Dependence of Solute Retention Parameters on Stationary Phase Molecular Weight in Gas-Liquid Chromatography Daniel E. Martire Department of Chemistry. Georgetown University, Washington, D.C. 20007
Prigogine’s treatment of chain molecule mixtures is used to derive a general theoretical equation for the infinite dilution molal solute activity coefficient (772“’) in nonelectrolytic solutions. From it, an expression is developed for n-alkane mixtures which relates 772- to the solute carbon number and solvent (stationary phase) molecular weight ( M I ) , and which yields values in excellent agreement with those determined through gas-liquid chromatography (GLC). The general equation is employed to examine in detail the dependence of 7 7 2 ~ (for n-alkane and other solutes) and the retention index (for two test solutes) on M I . The importance of the usually neglected structural contribution to solute retention parameters is demonstrated. Some generalizations and predictions are made. The possibility of utilizing GLC for accurate determination of the molecular weight of a polymeric stationary phase is investigated.
The worthwhile problem of obtaining and characterizing standard polymeric stationary phases for gas-liquid chromatography (GLC) has been one of the main concerns of Kovfits and coworkers for the past several years. Recently, Kovfits identified and treated an important aspect of this problem, i.e., the influence of the molecular weight of the stationary phase on gas-chromatographic retention data ( I , 2). He correctly pointed out that workers in the field often fail to separate energetic and chain length effects when discussing or analyzing the elution behavior of a given volatile solute on different stationary phases (solvents). In an attempt to account for the dependence of solute retention parameters on stationary phase chain length, Kovdts and colleagues utilized a solution model for the solute activity coefficient which considered only the combinatorial contribution that results from the molecular size difference between the solute and solvent ( I , 2 ) . However, largely because of the incompleteness of the model, it proved necessary for them to utilize extensive empiricism and curve fitting, and their final expression for the retention index led to physically unrealistic implications (see Table VI1 of ref. I ) . In this paper, the problem will be treated more completely and rigorously. Current developments in the theory of binary mixtures containing chain molecules will be utilized-namely, the average potential-corresponding states model of Prigogine and coworkers (3), as recently extended by Flory (4-7), Patterson (8-15), and us (16). It (1) G. A. Huberand E. sz. Kovats,AnaL Chem., 45, 1155 (1973). (2) D. F. Fritz and E. sz. Kovats. Anal. Chem.. 45, 1175 (1973). (3) I. Prigogine (with the collaboration of V . Mathot and A. Bellemans), “The Molecular Theory of Solutions, North Holland Publishing Co., Amsterdam, 1957, Chap. 16 and 17 (4) . , P. J. Florv. R . A . Orwoll. and A . Vrii. J , Amer. Chem. Soc.. 86, 3515 (1964). (5) P. J . Flory, J , Amer. Chem. Soc., 87, 1833 (1965). (6) R . A Orwoll and P. J. Flory. J. Amer. Chem. SOC.. 89, 6822 (1967). 17) P . J . Florv. Discuss. FaradavSoc.. 49. 7 11970). S. N Baitacharyya. D. Patierson, an’d T . Somcynsky, Physica. 30, 1276 (1964). (9) D. Patterson, Rubber Chem. Techno/.. 40, 1 (1967).
626
ANALYTICAL CHEMISTRY, VOL. 46, NO. 6, MAY 1974
will be shown that there are three important contributions to the solute activity coefficient: (a) combinatorial, (b) energetic, and (c) structural. The last contribution (also referred to as the “free volume” or “equation-of-state’’ contribution), neglected by Kovdts et al. ( I , 2 ) , is highly dependent on the chain length of the stationary phase. Explicit expressions will be obtained for mole fraction and molal based solute activity coefficients, allowing Kovdts results for the so-called p activity coefficients of n-alkane solutes to be derived directly. A more rigorous basis for differentiating between energetic and chain length effects on solute retention parameters (activity coefficient, specific retention volume and retention index) will be presented. Finally, the possibility of using GLC for quantitative determination of the molecular weight of the stationary phase ( I 7) will be discussed. Mole Fraction, Molal, and Weight Fraction Conventions. In GLC derivations relating the specific retention volume to the solute activity coefficient (mole fraction scale) it is usually assumed that the number of moles of solute (n2) is negligibly small compared to the number of moles of solvent ( n l ) . However, this assumption becomes untenable as the solvent molecular weight approaches infinity. Accordingly, let us consider a less restrictive derivation. Assuming ideal gas phase behavior, the partition coefficient ( K ) defined as the ratio of the solute concentration in the liquid phase ( c & ) to that in the gas phase (czg) is given by
where x 2 is the solute mole fraction in the liquid phase, V is the total liquid volume (approximately equal to VI), R is the gas constant, T is the column temperature. and p 2 is the solute partial pressure above solution. Taking the mole fraction convention, we have P? = Y2X?P?O
(2)
where y2 is the solute activity coefficient based on mole fraction and p2’ is the pure solute saturated vapor pressure a t T. The specific retention volume (V,”) is related to the partition coefficient by
where gl is the number of grams of solvent. Letting gl = nlM1, where M1 is the solvent (stationary phase) molecular weight, and combining Equations 1-3, one obtains D. Patterson. Macromoiecuies. 1, 279 (1969) D. Patterson, J . Poiyrner Scf.. Part C. 16, 3379 (1969). D. Patterson and J. M. Bardin, Trans. Faraday S o c . 66, 321 (1970). D. Patterson and G. Delmas. Discuss. Faraday S o c . 49, 98 ( 1970) D. Patterson, Y . B Tewari, and H . P Schreiber, J Chem. Soc Faraday Trans. 2. 68, 885 (1972) D.Patterson, Pure Appi. Chem.. 31, 133 (1972) G. M. Janini and D. E Martire. J. Chem. Soc , Faraday Trans. 2. in press. D. E. Martire and J H Purnell, Trans. Faraday Soc.. 62, 710 ( 1965)
