Stephen A. Wilson and James H. Weber University of New Hampshire Durham. 03824
Molecular Weight Determination of Weak Acids
In the nast several vears this Journal has contained several papers on the determination of molecular weights. Included among these are several that describe molecular weig-hts obtainedhy the measurement of colligative properties of;nelting point depression ( I ) , osmotic pressure (2,3), and boiling point elevation (4). However, in recent years there has been only one brief example of the determination of the molecular weight of a weak aeid ( 5 ) .In that note Shen showed that one can determine the molecular weight by measuring the [Ht] after each of a series of dilutions. However, this method requires unrealistically accurate p H measurements ( 6 ) . In this naner we will initiallv discuss a series of eauations .called vir;alkquations used f& the calculation of mhecular weights from collieative orooertv exoeriments. Then we will describe the correkou of t i e 06seGed colligative property change for the dissociation of mono and polycarboxylic acids. The dissociation correction will be made in a simple fashion from the measured p H of the solution. The discussion will be simplified in most cases by exemplifying colligative properties by freezing point depression. Molecular Weight Calculations for Nonelectrolytes Glover (7) has written an excellent review on the determination of polymer molecular weights by freezing point depression and other colligative properties. The review, which includes experimental details as well as computational approaches, is applicable to pure compounds as well as to polymer mixtures. The change in a colligative property (0) can be related to the grams of solute per kilogram of solvent ( W) by a power series in which a, b, and 80 are constants (eqn. (1)). 00 corrects for a non-zero B of the pure solvent. The a and b terms (1) 8=Bo+aW+bW2+. . . contain several constants including the molecular weiehts of the solute and wlvent and the frwzing point or boihng point constant for the solvent. The ronslanr a isdirertlv related to the molecular weight ( M ) by the apparatus constant Kspp (eqn. (2)). That is, after K s p pis determined
M = K,,&
(2)
(see below), the molecular weight can he obtained from eqn. (2) by using the value of a from eqn. (1). There are several ways of approximating the virial equation (equ. (1)) to obtain a. The simplest expression, B = Bo + a W, is called the first virial equation because only W to the first
Sample Data and Molecular
0
Wb
;10'. C
pH
Weight Calculationr for Tartaric Acida B(cotr)
((O(corr1
Molec-
X 10'. OC
- 8JNW)
o X ular 10% weightd
X 10'
9.12 X 10-+ana4.24 X respectively (91. 1.025. d#app i s 1.84'~m-'. The actual molecular weight i s 150. 'I K
b
a n d K , are
is g acia per kg water.
c 8 is
power is utilized. We used the first virial equation to obtain a value of K,,, with a sucrose standard in water. We measured the freezing point depression B values for a series of sucrose solutions and plotted B against W to determine the slope a . The K,, value of 1.84OC m-' was obtained by inserting a and the known sucrose molecular weight of 342.3 into eqn. (2).' However, the plot of 0 against W utilized in conjunction with the first virial equation often shows curvature due to the non-ideality of the solutions measured. When the non-linearity occurs, one can estimate 00 from the intercept of the least squares treatment, and utilize it in the second virial model, R = 80 a W bW2. The second virial equation is modified for easy graphical treatment by subtracting 00 from both sides of the equation and dividing both sides by W (eqn. (3)). The plots of (0 - 0o)lW versus W were linear, and the intercept
+
+
(0 - RoVW = o + bW (3) a was utilized in eqn. (2) to calculate the molecular weight. The above discussion reviewed the determination of the -
' The KO, value is our experimental determination of the freezing
point constant 1.855'C m-'. 'The pH meter actually measures the hydrogen ion activity, but in aereement with standard textbooks (reference (8). . .. for examole) . . the discussion will be in terms of hydrogen ion concentration. See Experimental section for additional details.
Volume 54. Number 8. August 1977 1 513
molecular weights of nonelectrolytes. The determination of the molecular weights of weak acids, however, requires the correction of 9 for dissociation. In the next section we will show that this correction can be made from the measured pH. Correcting 0 for Acid Dissociation
The usual relationship between a colligative property and the molalitv of a solution ex~ressedin e m . . (4) . . is aform of the first virial equation. 8 = ikm
(4)
In eqn. (4) the van't Hoff factor (i) is the number of moles of particles per mole of original solute, k is a solvent constant and m is the molality of the solute. 8 corrected for acid dissociation (9 (corr)) can be expressed as 8li. Thus, in order to determine the molecular weights of weak acids it is necessary to utilize 8 (corr) rather than 9 in the previously discussed equations. In all cases the correction factor can he obtained from the [H+] measured hv . a .D H meter and the calculated analvtical acid concentration (C.h2 For a monohasic acid C. can he ohtained without titration from the acid dissociation-constant K and the measured p H (eqn. (5)). This relationship for a
+
[H+12 [Ht]K -KC.
