzs
k X X ( 1 - X) - k X ( 1 - XC) (1 kX)2 C,X(l - X ) 2k"
+
where C,x denotes the value of C, a t a mole fraction X . Width of a Stationary Front. The width of a stationary front in units of time, wts,is defined as
(34) This expression is similar to that proposed for the widths of the fronts formed ANALYTICAL CHEMISTRY
(37)
and may be calculated from Equations 35 to 37. The width of the front in units of volume is wvS = uhA,wtS
(38)
where uh denotes the value of u a t X = 1/2Xt. u,,may be calculated from Equations 25 and 34 as Uh
= VI
+
Conditions under Which Stationary Fronts Are Formed. The preceding equations are valid only if a stationary front is formed. The condition for the formation of a stationary front is
=
b1
+-
b"X
> b"(1 - x.)
(45)
The formation of a stationary front under the above conditions is thus determined by the inlet mole fraction and the curvature of the isotherm, as characterized by the value of b"/b'. The results obtained in this section are in agreement with the semiqualitative predictions made above. Validity of Equation 28 and Comparison with Other Theories. The general expression for the width of a stationary front derived above may be compared with less general expressions obtained by other authors. The expressions derived by Schay (18) for nonlinear chromatography a t high sample inlet concentrations, with longitudinal diffusion as the only nonideal effect, may be found directly from Equation 29 by letting p -+ m. The case where longitudinal diffusion is negligible is now considered. When D = 0, Equation 35 yields Wf =
(18 )
dw (33)
= VfW/
(43)
(44) where b' and b" are constants. The condition expressed by Equation 41 then becomes k X
b'
v,"2 - X $ ) 2 k"1 - X it ) kh (2 - X i )
1, it is clear from Equations 40 and 66 that the front will be stationary if a d e m ,
UhAm(Wta)b
ANALYTICAL CHEMISTRY
(70)
kzX(1 - XZ)
> kz'(1 - XZ.) for 0 < Xz < Xz*
where D is the effective mobile phase diffusion coefficient, and X, the mole fraction of the solute at distance yr. Since the mass of solute in the stagnant layer will be far smaller that the mass of solute sorbed at equilibrium, fluctuations in the former can be neglected and it may be assumed that j is independent of y,. Integration of Equation 73 then yields
(72)
The value of kzX must be determined from the mixed sorption isotherm for solute 2. Linear isotherms clearly fulfill the condition expressed by Equation 72, and it can be shown that mixed Langmuir isotherms (23) also result in the formation of stationary fronts. Unfortunately, Schay has not proposed a type I11 isotherm for the simultaneous sorption of two sorbates, but if it is assumed that kzX can be described by an expression similar to Equation 44 in this case, the discussion following Equation
45 is also valid for the second front when a binary solute mixture is injected.
since the mole fraction at y, = 0, which is equal to the mole fraction in the central region of the mobile phase, can be taken as X, the (radial) average mole fraction in the mobile phase; X, is the mole fraction of the solute in contact with the sorbent. If a linear isotherm is considered, Y = kX,, and bY/bt is proportional to j, with the result that Equation 49 follows directly from Equation 74. An indication of the error which is made when Equation 28 is used may be obtained by comparing Equation 46 with Equations 47 and 48. Schay's (18) expression (Equation 47) is derived on the assumption that the isotherm is linear and sorption effects are important; under these conditions Equation 46 becomes Wt'
APPENDIX
The following simple example demonstrates that the deviation from equilibrium may be found more accurately from Equation 49 than from Equation 28. Consider the mobile phase in a chromatographic column as comprising a central region, in which there is radial
equilibrium, and a very thin, flat, stagnant region, of thickness y., in contact with the sorbent, in which there is a resistance to mass transfer (11). The resistance to mass transfer in the stationary phase is neglected. Let yr denote the distance measured from
=
2(2 - xl) pXi
(75)
whereas Schay finds that (Wt8)s =
8(1 - xl) ~pX"2 - Xi)
(76)
Glueckauf and Coates (9) have neglected sorption effects, but have taken nonlinearity of the isotherms into account. Their expression for the front width (Equation 48) becomes
when a single solute following a Langmuir isotherm is considered (see Equation 42). The treatment of the above authors is applicable to cases where Xi, but not b X i p , is negligible in comparison with unity. Equations 42 and 46 then yield
Equations 75 and 78 are compared with Equations 76 and 77, respectively,
= average
in Table I. It is evident from the data in the table that, in general, it may be assumed that
=
-
where / w / 1, is the standard deviation corresponding to the corrected front width, and u t s is the standard deviation corresponding to the front width found from Equation 35. The relation between wts and is obtained by combining Equations 13 and 79. NOMENCLATURE