V,o =
(4)
-
where x1 is the solvent mole fraction in solution. Under conditions where x1 1, 7 2 approaches y z r n (the infinite dilution solute activity coefficient), and Equation 4 yields the commonly used GLC equation. However, in the GLC experiment, n~ cannot be neglected with respect to nl when M 1 is very large, and Equation 4 applies in general. Nevertheless, it will be shown later that as long as the condition gl > > g2 is satisfied then xly2 = yzm,and 273.2R = M,y,"p," The awkwardness of utilizing the usual mole fraction convention for activities in GLC has been pointed out by Kovits et al. ( 1 , 2)--i.e.. as M1 becomes large, 72" approaches zero, and conceptual difficulties arise. However, since V,' is clearly finite for all M I , the product M1y2" must obviously be finite. This prompted Kovhts to advocate usage of a molal based activity coefficient (called 72 here), where
Vi
P? = q2m2P?' (6) and where m2 is the solute molality in solution. Since, in general, mz = 1000 xz/Mlxl, it follows from Equations 2 and 6 that
273.2R M2WZmPLO
where Mz is the solute molecular weight and u p is the infinite dilution solute activity coefficient based on the weight fraction convention (19, 20). y p and 72" from Prigogine's Solution Theory. The lattice theory (21, 22) and the more current theories (8-16) of chain molecule mixtures give the same general form for the mole fraction solute activity coefficient (yz), i.e.,
where x is the so-called "interaction parameter," rLrefers to the number of segments in molecule i and where the hard core volume fraction of component i, d,, is given by
Now, xlrl is proportional to glrl/ML,where g, is the number of grams of component i in solution and M , is the molecular weight of i. Further, rl/M1 and rz/M2 are of comparable magnitude (see later), while the usual case in elution GLC is that g1 >> g2. Therefore it is clear that xlrl >> x2r2 and that 41 = 1; leading to the following simplification of Equation 15:
(7) Thus,
or y2== q
In (y&)
1000 2
y
1
Also, from Equations 5 and 8, we have
(9) With most carrier gases (18), Equations 5 and 9 may be corrected for vapor phase nonideality by substituting the pure solute fugacity (fz") at Tforpz', where
and where B 2 2 is the second virial coefficient of the pure solute vapor at T. For comparison with the relations employed by Kovits ( I , 2 ) , recognizing that the Henry's law constant (H) is the product of the infinite dilution activity coefficient andp,", we have from Equations 5 and 9 273.2R H, = MlV,O and
H,
2732R 1000 V,"
___
where the subscripts x and m refer to the mole fraction and molal conventions, respectively. Note also that Kovhts' specific retention volumes are VJ quantities, where rn
For completeness and to acknowledge the fact that polymer chemists employing GLC were the first to recognize certain drawbacks with mole fraction based y p values (19, 20), it can be readily shown that (18) 0. E. Martire and L. 2. Pollara. "Advances in Chromatography," Vol 1 . J C. Giddings and R. A. Keller, Ed., Marcel Dekker, Inc., New York, N . Y , 1966, p 335. (19) D. Patterson, Y. B. Tewari, H. P. Schreiber, and J. E. Guillet, Macromoiecuies. 4, 356 (1971)
=
x 4-In? + (1 - ): r
-
= In yZrn
(18)
where yzrnis the infinite dilution (x1 1) solute activity coefficient. Hence, the replacement of 7 x 1 by y p in going from Equation 4 to Equation 5 is justified when gl >> gz. Note that the sum of the terms in r in Equation 18, the "combinatorial" or Flory-Huggins (21, 22) contribution, is purely entropic in nature. It becomes more negative as the ratio r1/rz increases and approaches minus infinity (Le., yp 0) as rl approaches infinity ( i . e . , as the solvent chain length becomes infinite). T o circumvent this problem, Kovits et al. (1, 2 ) wisely used the molal based solute activity coefficient (7zm), which, according to Equations 8 and 18, is given by
-
However, in treating the dependence of their activity coefficient terms on solvent molecular weight, Koviits et al. unfortunately neglected the important effect of the x term. It is now clear that in binary liquid mixtures involving chain molecules of different size, x is a composite of two contributions, energetic and structural (3-16). The latter contribution arises from the mixing of two pure components possessing different "free volumes," the lower molecular weight component (the solute) being in a more "expanded" state than the higher molecular weight component (the solvent). The greater the difference in the chain length between the solute and solvent, the more important this contribution becomes. Specifically, our recent extension (16) of Prigogine's average potential-corresponding states treatment of chain molecules ( 3 ) led to explicit theoretical expressions for yz,the excess enthalpy and the excess volume, which gave excellent agreement with the (20) F. H. Covitz and J. W . King, J . Polymer S o . . Par? A - 7 . 10, 689 (1972). (21) Y . B. Tewari. D. E. Martire, and J. P. Sheridan, J . Phys. Chem.. 74, 2345 (1970), and pertinent references therein. (22) Y. B. Tewari, J. P. Sheridan. and D. E. Martire. J. Phys. Chem.. 74, 3263 (1970), and pertinent references therein. A N A L Y T I C A L C H E M I S T R Y , V O L . 46, NO. 6, M A Y 1974
627
Table I. Physical Properties and Molecular Parameters for n-Alkane Solutes a t 80.0 "C Solute
a
-hz, cal/mole
CP,, cal/mole deg
n-Hexane n-Heptane n-Octane n-Nonane
0.0489 0 ,0320 0.0208 0.0124
6020 7180 8280 9500
12.5 14.2 15.9 18 .o
experimentally observed values for n-alkane mixtures. The expression derived for x was:
where hz and C p 2 ,respectively, the molar configurational enthalpy and heat capacity of the pure solute, are given by
h2 = -Ah,"ap' and
cp2 = cp2M. -C
+ RT
(21)
+R
(22) where Ahzvap is the pure solute molar enthalpy of vaporization at T, and Cp;iq and C,,gas are, respectively, the molar liquid and gaseous heat capacities of pure solute at T. Also PZCas
solvent chain length increases (see later). Hence, the increase in 12- with increasing stationary phase molecular weight is attributable both t o the combinatorial and structural terms, neither of which can be neglected with respect to the other. Evaluation of Solute Activity Coefficients in n-Alkane Mixtures. In what follows, only the pertinent information will be given. The finer details of the various equations used and a description of their derivation are available elsewhere (16). With n-alkane mixtures, it is valid (16) to utilize the geometric mean assumption for unlike interactions, i.e., c12* = v't11*t22*. This implies (3) that 20 = - 6 2 / 4 and, from Equations 18and 20,
The 6 values for the solutes n-hexane through n-nonane in higher n-alkane solvents are listed in Table I. It was found (16) that for n-alkane solvents containing at least twelve carbons (211 12) e l l * is a constant, independent of solvent chain length. This implies that, for a given n-alkane solute, 6 is constant for 21 1 12. Also, the following relation between the number of segments (rl) in an n-alkane and its carbon number (2,)was obtained: r , = 0.416
A+--] where c 1 1 * , c22*, and c12* refer to, respectively, the well depths of solvent-solvent, solute-solute, and solvent-solute potential energies of interaction per segment, and cl/rl is defined as one-third the total number of external degrees of freedom (translation, rotation of the molecule as a whole, and rotation about single bonds) per segment (3, 16). It is the second term on the rhs of Equation 20 which is referred to as the "structural" or ".free volume" term. Its magnitude is governed by the disparities between the solvent and solute in their external motional freedom and interaction energies per segment, as determined by the structure and reflected in the free volume (or expansion coefficient) of the pure substances. The first term on the rhs and Equation 20 is purely energetic. It is a measure of the difference between t12* and the arithmetic mean of (ell* + ~ 2 * )referred to a reference substance (t22*). It can give either a negative or positive contribution to x depending on whether t12* is, respectively, greater or less than the arithmetic mean value. (Note that h 2 is negative.) For a given solute component t22* is fixed, and within a homologous polymeric solvent series of sufficiently long chain length (negligible effect of end groups), both t12* and e l l * are approximately constant. Thus, for a given solute, the energetic term would be expected to vary only slightly with solvent molecular weight. This is essentially what was assumed by Kov6ts et al. (1, 2). However, the second term on the rhs of Equation 20, completely neglected by lattice theory (21, 22) and solubility parameter theory (21) and overlooked in the past by GLC workers, varies with X (6 assumed constant for a given solute, per the argument given above), whiih, as will be seen later, increases as the solvent molecular weight increases. Thus, the "structural" term gives an increasingly positive contribution to x as the solvent chain length increases. Also, the sum of the last two terms on the rhs of Equation 19 becomes increasingly less negative as the 628
A N A L Y T I C A L CHEMISTRY, VOL. 46, N O . 6, M A Y 1974
+ 0.5282,
(Z, >_ 3)
(27)
Further, h2 is determinable from Equation 21 and Dreisbach's compilation (23). Listed in Table I are the h 2 values for n-CeH14 though n-CgHzo at 80.0 "C. Previously (16), cl/rl as a function of r, was determined for n-alkanes by fitting reliable 7 2 " data (from GLC) using an equation of the form cL = d'r, e' (f'/rL). For the present purposes a two-parameter expression of the form c1 = dr, e, where d and e are constants, is both sufficient and physically reasonable (16, 24, 25). Accordingly, the aforementioned y p data were so analyzed using the plotting procedure described before (16), yielding the expression
+ +
+
Note that, with a given solute, as rl increases, (c/r)l decreases, hence, X (Equation 25) and the structural contribution to x (Equation 20) increase, as stated previously. The evaluation of C,, (Equation 22) at elevated temperatures (>40 "C) presents somewhat of a difficulty. The CPzl1q values are available only for n-C7H16 at 80.0 "C (26). However, the value of C,, as a function of T for nC7H16 and corresponding states arguments (16) can be readily utilized to obtain reliable C p , values for the solutes n-C6H14, n-CsHls, and n-CsHzo at 80.0 "C. These are listed in Table I. Finally, through Equations 25-28. and the data in Table I, yzm values were calculated at 80.0 "C for twelve n-alkane mixtures, where they are compared in Table I1 with the respective experimental quantities (21). On the average, the agreement between theory and experiment is fl.O%-i.e., within experimental error. Hence, Equation 26 not only incorporates all of the known contributions to solution nonideality, but it also provides a quantitative basis for analyzing 7 2 " values and other solute retention parameters (see later). The molal based infinite dilution activity coefficients (23) R. R . Dresibach. "Physical Properties of Chemical Compounds." Advances in Chemistry Series, ACS, Washington, D.C.: No 15, 1955; No. 22, 1959; No. 29, 1961 (24) J. Hijrnans, Physica. 27, 433 (1961). (25) J. Hijrnans and Th. Holiernan. Advan. Chem. Phys.. 16, 223 (1969). (26) J. F. Messerly, G. E. Guthrie. S . S. Todd, and H . L. Finke, J. Chem. Eng. Data. 12, 338 (1967).