=0
(5)
monohasic acid is based only on the assumption that [H+] >> [OH-] (8). For a monobasic acid (eqn. (6)), the listed mole balance and total molarity of particles
+
relationships are well known. Mole balance: C. = [HA] [A-1; Total molarity of particles: [HA] [A-] [H+] = C. + [H+]. Since the [OH-] is negligible, the total molarity of particles is the analytical acid concentration plus the [H+]. Thus the number of moles of particles per C, is 1 [H+]/C.. This relationship yields the value of i (eqn. (4)), which is used to calculate B (corr) (eqn. (7)).
+
+
+
n
0 (corr) = = L
n
1 + [Htl/Ca
(7)
Surprisingly the identical result can he derived for dihasic acids. The total acid concentration (C,) of a dibasic acid can he calculated (8)from a measurement of [H+] and the acid dissociation constants K 1 and Kg (eqn. (8)). [Ht]+
KKIIH+IZ+ (KIKz - KICa)[Hf] - ~ K I K ~ C = ,0
(8)
The only approximation in this equation is that [H+] >> [OH-]. For a dibasic acid H2A the correction for dissociation can he made on the basis of two dissociation steps (eqn. (9)). For a dihasic acid, as already determined for a KI
H2A oH+
+
+
+
ticles: [HzAJ [HA-] [A2-] + [H+] = Ca [H+].The results of eqn. (7) are general for any mono or polyhasic acid. Calculations and Results
We can summarize the molecular weight determination of mono and polyhasic acids by the following steps: (1) Utilize. the known acid dissociation constant(s) for the acid and the measured [H+] in eqn. (5) (monobasicacid) or eqn. (8) (dihasic acid) to calculate C. (2) Use C, and the [H+] in eqn. (7) to calculate 9 (corr). (3) Graph 8 (corr) against W (g of acid per kg of water). If the plot is linear, determine the slope a and go to step 5. For a nonlinear plot estimate the intercept 90 by the least squares method and utilize it in step 4. (4) Using the corrected values of 8, plot (8 - $)lW versus W (from eqn. (3)) to obtain the a intercept. (5) Finally a and Ksppare used to calculate the molecular weight M (eqn. (2)). As suggested by the excellent results for the tartaric acid samnle calculation shown in the table (observed molecular weight 150, theoretical molecular weight 1501, our experimental molecular weixhts for a variety of polycarboxylic acids and acetic acid are inexcellent agreemeniwith the theoretical values. The results are (observed, theoretical): trimellitic acid (211,210), oxalic acid (90.0,92.4), phthalic acid (168,1661,and acetic acid (62.2,60.0). The average deviation of the observed from the theoretical values is 1.6%. Experimental The p H measurements were carried out at 2 5 T using an Orion Model 407 Specific Ion Meter and a Corning Model 476050 combination p H electrode. The p H meter was calibrated with law p H buffers having an accuracy of *0.01 p H units a t 2 5 T . The freezing paint depression was measured with an Advanced Instruments Model 600-5 Osmameter, which measures temperature to f0.00l0C. Since we could not use a constant ionic streneth medium for the fre~zinr: p i n t dcprciiion txperimrnt\, w e did iwt uic m e 11.r o u r p H m e a w r e m e ~ C , I U . I ~ ! I I S 9 that $\ere determined a t experimental conditions as close as possible to our temperature and ionic strength conditions. The calculations of 0 (corr) were relatively insensitive to the exact K values used hecause of their relationship to C. (eqns. (5) and (8)). and because the [Ht]/C, correction term in the denominator of eqn. (7) is usually much less than 1. The weaker the acid. the lessimourtant is the exact valueafK used in the calculations. A trial caleulkan far tartaric acid showed that a doubling of the K I and Kp values changes the molecular weight only about 3%.
Acknowledgment
We thank the Water Resources Research Center (grant number OWRR A-035 NH) for partial support of this work. Literature Cited (11 Wawmnek,S.. J. CHEM. EDUC.dP.899 11972).
+ HA-
+
monobasic acid, the total number of particles per C, is 1 [H+]/C,, and 9 (corr) can he calculated by use of eqn. (I). Mole balance: C, = [HzA] [HA-] + [A2-]; Total molarity of par-
+
514 1 Journal of Chemical Education
171 Glover.C.A.,in "Polymer Molecular Woighw."Parl I, 1Edidilor Slsde. P. E., Jr.1, Marcel Dekker. Inc., New York. 1975.pp.80-159. (81 F r e i w . H.. and Fernsndo.9.. " h i e Equilibria in Analytical Chemirtry."&,hnWiley andsons, Ine. New York, 196'J.pp. 57-52, 81-86, 191 Kortilm. G.. Vugel. W., and Andrurmmu. K.. "DlrroelstionConrianm s f Oren& Acids in Aqueous Solution." Rutierwurths, London. 1981.