Unless otherwise stated, subscripts 1 and 2 refer t o the less and more strongly retarded solutes of a solute pair, respectively, and subscripts a and b refer to the first and second fronts which break through at the column outlet when a binary solute mixture is continuously injected. Other nomenclature is explained in previous papers (14, 1 5 ) . b, b’, b” = convenient parameters, Equations 42 and 44 C0 = equivalent i n l q concentration of a solut‘e = coefficient of u defining conC, tribution of resistaice to mass transfer in mobile and stationary phases to height equivalent per theoretical plate for a solute when radial permeability fluctuations are negligible C,“ = value of C, at a mole fraction
x
C
D H H,
j
KO kh
La
R
concentration of a solute in mobile Dhase at column outlet = effective diffusion coefficient in mobile phase = height equivalent per theoretical plate for a solute = contribution of resistance to mass transfer in mobile and stationary phases t o H when radial permeability fluctuations are negligible = mass flux of a solute = effective rate constant independent of nonideal mechanisms = ratio of amount of solute sorbed by stationary phase, at equilibrium, t o amount of solute in mobile phase, when mole fraction of solute in mobile phase, A- = 1112 = minimum column length required for formation of stationary fronts = front resolution of a solute pair =
x=o
value of u when mole fraction of more strongly retarded component in a stationary front is equal to half its mole fraction immediately behind front = progression velocity of mole fraction X on a front = width of a nonstationary front at column outlet, in units of volume = width of a nonstationary front at column outlet, in units of length = width of a stationary front neglecting radial permeability fluctuations, in units of volume = width of a stationary front neglecting radial permeability fluctuations, in units of time = width of a stationary front neglecting radial permeability fluctuations, in units of length = value of w : according to Glueckauf and Coates (9) = value of w: according to Schay (18) = value of wts corrected for theoretical approxima.. tions X, = mole fraction of solute in contact with surface of sorbent = mole fraction of solute a t distance yI in a flat, stagnant layer of mobile phase zI,zII = arbitrary points in column = value of 2 of point on a front z a t which concentration of more strongly retarded component within front is half its maximum value immediately behind front Y = product of X and ratio of amount of solute in stationary phase to that in mobile phase = distance measured in direcYr tion of sorbent from interface between a central ’ region of mobile phase, in which there is radial equilibrium, and a stagnant region of mobile phase, in contact with sorbent, in which there is a resistance to mass transfer = thickness of a flat, stagnant Y. region of mobile phase CY’ = relative retention of a solute pair, calculated from ratio of mass distribution coefficients corresponding t o their inlet mole fractions I.( = convenient parameter, Equation 28 V = convenient variable, Equation 20 =
(7 9)
flow velocity of mobile phase value of u in a region where
x,
Table 1. Comparison of Equations 75 and 78 with (the More Accurate) Equaiions 76 and 77
[Standard deviations corresponding to various front widths used in calculations (see Equation 13)] Linear Isotherm, Sorption Effects Important 0.25 0.50 0.75 1.00
5.58 2.39 1.33 0.80
1.22 1.18 1.05 0.64
5.47 2.13 0.85 0
h’onlinear Isotherm, Sorption Effects Negligible
1.0 3.0 5.0 7.0
2.13 0.85 0.55 0.41
2.39 1.33 1.12 1.03
= arbitrary value of
Y’
-1.18 -1.0j -0.95 -0.89
Y
integration constant, Equation 30 fJZ = standard deviation of a nonstationary front at column outlet, in units of length uts = standard deviation of a stationary front neglecting radial permeability fluctuations, in units of time ( u t S ) ~ c= value of ut according to Glueckauf and Coates (9) (utS)s = value of u t S according to Schay (18) (utJ). = value of u t s corrected for theoretical approximations w = convenient parameter, Equation 50 =
VI
ACKNOWLEDGMENT
G. J. Krige is indebted to the Director General of the Atomic Energy Board for permission to participate in this project. LITERATURE CITED
(1) Bosanquet, C. H., “Gas Chromatography 1958,” D. H. Desty, ed., p. 107, Butterworths, London, 1958. (2) Bosanquet, C. H., Morgan, G. O.,
“Vapour
Phase
Chromatography,”
D. H. Desty, ed., p. 35, Butterworths,
London, 1957. (3) Fejes, P., Nagy, F., Schay, G., Acta Cham. Acad. Sea. Hung. 20, 451 (1959). (4)
Fejes, P., Schay, G., Ibzd., 17, 377
(1958). (5) Giddings, J. C., “Chromatography,”
E. Heftmann, ed., Chap. 3, Keinhold,
New York, 1961. (6) Giddings, J. C., J . Chromatog. 5 , 46 (1961). (7) Glueckauf, E., J . Chem. SOC. 1947, 1321. (8) Glueckauf, E., Trans. Faraday SOC. 51, 1540 (1955). (9) Glueckauf, E., Coates, J. I., J. Chem. Soc. 1947, 1315.
