Activity Coefficient for Gas Chromatography Influence of the Molecular Weight of the Stationary Phase on Gas Chromat ographic Data D. F. Fritz and E. sz. Kovats Laboratoire de Chimie technique de I’Ecole Polytechnique FBdBrale, Lausanne, Switzerland
Henry’s coefficient and the activity coefficient defined in terms of mole fractions are shown to be inadequate for the interpretation of gas chromatographic data. Two equivalent parameters are proposed to replace them: gr(SP) and pl(SP), the first one being the hypothetical partial pressure over a one molal solution of a substance in a given solvent (in our particular case, the stationary phase SP), the second being an activity coefficient which is the ratio of g1(SP) of a given substance in a solvent SP to that of the same substance in a standard solvent, gl(SSP) (see Equations 3 and 9). The former is named Henry’s molal coefficient, the latter the p-activity coefficient. I n the second part, usefulness of these coefficients for theoretical considerations is demonstrated. With the aid of these two new parameters and the relationships developed by Flory and Huggins, equations are derived which show the dependence of gas chromatographic data on the molecular weight of the stationary phase.
similar units]. Therefore, hl(SP) must decrease with increasing molecular weight and reach a value of zero for a hypothetical solvent of MSP = m . A knowledge of Henry’s coefficient, therefore, does not give any direct information concerning the specific retention volume. For example: the value of h of a given substance in a polymer of MSP = 20000 is one-half that of the same substance in a polymer of the same structure but of MSP = 10000; however, its specific retention volume is the same on both phases. This point is also illustrated by the published Vg-values of nonpolar substances on polyethyleneglycols of different molecular weight in Ref. (2). Consequently, differences in hvalues do not necessarily mean differences in gas chromatographic behavior and vice versa, showing that Henry’s coefficient is unsuitable for interpretation and presentation of gas chromatographic data. Henry’s molal coefficient gl(SP), on the other hand, as defined by Equation 3, PI = g,(SP)m,
(m,
-
0)
(3)
The partial pressure of a solute over an ideal dilute solution is given by Henry’s relation
does not show such disadvantages. Concerning the thermodynamic statement involved, Equations 1 and 3 are equivalent. The relation between mole fraction and molality is given by
where hl(SP) is Henry’s coefficient of substance 1, dissolved in the stationary phase, and nl is its mole fraction in the solution. Suppose the validity of the theoretical treatment of James and Martin ( I ) , hl(SP) can easily be calculated from gas chromatographic data if the molecular weight of the stationary phase, MSP,is known:
m, 1000 m, M SP which is simplified for m l 0, i.e., a t ideal dilution to
nrn
P.Martin, Biochern. J . , 50, 679
(1952).
=
-
m,
where T, is the column temperature and Vg,l(SP) the specific retention volume of substance 1. Experience teaches that in homologous solvents of molecular weights over 10000, the quotient pl(SP)/cl(SP) is nearly constant and does not depend any more on the molecular weight [cl(SP) is the concentration of the solute in grams 1.-1 or (1) A. T. James and A. J.
x1
1000
= -X I MSP
+-
(x,
-
0)
Combination of Equation 5 with 1and 3 gives
which shows that the pressure gl(SP) is proportional to Henry’s coefficient for a solvent of given molecular weight. (2) W. 0. McReynolds, “Gas Chromatographic Retention Data,” Preston Technical Abstracts Company, Evanston, Ill. 1966.