Table IV. Values of the Solute Coefficients in Equations 33, 37, and 38
Table 11. Experimental (21) and Theoretical (in Parentheses) Values for n-Alkane Mixtures at 80.0 "C Solute
n-CsHir n-C?Hie n-CsHi8
n-CgHzo
n-Cdis?
n-CzdHaa
0.698 (0.702) 0.720 (0.725) 0.736 (0.750) 0,766 (0.777)
0.792 (0.792) 0.815 (0.812) 0.849 (0.834) 0.877 (0,856)
Solute
n-CseHw
0.633 (0.628) 0.650 (0.652) 0.679 (0.678) 0.716 (0.706)
n-C6H14 n-C7H16
n-CaHls n-C 9H20 Benzene
n-CjHiIC1 Table 111. Theoretical qam Valuesa and Smoothed Values from Equation 42, at 80.0 "C n-C~Hi~a ______ Solute
n-C6Hi4 n-C7H16 n-CaH18 n-CgH2o a
n-C8aH6?
a
B2'
-A?
- B?
C?
n2a
- B-.'
02
0.7690 0.6961 0.6251 0.5491 1.0815 0.8016
7.5911 8.3337 9.0515 9.8356 3.7295 4.9345
5.8205 7.3075 8.8720 10.7353 ...
0.767 0.878 0.980 1.092 , . . ...
201.65 221.38 240.44 261.27 99.07 131.08
29.41 32.36 35.07 38.03 ... ...
= 26.564 B2.
012 S
...
-GIB?.
qrm
proximate Equation 33 by
n-CasH~
-
Theor.
Eq. 42
Theor.
Eq. 42
Theor.
Eq.42
0.268 0.275 0.282 0.290
0.268 0.274 0.281 0.289
0.297 0.307 0.317 0.328
0.297 0.306 0.317 0.328
0.318 0.331 0.344 0.358
0.318 0.331 0.344 0.357
where r1 >> CYZ and C Y Z = -CZ/Bz. The CYZ values are given inTable IV. Furthermore, from Equations 30 and 37
Determined from Equation S and values in parentheses in Table 11.
( v p ) may now be determined with confidence for these same systems by utilizing the results in Table I1 (the theoretical values in parentheses) and Equation 8. The resulting v ~ = values are set out in Table 111. In order to derive Kovkts' ( I , 2) general n-alkane equations from this theoretical basis, the analysis must be carried several steps further. Combining Equations 8 and 26, we have In
+ (1 v2- = In ____ 1000 r1
):
RT4
Now, M I = 2.016 + 14.026 with Equation 27, gives
Ml = -9.034
- h,!? + CP -(A 21,
( r l 1 2)
(?
