Surface Electric IWential of Cylindrical Polyelectrolyte From a spectroscopic study of the SiH molecule, Douglas24 has shown that D(SCH) = 74 kcal/mol; this leads to a value of AH*"(SiH)= 87 kcal/mol which is consistent with our result above, Saalfeld and McDowell,26 from an electron impact study of AsSiHs have deduced ANf"(SiH2) = 81 kcal/mol, and from a study of pyrolysis kinetics, Bowrey and P u r n e l P have reported AH"(SiH2) = 59 kcal/mol. both values being significantly lower than our lower limil, However, it may well be that our lower limit IS high by a t least 92 kcal/mol, and perhaps by as much as 25 kcal/moil, because of uncertainty in the standard enthalpy of formation of SiH+. For this latter value ~ obtained from the apwe used 308 k c a l / m ~ l ,a~ value pearance pclteiitial of SiH+ from SiHI. Electron-impact processes that involve the breakage of so many bonds are notoraous for i ~ R ~ o l ~excess i ~ i g energy in some form and for leading to standard enthalpies of formation that are too ~ ~ that the ionhigh. In fact,, Potzinger and L a n ~ p eremark ization efficienc y curve of SiH+ had a long tail, an indica. tion of excess energy in its formation. Reasoning by analogy with ions derived from methane that the enthalpies of formation shoiild show a monatonic decrease in the se+ , MH2+, and MH3+ leads us to conclude that the ~ n t ~ ofa formation ~ ~ y of SiH+ should be in the range 283-296 kcal/mol. A value of A&"(SiH+) = 290 kcal/mol would be irn accord with the value of AHfo(SiHd reported by S,xalfeld and McDowelP and with the fact that (42) i s endothermic.
Acknowledggment 'This work was supported by the U. S. Atomic Energy commission under Contract No. AT(111)-3416.
References and Notes (1) U. S. Atomic Energy Commission Document No. COO-3416-12. (2) A. R. Anderson, "Fundamental Processes in Radiation Chemistry," P. Ausioos, Ed., Wiiey, New York, N. Y., 1968, Chapter 5. (3) G. G. Meisels in ref 2, Chapter 6. (4) H. F. Calcote, "Ion-Molecule Reactions," J, L. Franklin, Ed., Plenum Press, New York, N. Y.. 1972, Chapter 15. (5) S. A. Studniarz in ref 4, Chapter 14. (6) J. L. Franklin, ref 4 and references therein. (7) G. G. Hessand F. W. Lampe, J. Chem. Phys., 44,2257 (1966) (8) J. M. S. Henis. G. W Stewart, M. K . Tripodi, and P. 0 . Gaspar, J. Chem. Phys., 57,389 (1972). (9) T-Y. Yu, T. M . H. Cheng, V. Kempter, and F. MI. Lampe, J . Phys. Chem., 76,3321 (1972). (IO) J. M. S. Henis, G. W. Stewart, and P. P. Gaspar, J Chem. Phys., 58,3639 (1973). ( 1 1 ) D. P. Beggsand F. W. Lampe, J. Phys. Chem., 73,4794(1969). (12) G. W. Stewart, J. M . S. Henis, and P. P. Gaspar, ,I. Chem. Phys., 57, 1990 (1972). (13) G. W. Stewart, J. M S. Henis, and P. P. Gaspar, J . Chem, Phys., 57, 2247 (1972). (14) T. M. H. Cheng, T-Y. Yu, and F. W. Lampe, J. Phys. Chem., 77, 2587 (1973) (15) D. P. Beggs and F. W. Lampe, J. Phys. Chem., 73,3307 (1969). (16) D. P. Beggsand F. W. Lampe. J. Phys. Chem.. 73,3315 (1969). (17) T. M. H. Cheng and F. W. Lampe, Chem, Phys. Lett., 19, 532 (1973). (18) Landoit-Bornstein, "Zahlenwerte und Funktionen," 11. Band, I. Teii, G. Auflage, Berlin, 1971, p 335. (19) P. Potzinger and F. W. Lampe, J. Phys. Chem., 73,3912 (1969). (20) J. L. Franklin, J. G. Dillard, H. M. Rosenstock. J. T. Herron, K. Drexl, and F. 1-1. Field, Nat. Ref. Data Ser.. N a t , R u r . Stand.. No. 26 (1969). (21) H. E. Watson and K . L. Ramaswamy, Proc. Roy. Soc., Ser. A, 156, 144 (1936). (22) G. Gioumousis and D. P. Stevenson, J. Chem. Phys., 29, 294 (1958). (23) W. C. Steeie, L. D. Nichols, and F. G , A. Stone, J. Amer. Chem. SOC., 84, 4441 (1962). (24) A . E. Douglas, Can. J. Phys., 35,76 (1957). (25) F. E. Saalfeld and M. V . McDoweil. inorg. Chem., 6 , 96 (1967). (26) M. Bowrey and J. H. Purnell, Pfoc. Roy. SOC., Ser. A, 321, 341 (1971).