A N A L Y T I C A L C H E M I S T R Y , VOL. 45, NO. 7, J U N E 1973
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In fact in the latest Manual of Symbols of IUPAC [page 39, Ref. (3)] the use of (Al/Ml)C is advanced (where X is the absolute activity) in defining a n activity coefficient of a solute substance in a solvent. This quotient is identical with our Henry's molal coefficient, gl(SP). Contrary to the behavior of Henry's coefficient, hl(SP), the value of the Henry's molal coefficient tends t o a limiting value with increasing molecular weight of t h e solvent. Furthermore, it is directly related to the specific retention volume and can be calculated from this without the knowledge of the molecular weight of the stationary phase, as shown by combination of Equations 2 and 6: *g,(SP) =
RT, lOOOV, ,(SP)
7)
Also gl(SP) can be visualized as the partial pressure of the substance in question over a one-molal solution assuming that the proportionality expressed in Equation 3 would be valid up to this concentration. Not only Henry's coefficient but also the activity coefficient of usual definition is of little value, if a t all, for interpretation purposes. The activity coefficient of current definition compares the partial pressure of a substance over a real solution with that over a so-called ideal mixture. Its limiting value for ideal dilute solution is
(An analogous expression could serve for the pressure gl(SP):rl(SP) = gl(SP)/pl", where rl(SP) = (MsP/ lOOO)f1'(SP).) In fact, such a function compares Henry's coefficient with the partial pressure of the pure substance at the same temperature, Le., the pure substance serves as standard state. In the domain of the validity of Henry's law, the molecules of the substance are surrounded by molecules of the solvent only. In the chosen standard state, the molecule is in the field created by molecules of its own kind. This means t h a t activity coefficients of a series of substances determined in the same solvent compare interaction' forces of a n environment which remains the same from measure to measure, with those acting on the molecule in the pure substance, i . e . , a different environment for each substance. An activity coefficient as defined by Equation 9
compares the Henry's molal coefficient in a given stationary phase with that of the same substance in a standard stationary phase (SSP).Consequently, it is a very convenient function in the study of differences between interaction forces acting on a molecule in two defined environments. An analogous activity coefficient defined with the aid of Henry's coefficient allows the use of published data:
With the aid of the Henry's molal coefficient, or the related chemical potential, the interpretation of gas chromatographic data would be possible if the entropies of mixing with the stationary phase were comparable for all substances. A proportionality between excess enthalpy and entropy is a good and logical assumption except for the entropy term due to the great differences between the molar volume of the solutes and that of the solvents used in gas chromatography. Therefore. the knowledge of the magnitude of this entropy part is absolutely necessary not only, as will be seen, for a proper interpretation of gas chromatographic data, but also for the choice of standard stationary phases. The arguments of the following chapters are based on a n equation derived by Flory (4) and Huggins (5) which gives the entropy of the components in a binary mixture of two substances of different molecular sizes but with zero heat of mixing. Thereby the usefulness of both the concept of the molal Henry's coefficient and the p-activity coefficient will be demonstrated. As will be seen, the effect of the molecular weight of the stationary phase on the gas chromatographic behavior is surprisingly high. Experimental d a t a confirming this point will be presented in subsequent communications. I N F L U E N C E O F T H E MOLECULAR WEIGHT O F
THE SOLVENT O N S O L U T I O N P A R A M E T E R S A T IDEAL D I L U T I O N The p-Activity Coefficient of Substances i n Solvents of the S a m e S t r u c t u r e but with Different Molecular Weights in the Light of the Model of Flory a n d Huggins. Flory (4) and Huggins (5) have shown that activity coefficients of the type f1' of substances which dissolve in solvents of high molecular weight without heat of mixing are fairly well described by (13)
where VI and Vsp are the molar volumes of the substance and the stationary phase, respectively. Substituting Equation 13 into 11 and assuming that the partial molar volume of the substance, VI, is the same in both stationary phases, we obtain
P,(SP) =
The molar volume of a substance can be calculated from its specific volume, u1, In the following, a few useful relationships between the specific retention volume and the activity coefficients are compiled in Equations 11 and 12.
(3) "Manual of Symbols and Terminology for Physico-chemical Quantities and Units" IUPAC Publications, M. L. McGlashan. Ed., Butterworth, London, 1970.
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A N A L Y T I C A L C H E M I S T R Y , VOL. 45, NO. 7, JUNE
1973
v,
=
M1ul
(15)
Substituting this into Equation 14, we obtain for the pactivity coefficient
(4) P. Flory,d. Chem. Phys.. 9 , 660 (1941); ibid.. 10, 51 (1942). (5) M . L. Huggins, J . Phys. Chem.. 9 , 440 (1941).
