Eugene E. Schrier State University of New York Binghomton, New York 13901
I
Intermolecular Association
I
by Vapor Pressure Osmometry
I
A physical chemistry experiment
The study of molecular association in solution has provided important thermodynamic information relating to hydrogen bond formation and other noncovalent interactions. The data derived from these investigations, while of interest in themselves, have found additional application in discussions of the stability of the secondary and tertiary structure of proteins and other biologically important macromolecules ( I $ ) . A previous experiment (3) described an examination of the concentration dependence of the molar polarization of two compounds containing the amide group. The compounds investigated were 2oxohexamethyleneimine (caprolactam), I, and Nmethylacetamide (NMA), 11,in benzene solution: 0 H
L A\
/
cH,
c H L N  Ic n ,
The molar polarization, P*M,of each of the solutes in benzene exhibited contrasting trends with increasing solute concentration. The value of P2Nfor NMA increased with increasing solute concentration while this quantity showed the opposite behavior for caprolactam. These trends were interpreted (3) by assuming that an extended series of multimers is formed with NMA while only monomers and dimers exist in benzene solutions of caprolactam. The diierent modes of association are considered to arise from the diierent stereochemical configurations of the amide group in the two compounds. The interpretation of the molar polarization curves in accordance with the hypothesis of association in these systems has been discussed (3). The experiment to he described deals with a further examination of association in these systems. Measurement of the effective concentration of all molecules in various solutions containing amide1 and lactam are made using vapor pressure osmometry. Equilibrium constants for association are calculated from the data utilizing two models. One model supposes the existence of a monomer and a single associated species. The other assumes that an extended series of multimers is formed with each stepwise association, A , A SA , + ,, characterized by the same equilibrium constant. Use of each scheme in turn with the data for the amide and the lactam allows the student to choose the model which
+
1 Nnbutylaoetamide was substituted for Nmethyl acetamide for reasons mentioned below.
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lournal of Chemical Educafion
yields the most selfconsistent set of equilibrium constants. The results lend support to the proposal of different modes of association in amide and lactam solutions. Vapor Pressure Osmometry
The principles and practice of vapor pressure osmometry have been reviewed previously (4, 5) and the capabilities of the commercial instruments have been discussed (6). Therefore, only a brief discussion of the method will be given here. The diagram given in Figure 1 illustrates the principle. A matched pair of thermistors is located in an isotherma1 chamber which is saturated with solvent vapor. A 75;E;'g; drop of puresolvent @Em is placed on one of o t h e thermistor I 2t;iEm;b~ heads while the ! other bead holds a I , drop of solution. I Since the vapor I ! pressure (and the chemical potential) Figure 1. A rchernotic diogram of a vapor of the solvent in the pre$lure osmometer. drop of solution is lower than that of the pure solvent, solvent vapor will condense on the solution drop a t a greater rate than on the drop of pure solvent. A steady differential rate of condensation is soon established. Since condensation releases heat to the surroundings, the solution drop is warmed relative to the solvent drop. This differential heating is reflected in a steady temperature difference between the drops. I t has been established empirically (4) that this temperature difference is a colligative property. Studies of association in solution may therefore be carried out using this method. The equations used to calculate equilibrium constants from the experimental data are developed in the following section. Theory of the Calculation of Association Consknts
The quantity obtained from vapor pressure osmometry is the effective concentration of all species in solution. Each particle is counted as one unit whether it is monomer, dimer, or higher multimer. Using the mole fraction concentration scale, the effective mole fraction may be written as
equilibria are occurring simultaneously in the system: 2A1 A,, 3A1 e A,, . . . ., pA, a A, (13) The equilibrium constants for this case may be denoted asfollows:
*
where q indicates the stoichiometry of the largest complex. The stoichiometric mole fraction, X., is defined in terms of the actual concentration of solute which is mixed with the solvent to prepare the solution. We may then write X.