Polymer Concentration Dependence of Surface Electric Potential of Cylindrical Polyelectrolyte in Aqueous Salt Solutions atsiitoshi Nitta" and Shintaro Sugai Department of Polymer Science, Faculty of Science, Hokka/do University Sapporo, Japan (Received December 28. 1973) Publicdtion costs assisted by Hokkaido University
he surface electric potential of cylindrical polyelectrolyte is calculated a t finite dilution of polymer by use of the Poisson-Boltzmann equation with an electronic computer. Dependence of the potentiometric titration curve of poly(acry1ic acid) or poly(g1utamic acid) on its concentration is well explained by assuming its rod radius including counterion size to be 5.5 or 12. 5 A, respectively. Also the calculated dependence of pH on the polymer concentration is compared with theory of Maeda-Oosawa and ManningHoltzer. In the presence of excess salts, the pH becomes independent of the polymer concentration as shown by Maeda-Oosawa. At any salt concentration and any degree of ionization, the pH is apparently lineal. us. logarithm of the polymer concentration, if a change of the polymer concentration is caused by removal or addition of solvent. The slope is similar to that expected from these theories.
Introductioin Numerical calculation of surface electric potential of cylindrical polyelectrolyte in aqueous salt solution has been performed by solving the Poisson-Boltzmann equa-
tion with electronic computers.lJ The surface electric potential obtained from potentiometric titration and electrophoretic mobilities of various linear weak-acid polyelectrolytes have been compared with the theoretically The Journalof Physical Chemistry, Vol. 78, No. 12, 1974
1190
Katsutoshi Nitta and Shintaro Sugai
calculated v a h e s ai, infinite dilution of the polymers.2-5 However, to compare with experimental data a t finite dilution of the polymers, dependence of the surface electric potential on the polymer concentration should be taken into the theoretical consideration, because of long-range electrostatic interaction between the molecules. Indeed, the remarkable discrepamcies between the theoretical surface electric potent iai a t infinite dilution and the experimental at finite dilution have been pointed out, especially when the polymer concentrations were comparable or high as compared with the added salt concentrations.2-4 Recently, t w o groups of researchers have published theories of polymer concentration dependence of pH of the solutions of poly weak acid^.^,^ In their papers, the Poisson-Boltz mann equation has not been used, and the relationsh ps of the theories to the Poisson--Boltzmann approach h a w been discussed, although few studies of the rigorous analytical or numerical calculation of the equation at finite dilution have been found.S Here, the wmerical calculation of the surface electric potentiial of a cylindrical polyelectrolyte in aqueous salt solution sf the finite polymer concentration is presented by solving the Poisson-Boltzmann equation without the Debye -Muckel approximation. A model similar to Alexandrowici and Katchalsky was used and the calculation has beerr performed, without any complicated assumptions used by them, with an electronic computer. The resulis are applied to interpretation of the polymer concentration deplendenice of potentiometric titration curve of some ~ ~ oweak l y acids, and compared with other theories for the same pmlolem Theory Consider polyelectrolyte molecules in aqueous salt solution as an array of parallel rod-like cylinders a t an average distance 2R between the molecular a x e ~ . ~ - Then, ll the polymer conceatration C, (monomol/l.) is represented by the distance R ,as
1000/.ArACp= 5rA2L
(1)
where L is length of a monomeric unit in the polymer and NA Avogadro’s number. The polyelectrolyte solution is assumed to be separated with a semipermeable membrane from an aqueous dialysate with the same salt, of which concentration is Cs’. The reduced electrostatic potential 4 ( = t+/kT) in the aqueous polymer solution can be determined from the Poisson-Boltzmann equation of cylindrical symmetry 1 6”-t- &;’ - sinh 6 = 0 (2) where e, h, and T are defined in the usual manner, and # is the electric potential. Also, x ( = Kr) is the reduced distance, r the distance from the axis, and K the DebyeHuckel paramcter, An assumption of the uniform charge distribution on the cylinder leads to the following boundary conditions9Jo
$ ’ ( ~ a= ) -Ze2/L)k1TKaL
(31
6’(iR) = 0 (4) where 2 is the averalge number of charges on the monomeric unit, a distance of closest approach of mobile ions, and D the dieltxtric constant of medium. The electric potential in the outer dialysate solution a t a distance suffiThe Journal of Phyrical Chsmistry, Vol 78, No. 12, 1974
ciently far from the membrane is defined to be zero, and therefore the Debye-Huckel parameter K should be calculated with the salt concentration C,’. Assume 4 may be expressed in the series
4
+A3q$3-l-A54j
=
+- ...