+ 26.564)
A , = -1.208
(39)
/32
(41)
E
(40)
Introducing Equations 39-41 into Equation 38:
(30)
and In (M1) rl = In
+ 0.0731Z2 B2' = -(82.75 + 19.79222) 12.29 + 285722
+ 6)2 (29)
2R
which, when combined
+ 26.564r1
where Bz' = 26.564 B Z and PZ = 26.564 CYZ + 9.034 (see Table IV).Finally, least-squares analysis of A z , Bz', and P Z leads to the following linear functions of solute carbon number (&):
(31)
For
2 , 2 12 ( i e . , r1 2 6.752), 0.3401 In 26.564 - _ _ r1
In
= (-1.208
19.792(4.181 + 2,) + 0.073122)- Ml + 2.857(4.302 +Z2)
where, to summarize, 21 > 22, 2 1 2 12, and ZZ 2 3. The goodness of fit of Equation 42 is illustrated in Table 111. Assuming now that Equation 42 applies generally to nalkane mixtures, the solvent chain length dependence of In vzm is demonstrated graphically in Figure 1 for representative n-alkane solutes at 80.0 "C. Kovats et al. ( I , 2) defined a normalized molal solute activity coefficient ( 0 2 ) as follows: (43)
Inserting Equations 25, 28 (for c 1 / r 1 ) and 32 into Equation 29, expanding in powers of l/rl, and collecting terms, one can write an equation which describes the dependence of qzm on the number of segments (rl) of the n-alkane solvent for a given n-alkane solute, ie.,
where pp 1 as M I -* m , and where qzm(M1) and vzm(m ) are the molal activity coefficients in solvents of molecular weight A41 and infinity, respectively, From Equations 38, 42, and 43:
where Az, B Z and C Z are the constants for a given solute, and where
which may be compared with Kovats' empirical expression (27):
+
In (34)
The values of Ap, Bz, and Cp, calculated using Table I and Equations 27 and 28, are listed in Table IV for n-hexane though n-nonane. For Z1 2 12, the contribution of the term C z / r 1 2 to In 02" is small. Hence, it is valid to ap-
p? =
+
-20.492(3.095 22) Mi 31.7
+
145)
Since Kovdts et al. ( I , 2) studied branched chain alkane stationary phases, one would not expect the agreement between Equations 44 and 45 to be perfect. Nevertheless, the plots of p z us. M1 shown in Figure 2 are remarkably similar to those given in Figure 3 of ref. 1 . To return to a main point of the present study, note that the increase in pz with increasing M I is governed primarly by Bz' (Equation 38) which is proportional to Bz (27) E. sz. Kovats. Laboratoire de Chemie technique de I'Ecole Polytechnique Federale. Lausanne. Switzerland, private communication, July 9, 1973. A N A L Y T I C A L C H E M I S T R Y , VOL. 46,
NO. 6,
M A Y 1974
629
I
*oo~
,OOt 500
//
-1.2-1.4-
t
1 .o
0.0
L
176 86
2.0
(M,+,Q-'
x
54
40
3.0'
loot
io3 30
24
c, c,
g - c
"- . .
-,.
20
2 , (for 2 , = 10)
Figure 1. In ~p (molal solute activity coefficient) vs. the variable (M, @2)-l X l o 3 for n-alkane mixtures at 80.0 "C (see Equation 42). Lines for five representative solutes shown. Indication of stationary phase carbon number (21) corresponding to (M1 & ) - I X lo3 for n-decane solute (2, = 10)
+
+
P2
LOA
0.8 I 0.9[ 0.71 0.61
0.5
1
c,
1
8
0.41 10
::!
0.31 0.210.lt :
0
.
0
0
2000
1000
>
3000
MI
.Figure 2. Normalized molal solute activity coefficient (p,) vs.
stationary phase molecular weight ( M I ) for n-alkane mixtures at 80.0 "C (see Equation 45). Five representative solutes shown (Equation 37). Referring to Equation 35, the combinatorial contribution to -B2 is r2 0.3401, while the structural contribution is the remainder. Thus, the fraction of Bz which is due to the combinatorial contribution is found by Equations 27 and 40 to be (0.756 + 0.528 &)/(3.115 + 0.745 &), from which one calculates that the structural term contributes anywhere from 50% ( Z z = 5 ) to 40% ( Z 2 = 14) of the total. This verifies the previously stated contention that both contributions must be considered. The same applies to the observed decrease in V g owith increasing M1 (see Figure 3), because, from Equations 43 and 44,
+
From the above and Equation 44, one obtains
M I ' = 4.024 630
+ 71.06Mz
(47)
ANALYTICAL CHEMISTRY, VOL. 46, NO. 6, MAY 1974
where MI' is the molecular weight of stationary phase necessary to obtain a V,' value differing by not more than 2% of the limiting value (2). With n-decane, e.g., MI' = 14,000. If, instead of a normal paraffin, one employed a polymeric stationary phase of the type X-(Y),-X, where Y is more "polar" and/or polarizable than methylene groups, one would expect e l l * to increase. Also, if, as is likely, Y contained double bonds or rings or other parts which diminish external motional freedom, c l / r l would be less than the value for an n-alkane solvent with the same rl. Thus, with a given n-alkane solute, X-(Y),-X would exhibit both a more positive 6 (Equation 24) and a more positive X (Equation 25) leading to a larger q p and smaller V,'. Furthermore, B2 would become more negative (due to an enhanced structural contribution), implying a greater dependence of p 2 (or Vgo)on stationary phase molecular weight. It would be interesting to test this prediction with reliable thermodynamic data from GLC. Retention Parameters of Solutes Other t h a n n-Alkanes. If we assume that other solutes follow the same corresponding states behavior as n-alkanes, then the general equations listed and derived thus far would be applicable. The only major difference would be that the geometric mean assumption for unlike interactions might not apply. Therefore, the unmodified expression for x (Equation 20) should be used, i e . , P / 4 should be replaced by -20 in Equations 26, 29, and 34. Activity coefficients ( 7 2 " ) have been determined for benzene (21) and n-pentyl chloride (22) in n - C 2 4 H 5 0 , nC30H62, and n-C36H74 a t 80.0 " C . These can be readily converted to q 2 m values (Equation 8 ) . At this point, rather than attempting to determine all the molecular parameters for these two test solutes, it will be assumed that 772" has the same f o r m (see preceding paragraph) as for n-alkane solutes-ie., that Equation 38 applies. Further, since only three data points are available, it will be assumed that & (which is small in any event) is zero. The values of A2 and Bz' determined through least-squares analysis of 72" as a linear function of M I - l are listed in Table IV. With these solute coefficients and Equation 38, 72- values are retrieved which agree with the measured values to within 0.5%. The retention index I, of componentj is given by
~~
690
I
\.