= 21
+ 2 2 ~+ 3zs + . .. + qz.
(3)
where the x's again are mole fractions of the various species. In this case, we may write by analogy to eqn. (8) X. = z, 2K12 (21)' 3 K e ( z 2 + . . . qKl,(u)* (15) and, similar to eqn. (9) X. = a K d z d ' K I ~ z I ) ' . .. KdzJ' (16) A set of equilibrium constants which are related to the Kl,'s may be written as
+
+
The experimental values of X , and X , are the quantities which are utilized to obtain equilibrium constants for association. The calculation of equilibrium constants using these data requires the postulation of a model for the possible stoichiometry of the complexes formed. In addition, an assumption which is necessary for any treatment is that differences between X , and X, must be considered to arise solely from molecular association. Any other effects leading to nonideal behavior are assumed to be of lesser importance. With this assumption in mind, we consider the following models. In Model I, the monomer and one other polymeric species of degree of association q are assumed to be in equilibrium in the system. We write the equilibrium constant for the reaction
+
+
+
+ +
.
K,= K,, = A,;K,, = (XI) 21.2,'
"
.
Z4 (17) za.z,, K (qll(d = 2 ~ 1 . 2 1
Using the equilibrium constants of eqn. (17), eqn. (15) may be written as X.
=
n
+ 2K~121'+ ~
.. . +
K U ~ Z I qK
while eqn. (16) becomes X.
=
zr
+ Rtu12+ K d a P + ... + KWZI(.I*'Z~Q(19)
Equations (18) and (19) can be solved to obtain equilibrium constants for association using one of several possible simplifying assumptions. One such simplification is to assume that Klz = Kpd = ...K+,)(d K.. Then we may write eqns. (18) and (19) as

qA, Ft An
Here x and x, are the mole fractions of the monomer and polymer of order q, respectively. On the basis of eqn. (4), the stoichiometric mole fraction may be written for this model as X.
=
21
+ 92.
(6)
Equations (20) and (21) may be divided by x, to yield
and
while the effective mole fraction is [cf. eqn. (2) ] X,
= 2,
+ z,
(7)
Combination of eqn. (5) with eqns. (6) and (7) in turn gives X. X.
= 21 = ZL
+ qK.(xd* + K,(zdq
The righthand sides of eqns. (22) and (23) are infinite series of standard form in K.x,. The sums of these series are given as
Subtraction of eqn. (9) from (8) leads, with rearrangement, to X.  X ,
=
K,(q
 l)(z+
(10)
and solving eqn. (10) for XI
Substituting this result in eqn. (8) and rearranging, we obtain
This may be verified by dividing the numerator by the denominator in the righthand sides of eqns. (24) and (25). The quotient X , / X , may beobtained from eqns. (24) and (25) as 1 x, = 
X.
(26)
1  K&
Solving eqn. (26) for K. gives Model I1 assumes that an extended series of multimers is formed, i.e., we consider that the following Volume 45, Number 3, March 1968
/
177
while, from combination of eqns. (23) and (25)
This equation may be solved for X I as
Combination of eqns. (27) and (29) finally give an explicit expression for K , in terms of the experimentally available quantities: A calibration curve for the vapor pressure orrnorneter, Model 301, run with biphenyl as solute in benzene solution.
Figure 2.