We presume that this relationship must also be valid if the substance shows a nonzero heat of mixing on dissolving in SP and/or SSP providing the excess enthalpy is the same in both solvents. This can also be seen by using, instead of Equation 13, the "extended Flory-Huggins equation" for solutions with nonzero enthalpy of mixing [see page 294, Ref. fs)]. If the excess enthalpy is the same in the solvents in question, this treatment also leads to Equation 16. Obviously for our purpose, the condition of "zero heat of mixing" must be replaced by "equal heats of mixing in SP and in SSP." Expressed in a different way, there must not be any temperature change if a substance passes from its ideal dilute solution in SSP into SP forming a n ideal dilute solution in the latter also, the two solvents being kept in thermal contact. This condition is fairly well satisfied for e u e p substance, if t h e soluents are members of a homoiogous series, i.e., their chemical structure is the same, but differing in molecular weight. For polymers, the effect of the end groups must be negligible. This is the case at higher molecular weights. For the lower homologs, the end groups must be blocked t o form functional units similar to those found in the chain. The equation giving the p-activity coefficient for a series of homologs will be simplified by choosing a hypothetical homolog of infinite molecular weight as a standard within the group. Equation 16 will then give 17) and for the pressure gl(SP),we get
As a first approximation. gl(SP) is at constant temperature a n exponential function of 1/Mh.p since the specific volumes of higher homologs are nearly equal. A better approximation can be obtained by the use of Equation 19. The specific volume of homologs can be given with a reasonable precision as
(19) where is the specific volume of a homolog with infinite molecular weight, MI is the molecular weight. The constant lz can be calculated by a simple linear regression, and its value is different for each homologous series. Substituting Equation 19 into 18, we obtain for the dependence of the pressure gl(SP) on the molecular weight of the solvent:
This formula can be simplified by introducing the variable f = 1/(MsrJ+ k s p ) r
T I
-.
Obviously the approximation (1 - h h ~ f )= e x p [ - h ~ { ] is justified since the value of f is small even for stationary phases of lower molecular weight so t h a t
The constants ~ S and P V S P depend on the structure of the stationary liquid but are, of course, independent of the molecular weight. The equations derived describe a t least qualitatively the behavior of the specific retention volume of a substance. i.e., it decreases on homologous stationary phases of increasing molecular weight and approaches a finite limiting value. Equation 22 allows a rough estimation of the molecular weight of a stationary phase which is necessary to obtain a specific retention volume differing not more than 2% of its limiting value. Supposing that l z s ~is small, and the specific volume of the substance is equal to that of the stationary phase, one obtains
and ln(1 - 0.02) so that
=
-0.02
MSP = 50
-(M,';M~~)
M,
Hence for a substance of a molecular weight of 300, a stationary phase of a molecular weight of 15000 should be used. The Standard Chemical Potential of a Substance in an Ideal Dilute Solution and the Molecular Weight of the Solvent. The standard chemical potential of a substance in ideal dilute solution depends not only on the standard state chosen but it is also a function of the variables which characterize the equilibrium between solution and gas phase. It is evident that the most practical standard state to be chosen for the substance is t h a t a t low pressure in the gas phase where fugacity and pressure are equal because there are no interaction forces between molecules and environment. Usually the unit chosen to characterize the concentration in solution is the mole fraction so that the chemical potential difference becomes l p l + , L = RT In h l . By the nature of this definition, l f i l + , i of a substance becomes more negative in a solution going from a low to a high homolog of the solvent suggesting the false impression that it is a t a more stable energy level in the higher homolog. This chemical potential even reaches a value of minus infinity in the hypothetical solvent of infinite molecular weight. The expression RT In gl is also a chemical potential difference where the equilibrium between gas phase and solution is characterized by the variables molality and partial pressure ( i n units of a t m ) . It is obvious that this characteristic value does not show the disadvantages discussed above. In order to avoid any confusion this chemical potential difference will be labeled by a n asterisk ( & L I * . ~ ) . The function 1 ~ 1 ' 9 ~ is known to be nearly linear in a wide temperature range, l f i l * . L must show the same property. T h e linear dependence can also be expected in the hypothetical stationary phase of infinite molecular weight. and so ooA,u,*~'(SP) = R T lnmg,