L
j750
n-CsHnC1
Benzene
650
I L-.L
1.o
M-' x
Solvent
Observeda
Calculatedb
Observeda
Calculated5
n-Cz4Hjo n-CaaHss n-CasHir n-C,H,
643.9 649 . 3 653 . 4
643.9 649.4 653.5 677.8
722.5 725.6 728.2
721.9 726 .O 728.9 749 0
...
2 .o
3.0
io3
Retention index (1.)) vs. reciprocal of n-alkane stationfor benzene ( a ) and nary phase molecular weight (MI-') pentylchloride ( b ) at 80.0 "C. Curves generated through Equations 38 and 50, using coefficients listed in Table I V and available f2' data (27, 22) Figure 4.
Table VI. Physical Properties a n d Molecular P a r a m e t e r s for Benzene a n d n-Pentylchloride at 80.0 "C Benzene
C
(cal/mole deg) h (cal/mole)
10 - 6650 -0.27 -0.031
C p
and with Equation 38. we have =
FI
100
F ,+ A , + ( B , ' / M l )- A - (B2'/bvl+ b' 1) + ( B , - m m )- A; - ( B :
+
+ 1002
m (50)
where F, = ln(fjo/fz") and F,,1 = ln(fi-~"/fz") are constants at a given T Listed in Table V are the observed I, values calculated through Equation 48 using experimental Vgo's(21, 22), and the I ] values generated through Equations 38 and 50 using the coefficients in Table IV and available f2" data (21, 22). Shown in Figure 4 is the generated plot of I , us. M1-l, which is nonlinear as expected. Over the range 20 5: Z , 5 m , I, varies from 639 to 678 for benzene, and from 718 to 749 for n-C5H11Cl. Taking the derivative of Equation 50 with respect to M I - l and examining the contribution of the various terms, it becomes apparent that the primary reason for the large increase in I, in going from Z1 = 20 to Z1 = m is the fact that -B,' is smaller than -Bz' (see Table IV). In other words, the solvent chain length dependence of 7," is less pronounced than that of vZm.To explain this behavior and to account quantitatively for the observed A , and Bj' values of benzene and n-C5H11Cl, the following model is introduced. The number of segments (r,) is determined from available physical properties (21-23) by Kreglewski's procedure (16) for finding hard core volumes (V,*), where V,* = 37.28 r, (16). The configurational enthalpy (h,) is calculated from Equation 21 and available data (23).One-third the number of external degrees of freedom (c,) is obviously 2.0 for benzene, and is assumed to be the same as nhexane's value (c = 2.756, as calculated through Equations 27 and 28) for n-C5H11C1. Now considering Equation 34 (with - ~ 5 ~ /replaced 4 by 20) and Equation 35 (where B 2 = B2'/26.564), it is apparent that the only remaining unknowns are 6, 0, and C ., Accordingly, values of 6 and 6 are found which produce agreement with the observed A , and R,' values and also give self-consistent C,, values as estimated from corresponding states treatment of the C,,
3.4 2.756 0.81 14 - 7560 -0.28 -0,025
0 .so
c/r
where z refers to the carbon number of the lower bracketing n-alkane solute. Replacing pzO by f 2 " (see Equation lo), Equations 9 and 48 give
n-CsHuC1
2.5 2 .o
r
I,
...
a Determined through Equation 48 and experimental Veo data (21, 2 2 ) . Calculated through Equations 38 and 50, using coefficients listed in Table I V and availablef?' data (21,22).
6401
0
~~
T a b l e V. Comparison of Observed and Calculated R e t e n t i o n Indices (I,)a t 80.0 "C
6 0
data of n-heptane (16). The findings are summarized in Table VI. Note that the necessary series expansion conditions ( 3 ) ,101 < 0.3 and 161 < 0.3, are satisfied. Accepting the numbers in Table VI a t face value, it becomes possible to rationalize matters. Comparing the -B2 value of benzene (j) with that of n-hexane ( z ) , the combinatorial and structural contributions are 2.84 and 0.89, respectively, for the former solute, and 3.92 and 3.67, respectively, for the latter solute. Note that both terms contribute to the observed difference in B2' values, the structural term more so. For benzene, the smaller combinatorial contribution is clearly due to a smaller 7-2 value, while the smaller structural contribution is due partly to a lower C ,, value, but mainly to a large negative 6 value. A 6 of -0.27 means that the segmental interaction energies of benzene and the n-alkane solvent are related by ell* = 0.73 ell*, while those of benzene (j) and n-hexane ( z ) may be compared through Equation 24, which yields e,,*
-=e::*
1 + 6; 1 6,
+
from which one obtains the estimate that el,* = 1.44 c Z Z * . Turning to A2 (Equation 34), a breakdown of the four contributing terms gives -2.71 + 1.00 0.58 0.05 for benzene, and -2.36 1.00 + 0.01 0.58 for n-hexane, where the first two, third and fourth terms are, respectively, the combinatorial, energetic, and structural contributions. Clearly, the more negative A2 value for benzene is mainly a result of its smaller r2 value. Also, for n-hexane the energetic term is negligibly small, since ell* := e l z * = tzz* (16), while for benzene the appreciable energetic term is due to a negative 0 value, which indicates that more energy is expended in breaking segmental 11 and j j interactions than is retrieved from segmental interactions. In fact, from Equations 23 and 24, we have
+
+
+
+
from which one obtains the estimate that elJ* = 0.834 t j j * . Thus, c j j * > ~ 1 j *> ell*. Very similar comparisons and findings prevail for nC5H11Cl(j) and n-heptane(z). A breakdown of -B2 gives ANALYTICAL CHEMISTRY, VOL. 46, NO. 6, M A Y 1974
631
Table VII. Dependence of vzm on Molecular Parameters Increasing
X + 6
r2 ED*
ell*
em*
cdrz cdr1 a
+:
112-
Positive Negative Positive Negative Positive Negative Positive Negative OL_--.---
increases; - : q13' decreases; i:72" niay increase or decrease.