The Experiment
Reagent grade benzene was used as the solvent. Eastman practical grade caprolactm was recrystallined from ethanol. Nnbutyl acetamide (NBA), Eastman white label, was used without further purification. As mentioned previously, this compound was employed instead of NMA, which has a higher vapor pressure in bensene and consequently did not give reproducible osmometer readings. The additional. carbons of NBA should not alter the comparison of these results with the conclusions derived from the previous experiment (3) since the stereochemical configuration of the amide group is preserved. Eastman biphenyl was employed in the calibration of the vapor pressure osmometer. Benzene solutions of biphenyl are known to behave ideally in the concentration range considered (7). The instrument used was the Mechrolab Model 301A Vapor Pressure Osmometer which was obtained from the F. & M. Division of the HewlettPackard CO. Solutions of biphenyl in benzene were prepared on a mole fraction basis covering the range of approximately mole fraction in five approxi1 X 103 to 16 X mately equal intervals. Since very little material is used in the measurements, 100 g of each solution will suffice for a class of 3 M 0 students. The calibration of the instrument was carried out in the following manner. Solvent was placed in the two syringes provided to contain it and a sample syringe was filled with the least concentrated biphenyl solution. While there are syringe positions provided for holding four different samples, we have found that it is best to equilibrate only one sample a t a time since evaporation of the solvent occurs if the samples are left in the syringes for an extended period. After the syringes reached thermal equilibrium, the machine was zeroed with solvent drops on both thermistor beads using the prescribed operating procedure. Three consecutive rnns of 2min duration were then performed using the biphenyl solution. The Zmin time is considered to be sufficient for a steady temperature dierence between the thermistors to be established. The machine was then rezeroed and any drift from the initial zero reading was noted. Three more 2min runs were then made and the macbme was zeroed with solvent again. A new biphenyl solution of higher concentration was then introduced, allowed to reach thermal equilibrium, and the above steps were repeated. The experimental values obtained are readings of AR, the resistance dierence between the thermistors, which is proportional to the corresponding temperature difference between them. 178
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Journal of Chemicol Educotion
The six values of AR obtained for each sample were averaged and used to prepare a calibration curve (see Fig. 2). The results of the measurements on the benzene solutions of biphenyl are given in Table 1. The first column gives the stoichiometric mole fraction of the solute while the second gives the average AR value obtained for the sample. The third column gives the Table 1. Vopor Pressure Osmometer Dato for Solutions of Bi~henvl . . in Benzene
Stoiohiometric Mole Fraction Solute X.
AR (w.)
AR 2
calculated value of ARlx. These values are instrument constants for particular AR values. In principle, AR/x should be constant for any value of AR as long as the solute is ideal. However, the instrument response varies over the range of AR. This is the reason why calibration a t more than one concentration of ideal solute is required. Figure 1 shows the experimental calibration curve; values of AR/x are plotted versus AR. Solutions of the amide and the lactam in hensene were prepared and run according to the procedure used for biphenyl.% Five solutions each of NBA and caprolactam which had mole fractions in the range, 3 X to 27 X 103 were utilized. Table 2 gives the experimental data for these compounds. In addition, the third column lists the values of AR/x obtained from the calibration curve while the fourth column gives the value of the effective mole fraction X., calculated by dividing AR/x into the corresponding value of AR. The equilibrium constants for the association of the two amides in bensene solution may be calculated using Models I and 11, eqns. (12) and (30). The results are shown in Table 3. For Model I, calculations for q = 2 and q = 3 are shown. Further calculations for higher q values can also be done conveniently if a computer is available. Inspection of these calculated results shows that a set of internally consistent equilibrium constants is oh3 The colleotion of data for the biphenyl, smide, and lactam solutions takes about two 4hr laboratory periods.
Mole Fraction Solute X.(X 109
AR
(av.)