The relationship with gas chromatographic d a t a is then given by
2
=
a + bT +
cT'
(24)
where the term cT2 gives only a small correction. The combination of Equation 18 with 24 gives
Al,*.'(SP) = RT In g,(SP)
( 6 ) J . M. Prausnitz. "Molecular Thermodynamics of Fluid-Phase Equilibria." Prentice-Hall. Englewood Cliffs. N.J., 1969. A N A L Y T I C A L C H E M I S T R Y , V O L . 45, N O . 7. JUNE 1973
1177
Using Approximation 19 for the specific volume of the stationary phase and accepting the substitution and approximation on deriving Equation 22, we obtain by combination of Equation 22 with 24 R T lng,(SP)
= a
+
bT - R kSP +
(
(1 + $7)
L ' , ~ - ~ ~ = ~ ~ 1.155 : + ~
for T = 20°C (27)
=
CY,
CYZ
+ (PO- R14.027~'S)T
With the aid of Equation 35, we can easily calculate the chemical potential difference between two n-paraffins with z i and z carbon numbers: 1.155 R T ln(gZ+i/gZ) = ia + i ( p - R 1 4 . 0 2 7 7 05" (36)
+
L' SP
where the numerical values of and k11 have been calculated with the data published in Ref. (7). Substitution of Equation 27 into 15 gives the molar volume of n-paraffins 1.155 (M, + 31.7) for T = 2OoC Combination of Equation 28 with 22 then results in
V,
RT lng,(SP)
f T icT2 (26)
Influence of'the Molecular Weight of the Stationary Phase on the Gas Chromatographic Data of Paraffinic Hydrocarbons. The specific volume of a n-paraffin of the elementary formula CzHzZ+2is given by clI =
Comparing Equation 24 with 34 and using 32, we obtain the formula which gives the chemical potential, &L~*.L(SP), as a function of 2, T, and {:
=
(28)
Finally Equations 35 and 36 become somewhat simpler for the chromatographic behavior of n-paraffins on paraffinic P K I I = 31.7 and - U S P stationary phases. In this case ~ S = = giving m~~~
V I + -(Mz + hlI)]
-j-[kll
TI1
= -c(2hII
+ M:) = ~((63.4 +
M2)
(37)
and also as a function of the carbon number where
&'I = hsp
+ '=31.7
= h,,
=usp
-((63.4
+-, 36.6
(30)
L'SP
The molecular weight of a n-paraffin can be calculated if the number of carbon atoms, 2, is known
M, = (14.0272 + 2.016)
= 14.027 (2
+ 0.144) (31)
Using Equation 31 together with 29 gives Henry's molal coefficient as function of the molecular weight of the stationary phase and the carbon number of the n-paraffin
+
14.0272 + 2.016) =
(14.027
( K 11
+
(38) where the value of K ~ ,is 4.66 carbon number units. By using Equation 38, Equation 36 becomes
R T ln(g,+Jg,)
=
;[a
+ (/3
-
- R14.02707'1
2)
(39)
Influence of the Molecular Weight of the Stationary Phase on the Retention Index. The retention index of a substance is defined (8, 9) as:
By simple substitutions in Equation 40, it can easily be seen t h a t (10, 11): where K'
=
ksP -
+
14.027
1.155(31.7 0.144) "Usp 14.027 = 0.0713kSp
+7 2622 (33) USP
In the preceding formulas, 1.155 has been used for This value, valid a t 20"C, is obviously temperature dependent. However, occurs in all the equations as quotient in connection with another specific volume. Knowing that the coefficient of thermal expansion is about the same for all organic substances (-0.001 degree-I), the quotient should remain constant over a considerable temperature range. It is a well known fact that the logarithm of the retention volume of normal paraffins is a linear function of the carbon number, z, if 2 > 5 [See Ref. (8) and the bibliography discussed therein]. The same must be also true on the hypothetical stationary phase with infinite molecular weight so that for the homologous series of a-paraffins, Equation 24 can be re-written in the following form. aoApz*s'(SP)= RT InCDg,(SP) =
a,
+
a2
+ POT + PzT +
Cl,T2
I>+,(SP)=
6 - R14.027+l5j5{) USP T
(42)
+ 1002 = 100 ( 2 + i)
Relationship 42 is easy to interpret: if the molar volume of a substance is equal to that of a hypothetical n-paraffin of the elementary formula C l + z H 2 ~ L + r ~ -its 2 , retention index is independent of the molecular weight of the stationary phase. Isoparaffins satisfy this condition approximately. In order to find the influence of the molecular weight of the stationary phase on the retention index of a substance other than a hypothetical n-paraffin, let us first calculate its retention index on the stationary phase of infinite molecular weight:
(34)
(7) National Bureau of Standards. Selected Values of Properties of Hydrocarbons, AP1 Res. Proj. 44. (8) A. Wehrli and E. sz. Kovats. Helv. Chim. Acta. 42, 2709 (1959).