2000
4000
6000
8000
M,
+
3.74 1.19 for the combinatorial and structural terms of n-C5H11C1, and 4.45 + 3.88 for the corresponding terms of n-heptane. Note a t this point that the dependence of I j on solvent molecular weight (Figure 4) is more pronounced with benzene than with n-C5H11C1 because of a greater difference between B , and B , for the former. With both solutes, the structural terms contribute more to this difference than the combinatorial terms. To continue, a comparison of n-C5HllCl and n-heptane through Equation 51 gives t j J * = 1.43 e***, and, as before, one can show that c 1 1 * = 0.72 and e l J * = 0.835 el,*. Thus, the polarizable benzene and the "polar" n-C5Hl1Cl seem to exhibit remarkably similar energetic behavior. While a reasonable molecular interpretation of solute retention quantities can be achieved in certain specific cases, such as the preceding ones, generalizations based on the current solution model are more difficult to make. However, given certain restrictive conditions, it is a t least possible to distinguish certain trends. Consider, for example, the general case of a solute (any solute) dissolved in a polymeric stationary phase of structure X-(Y),-X. To a reasonable approximation, corresponding states arguments (24, 25) indicate that (53) where s and t are constants and the other symbols are as defined before. With the above and Equations 23-25 and 29 (again replacing by 2 e), one obtains
I t is assumed that: (a) 0 5 0, (b) rl > r2, and (c) c2/r2 > cl/rl, which would be the most likely set of conditions encountered in GLC systems. Also, since Equation 54 is based on a Taylor series expansion (3, 16), the additional conditions, (d) (01 < 0.3, and (e) 161 < 0.3, must be met. Thus, highly dipolar and/or complexing compounds must be excluded from the current discussion. Generalizations based on the above conditions and Equation 54 are set out in Table VII, where the effect on 112" of increasing a given molecular parameter, while holding all the others constant, is indicated. Note that, since 6 may be either negative or positive, the sum (6 A) may be either negative or positive ( A assumed positive by condition (c) above). Hence. in some cases, it is not possible to ascertain whether qzm increases or decreases, because of competitive effects. For example, when (A + 6 ) is positive, an increse in c22* would concurrently decrease the structural contribution and increase the energetic contribution.
+
632
ANALYTICAL CHEMISTRY, VOL. 46, NO. 6, M A Y 1974
Figure 5. Relative error ( U M I I M , )V S . n-alkane stationary phase molecular weight ( M , )for three alternative methods of determining M , : ( a ) from ~ 2 a(Equation 5 5 ) , ( b ) from Rub (Equation 5 7 ) , and (c) from I, (Equation59)
Note that t * increases as dipolar and/or polarizable groups are added, and r2 increases as the solute molecule becomes larger. Also note that all nonlinear molecules possess a t least 6 external degrees of freedom (Le., 3c 1 6). Extra external degrees of freedom are gained by rotational motion about bonds (24, 25). Therefore, any structural change which diminishes this latter motional freedom (e.g., chain branching, unsaturation, rings, etc.) would act to decrease c/r. Similar manipulation and examination of Equation 35 would indicate that -Bz should increase as r2, e l l * or cz/rz increases, or as t 2 2 * decreases. An increase in -B2, of course, suggests a more pronounced dependence of I, on solvent chain length. Finally, note that an increase in temperature ( T ) should affect ~ pbut , should have negligible effect on -Bz (see Figure 5 of ref. 1 for confirmation of the latter). Determination of Stationary Phase Molecular Weight. It was once suggested (17) that GLC might provide a suitable method for the determination of polymer molecular weight in an intermediate range ( M I < 10,000). A reexamination of this possibility based on a more rigorous solution model is desirable. Three alternative methods will be considered and the accuracy of each will be analyzed. The "test polymers" will be n-alkanes a t 80.0 "C. (1) From Vg" or qZrnof a n n-Alkane Solute. Equations 3%41 relate 02" to the polymer molecular weight ( M I ) and the n-alkane solute carbon number ( 2 2 ) . By measuring V g o ,r/2m can be determined through Equation 9 (replacing p ~ by " f z " ) . Hence, M I may be determined. To assess the precision in the measurement of M I , statistical error analysis is utilized, giving
- [MI OII
-
+
+
2.857(4.302 ZJ]' 19.792(4.181 Z2)
u'nq2a
(55)
where u refers to the standard deviation. Making the estimate (21) that u(lnq2") = 0.007, oM1/M1 is determined with n-CloH2z as the solute probe. Figure 5 indicates that u . ~ 1 / M 1is a nearly linear function of M1 (see Equation 55) and that a relative error of 15% is reached a t about M1 = 5,950. ( 2 ) From Relative n-Alkane Retention. This approach has the advantage of not requiring the more troublesome measurement of absolute retention quantities. Taking nCsHls (a) and n-CloHzz ( b ) as the solute probes, the relative retention ( R a b )as a function of M I is given by In Rob = In [(VgO),/(VgO)J= 1n