ARG/z
Mole Fractmn Solute X , ( X loea)
om oallbratxon curve
tained for caprolactam only with Model I where q = 2. Correspondingly, internally consistent equilihrium constants are obtained for NBA with Model I1 alone. The average values of the equilibrium constants obtained from these sets are 154 6 for caprolactam and 45.6 3.8 for NBA. The uncertainties given are the standard deviations of the mean. A propagationoferror treatment was also carried out based on an estimated devie tion in the AR values of 1%. This value is given as the manufacturer's specification of precision for the instrument and has been verified in our experience with it. Errors in the stoichiometric mole fraction, X,, are considered to he insignificant compared to the uncertainty in AR. The estimated uncertainty in the equilihrium constant based on this uncertainty in AR is *9 for caprolactam and *3 for NBA. These values are in good agreement with the standard deviations reported above. We can compare these equilibrium constants with correspondii values in the literature. Huisgen and Walz (8) have reported the equilibrium constant (mole fraction scale) for the diierization of caprolactam in benzene as 280 a t 25°C. To compare this quantity with the value ohtained in the present experiment at 37"C, the enthalpy of association is required. Davies and Thomas (9) have ohtained enthalpies for the association of various trans amides in benzene. A value of 3.75 kcal/mole appears to be a reasonable estimate from their data for the enthalpy of formation of a single amideamide hydrogen bond. Smce caproIactam can form two hydrogen bonds when it associates, U should be approximately 7.50 kcal/mole for this process. Using this estimate the equilibrium constant a t 37°C calculated from the data of Huisgen and Walz is 172. Since no error measure is given for their data, comparison of their value with the diierization constant obtained here is diflicult. If it is assumed that their number has the same estimated uncertainty as our
value, i.e., *9, the equilibrium constants agree within experimental error. The equilibrium constant for a step in the chainassociation of NBA has not previously been reported in the literature. However, Davies and Thomas (9) have studied the association of Nnpropylacetamide in benzene a t 21.8°C. They cdculated their results on the basis of Model I1 and ohtained an equilibrium constant (mole fraction scale) of 52 2 for the association process. Using 3.75 kcal/mole for the enthalpy of association, the equilibrium constant for Nnpropylacetamide is 38 at 37'. Although a quantitative comparison of this value with that for NBA is not justified, there appears to be order of magnitude agreement between them. This would he expected for compounds diiering only by a CH2group. We have concluded that selfconsistent sets of equilibrium constants are obtained in this experiment only when the existence of monomer and dimer is postulated in the case of caprolactam and an extended series of multirners is assumed to exist in the NBA solution^.^ This conclusion lends support to the hypothesis advanced to explain the shape of the curves of molar polarization versus solute concentration obtained in the previous experiment (3). The experiment on molar polarization and this one, when done consecutively, allow the student to develop an interpretation regarding physical data which he has obtained and then to test this interpretation using an entirely different experimental approach. The fact that a t least two models are available with which to fit the data ohtained by vapor pressure osmometry lends further interest to the experiment. Finally, the experiments serve as an illustration of the effects of hydrogen bonding in solution, something which biologicallyoriented students find appealing.
'It should be pointed out to the students that the selfconsistency of a set of equilibrium constants does not prme that the species to which the oonstants refer exist in solution. Such proof must be provided by the use of a direct method, e.g., infrared spectroscopy (10).
J. A., Compt. rend. trav. lab. Carlsberg Ser. (1) SCHELLMAN, Chirn., 29,223 (1956). (2) Ts'o, P. 0.P., MELVIN,I. S., AND OSEN, A. C., J. Am. Chm. SOL, 85, 1289 (1963). (3) SCHRIER,E. E., J. CHEX.EDUC., 43,257 (1966).
*
*
Literature Cited
Volume 45, Number
3, March 1968
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179
(4) BWY, A. P.,H m , H., AND MCBNN, J. W., J. Phys. and Colloid Chem., 55,304 (1951). , E.,J . Phys. Chem., 67,2590(1963). (5) B n a a ~ D. (6)Lorn, P. F.,AND MILLICB,F.. J. CHEH. EDUC., 43, A295 (1966).
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Journal of Chemical Education
(7) TOKZA, H., J. Chnn. Phys., 16, 292 (1948). (8) HUISQEN,R., AND WALZ,H., C h a . Bw., 89,2616 (1956). (9) DAVIES, M., AND THOMAS, D. K., J . Phys. c h a . , 60, 763 (1956). (10)LUCK,W., Notu~issenschojta,52, 25 (1965).