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Let us suppose that a hypothetical n-paraffin of z + i carbon number is chromatographed where 0 < i < 1, then by combination of Equation 36 and 41, we get
A N A L Y T I C A L CHEMISTRY, V O L . 45, N O . 7, J U N E 1973
(9) E. sz. Kovats, Heiv. Chim. Acta. 41, 1915 (1958). (10) E. sz. Kovats, Advan. Chromatogr.. 3, 229 (1965). (11) P. B. Weiss and E. sz. Kovats, Ber. Bunsenges. Phys. Chem.. 69, 812 (1965).
or, by using Equation 24
+
1002 (44)
Let us suppose for simplicity t h a t the coefficients a1 and b l are proportional to the coefficients a, and b,; furthermore t h a t the coefficients c1 and C, are equal, meaning t h a t the indexmI1(SP) is independent of the temperature. In this case, Equation 44 can be written in the following way
"I,(SP) = 100
=i,a CY
+ 5,pT + 1002 = l O O ( 2 + "i,) PT
(45)
+
graphic data, due to the molecular size of the volatile solutes and the stationary phases as well. The formulas obtained can serve as a guide in evaluating experimental data in order to determine the exact magnitude of the effect of the molecular weight of the stationary phase on the retention index and on similar characteristic values. It is evident t h a t the reproducibility of the molecular weight of a polymer influences the reproducibility of gas chromatographic data. With the necessary information a t hand, we will be able to give the desired precision for the synthesis of a polymer which should be used as standard stationary phase. Furthermore, it seems that for theoretical purposes the most convenient data would be those converted to hypothetical solvents of infinite molecular weight. Our results let us hope t h a t adequate terms, which are necessary for correcting data obtained on real solvents, can be calculated once they are known in detail on well defined series of solvents.
The retention index on an SP of finite molecular weight can be given by using Equation 26
Z,(SP) = 100 CY
+ PT
L155
- RT 1 4 . 0 2USP 77l
+ 1002 (46)
The linear term in {in the numerator can be re-written as
-RT('~
-Vz+"i ~
USP
+ Vzt-i - V,){ mUSP
-RT7j-AV,
-"ilRT
1155 14.027 [ WSP
USP
The combination of Equation 46 and 47 gives
RT-AVl %P
I,(SP) = "I,(SP) - 100 CY
+ PT - RT 1 4 . 0 21157 57 c USP
(48) Since the negative term in the denominator is smaller 1 - 6 is than ( a + PT), the approximation (1 certainly justified:
+
I,(SP) = "I,(SP) - 100-
CY
RT AVl PT
+
c'
Equation 49 is actually of the simple form
The quadratic term in { is small amounting to about 5% of the linear term for a stationary phase of M S P 400. It will even be smaller on a stationary phase of higher molecular weight. In conclusion, with the aid of the relationships derived, we can account for a contribution affecting gas chromato-
NOMENCLATURE Symbols in Alphabetic Order: a(Kca1 mole-l), b(Kca1 mole-1 degree-I), and c(Kca1 mole-1 degree-*): constants of the quadratic equation of the chemical potential of the substance in a hypothetical solvent of M = a ;c(g 1.-1) or similar units: concentration; cyo and 01 (Kcal mole-'), Po and P(Kca1 mole-1 degree-1): constants of the quadratic equation of the chemical potential of n-paraffins in a hypothetical solvent of M = as function of the carbon number; $(g ml-1): constant of Equation 50; f ( -): activity coefficient of Raoult's law; I$( -): activity coefficient of Equation 10; g (g a t m 103): Henry's molal coefficient; h (atm): Henry's coefficient; i (-): additional carbon number of paraffins; I (-): retention index; k (8) constant of Equation 21; k' (g) constant of Equation 31; K~~ ( - ) in units of carbon number: constant of Equation 31; 39; K' (-): constant of Equation 34; m (103 g-1): molality; M (9) molecular weight; p(Kca1 mole-l): chemical potential; IZ ( - ) : number of moles; p ( a t m ) : partial pressure; r (10-3 g): activity coefficient defined in brackets following Equation 8; R (Kcal mole-' degree-l) in equations of type 19 and (ml a t m mole-1 degree-1) in equations of type 7 : gas constant: p ( -): the p-activity coefficient; ?(degree): absolute temperature; V (ml): molar volume; V, (ml): specific retention volume; u (ml g-1): specific volume; x ( - ) mole fraction; z ( - ) carbon number of paraffins; f (g-1): variable defined in Equation 23. Logic of Symbols: a superscript right or a symbol in parentheses means "in"; a subscript means "of"; finally the superscript left " m " refers to a hypothetical solvent of M = m. List of Symbols Used as Superscript or Subscript: i: ideal dilute solution; SP: stationary phase; SSP: standard stationary phase; 1,2 etc.: substance in general; z : paraffin with carbon number t ; 0: pure substance; I1 paraffins in general.
Received for review December 7 , 1972. Accepted January 29, 1973. The present work is part of a project supported by the "Fonds national suisse de la recherche scientifique."
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