VOL. 37, NO. 10, SEPTEMBER 1965
1201
>r. .T.
fln’i Gnlav.
E.. (‘Gas Chroma-
Cniversity of Pretoria, Pretoria, 1962. (12) Haarhoff, P. C., Pretorius, V., J. 8. Afracan Chem. Inst. 14,22 (1961).
(13) Klamer, K., Krevelen, van, D. W., Chem. Eng. Sci. 7,197 (19%). (14) Krige, G. J., Pretorius, V., ANAL. CHEM.37,1186 (1963.
1151 Ibid..
D.
Acta Chim. Acad. Sci. Huna. 22. 285
1191.
Ashley, J. W., Jr., ANAL 1198 (1962). (18) Schay, G., “Theoretische Grundlagen der Gaschromatographie,” Chap. 4, VEB Deutscher Verlag der Wissenschaften, Berlin, 1961. (19) Schay, G., Petho, A., Fejes, P.,
“Physicd Adsorption of Gases,” Chap. 1, Butterworths, London, 1962. (23) Ibid., p. 373. RECEIVEDFebruary 23, 1965. Accepted July 1, 1965.
Frontal Analysis Chromrtography Experimental Study of Stationary Fronts on Columns with Negligible Radial Permeubility Fluctuations G. J. KRIGE’ and VICTOR PRETORIUS Department of Physical and Theoretical Chemistry, University o f Pretoria, Pretoria, South Africa
b The existence of stationary fronts in frontal analysis at high sample inlet concentrations is experimentally demonstrated. Continuous introduction of samples of both pure methane and a mixture of methane and carbon dioxide, into a column packed with activated carbon, i s considered. A clear indication is given of the sources of error arising from limitations in the experimental technique, and their effect on the values of the observed front widths is estimated. The results are correlated with the theoretical expressions derived in an earlier paper by using an equation for the plate height in gas-solid chromatography which takes microporous diffusion in the adsorbent into account. Good agreement with existing data in the literature is obtained.
A
the assumption that stationary fronts are formed has been employed to determine distribution isotherms ( 7 ) , no attempt has been made to prove that the front width is actually independent of the column length. Furthermore, the interpretation of experimental front widths in terms of theoretical expressions for the effect of nonideality has attracted very little attention (2$), possibly because existing theories were too specific to be applied under the conditions normally encountered in practice. However, in a previous paper ( I S ) general theoretical expressions were derived for the widths of the stationary fronts formed when either a single solute or a binary solute mixture is continuously injected into a LTHOUGH
1 Present address, Department of Chemistry, University of British Vancouver, Canada. Permaner Atomic Energy Board, Pelindabs . of South Africa.
1202
ANALYTICAL CHEMISTRY
chromatographic column, assuming that radial variations in the permeability of the column packing are negligible. As a result, a meaningful experimental investigation of the phenomenon of stationary fronts is now possible. In the investigation described the existence of stationary fronts is experimentally demonstrated on analytical scale columns. The mechanisms which determine the widths of such fronts are then analyzed in terms of the theory developed in a previous paper ( I S ) . The verification of theoretical predictions regarding the basic nature of processes, rather than exact agreement between theory and experiment, is stressed.
The general expressions for the widths of stationary fronts (IS) are written in the form which they assume when the sorbates exhibit Langmuir type sorption isotherms on the sorbent. This behavior has been observed for the methane- and carbon dioxide-activated carbon systems (12). Single, Undiluted Solute. The width of the stationary front formed when a single solute is continuously injected into a column is ( I S ) :
y
-
For a Langmuir type isotherm ( l a ) ,
k
--
1
+ bXp
-
pabQJnT
tT,(1
+ bXp)
A,.-A
dX.
’
V/
l+bAp
(2)
=
UO 1 + ki
Equations 1, 3, and 4, with Xi yield Wt’
THEORY
kX=--
The subsequent discussion is further facilitated by expressing the progression velocity of the front in terms of the flow velocity through the column once the last front has emerged, uo. This is achieved by using the relations derived for the rate of flow during frontal analysis (11). For the situation under consideration, the mass balance equations a t the first front of a binary solute mixture are used, t o obtain
+
=
2(1 kh(2
=
At*
C,h
+ bls)
(4) =
1,
+
+ Bt*/u02
for the width of the stationary front. Binary, Undiluted Solute Mixture.
I n this case the region between the first and second fronts contains pure solute 1 (11), so that the front width is described by Equation 1. The progression velocity of the front at the
(2 - x.) moment the front breaks through may be obtained from a n existing expression ( 1 1 ) by setting sa = L:
ax
II
Uolli