( ~ b ~ / v ~ (56) ~ )
where, again, the 72"'s are given by Equations 38-41 with Z, = 8 and Z b = 10, and
+
- _ 19.792(4.181 Z,,) [ M , f 2.857(4.302 -t- Z , )I2}
'+f1
(")
where cr(lnRab) is estimated to be 0.001. The almost linear plot of C M ~ / us. M ~M I , as generated from Equation 57, is shown in Figure 5. Note that a relative error of 15% is reached at about M I = 5,800. (3) From Retention Index of Benzene. One finds that the retention index (I,) of benzene fits the following empirical function of stationary phase molecular weight: 14580 I , = 677.8 - ____ (Mi+ 93) from which one can derive that G\T
=
+
( M I 93)2 14580 r'
(59)
where we estimate u1, to be 0.3 unit. Illustrated in Figure 5 is the dependence of c r ~ l / M 1on M I (also nearly linear for M I > 1000) as deduced from Equation 59. This alternative method for determining M I seems to give the best precision (a relative error of 15% for M I = 7,150). However, it also requires utilizing several calibration solvents to obtain the working equation (Equation 58). It would appear that, if one is to seriously consider using GLC for the stated purpose, this last alternative is
the best of the three. For one thing, it involves only relative retention parameters. More important, it can be more readily adapted to studying other polymeric series. As mentioned before, compared to n-alkane solvents, other polymers should produce a more marked chain length dependence of I,, thus reducing the relative error in the measurement of MI. CONCLUSION Equations derived through refinement of Prigogine's theory of chain molecule mixtures have been successfully employed in interpreting and predicting the solute activity coefficient and its dependence, and that of the retention index, on the stationary phase molecular weight. The importance of the so-called structural contribution (heretofore neglected by GLC workers) to these retention quantities has been demonstrated. These theoretical developments are currently being applied to the problem of selectivity in gas and liquid chromatography and to the analysis of Vgovalues of a variety of monofunctional alkyl solutes in monofunctional n-alkyl solvents, where the solvent chain length is held approximately constant. ACKNOWLEDGMENT Helpful discussion with G. M. Janini and J . W. King are gratefully acknowledged.
Received for review September 5, 1973. Accepted January 10, 1974. This research was supported by a grant from the National Science Foundation.
Role of Nitric Oxide in Positive Reactant Ions in Plasma Chromatography Francis W. Karasek and Donald W. Denney Department of Chemistry. University of Waterloo, Waterloo, Ontario
Ions of the form (H20)nH+ and (H20)n,NOC have been observed among the positive reactant ions using a coupled plasma chromatograph/quadrupole mass spectrometer, but only limited work on identification of these ions in the mobility spectra has been reported. Addition of nitric oxide to the nitrogen carrier gas in the PC instrument results in a large increase in relative abundance of the reactant ion attributed to (H20)NO+ in the mobility spectrum, confirming its mobility. An increased abundance of the (H20)NO+ ion in the reactant ion group affects the product ion spectra of different classes of organic compounds, and its greater reactivity leads to increased sensitivity of the plasma chromatograph.
Functioning at atmospheric pressure, the plasma chromatograph first creates both positive and negative ions in a carrier gas using a nickel-63 beta source. The reactant ions undergo ion-molecule reactions with trace molecules injected into the carrier gas stream. The resultant product ions are separated in a coupled ion-drift spectrometer to
give positive and negative mobility spectra characteristic of the organic molecules involved and the reactant ions generated. Both the technique of plasma chromatography (PC) and instrumentation have been described previously (1-4).
The type, reactivity, and relative concentrations of the reactant ions generated are of basic importance in the PC technique. Although a number of limited studies of their identity have been made, further work to understand and explore their function is needed. Using a combined plasma chromatograph/quadrupole mass spectrometer, various positive ions have been identified as the reactant species, using either air or nitrogen as a carrier gas. These are (HzO)H+, (HzO)zH+, (Hz0)3H+, NO+, (Hz0)NOf and (HzO)zNO+ (5-7). Their mobility spectra are the same whether air or nitrogen is used as the carrier gas. (1) F. W. Karasek, Res./Deve/op., 21 (12), 25 (1970). (2) F. W. Karasek, W. D. Kilpatrick, and M . J. Cohen, Anal. Chem., 43, 1441 (1971). (3) F. W. Karasek, Inf. J. Environ. Anal. Chem., 2, 157 (1972). (4) F. W. Karasek, 0. S. iatone, and D. M. Kane, Anal. Chem., 45, 1210 (1973).
ANALYTICAL CHEMISTRY, VOL. 46, NO. 6, MAY 1 9 7 4
633