(5) where A is a constant to be determined from the boundary conditions. The function & ( r ) can be obtained by the same method2q9 as in the case of a spherical or rod-like polyelectrolyte at infinite dilution with the boundary condition eq 4 as
=
41,
fn,(x?
( n = 3,5,7, ... ) (6) where I n ( x ) and K n ( x ) are modified Bessel functions of the first and the second kinds, respectively, and the function g n ( x ) is defined in the literature.12 Then 4 can be written as 1=1
(7) (J1(x) = 1) Equations 3 and 7 determine the constant A in eq 5 or 7 with the same method as in the case of infinite dilution.2 The concentration of added 1-1 salt in the polyelectrolyte solution, C, is not equal to C,’ but represented in terms of C,’ as
where V represents the volume of the polyelectrolyte solution per polymer molecule. To calculate 4 in the polyelectrolyte solution, where the salt concentration is C, i t may be necessary to take the following procedure: first, calculate a set of $, i.e., 4(1), 4(2), $(3),. . . , for a set of C,’ near the desired C,, Le., Cs’(l),C,’(2), C,’(3), . . . , respectively. Second, calculate the salt concentration C,(i) in the polyelectrolyte solution in each case of the salt concentration C,(i) in the outer dialysate with eq 8. Finally the reduced electric potential $ in the polymer solution with the salt concentration C, is determined by use of a suitable interpolation method with the sets of 4(i) and
GN. Application to Interpretation of t h e pH Titration Data The potentiometric titration data are related in a direct way to the surface electric potential of the polyelectrolytes. The pH of a weak polyacid solution is given byll
OK
= pH =
CY - log __ 1-CY
(9)
pK, -0.434(@(~a)- @ ( K R ) )
where pKo is the intrinsic dissociation constant and a the degree of ionization of ionizable group. Calculations were performed with an electronic computer FACOM 230-60in the Hokkaido University Computing Center, as the same
Surface Electric Potentia.!of Cylindrical Polyelectrolyte 6.
5.
0
= I
5.
4. 0.1
0.2
0.3
Degree of Ionization
__MI/
I
0.2
0.4
0.6
0.8
1.0
Theoretical a’nd experimental titration curves for poly(acrylic acid) at 14”3 with a = 5.5 A and L = 2.54 A. The numbers adjacent to the curves refer to the concentration of added 1-1 salt in M . The heavy lines are computed for infinite dilution and the light lines are for the corresponding polymer concentrations. She polymer cogcentrations are 0.00829 (small open circles), 0.193 (small filled circles), 0.0335 (large open circles ) , and 0.0419 moiiomoI/l. (large filled circles), Figure 1,
manner as that of infinite dilution.2 The results are computed with potentiometric titration data of relatively high concentration of poly(acry1ic acid) and of a-helical poly(glumatic acid) in Figures 1 and 2, respectively. The very precise data of Yagarsawa, et a L , 3 J 3 were reproduced by enlarging photographs of the figures in their papers. Common pKo is employed for different C, so far as C, is common. The parameter a is assumed to be 5.5 or 12. 5 A, respectively. The polymer concentration dependence of pH titration curve is fairly well explained by introducing the present concentration effect, especially a t low added salt concentration. Jn Figure 3, the theoretical relationship between pK and Cp1/2 is compared with the data of poly(acrylic acid). Although the experimental data seem to indicate the linearity of pK to Cp1/2, linear extrapolation should not be done tcr evaluate the limiting pK a t infinite dilution.
Discussions Many previous theories of the titration curve have been based on solution of the Poisson-Roltzmann equation for a uniformly charged cylinder in a uniform dielectric, although its usual treatment is within the regime of the Born approximation. Nagasawa and the present authors have recently performed the numerical calculation for the rod-like poly weak acids a t infinite dilution, and by adjusting the radii of polymers and counterions to coincide with the experimental data of their dilute solutions, they have found the radii of polymer rods including counterion sizes (Na or K ion) t o be 5.5 and 14-15 A in the cases of poly(methacry1ie acid) and a-helical poly(g1utamic acid), However, few studies of the rigorous solutions of the Poisson -Boltzmann equation a t finite dilution
Figure 2. Theoretical and experimental titration curves for helical poly(g1utamic acid)I3 with a = 12.5 A and L = 1.5 A . The numbers adjacent to the curves refer to the concentration of added 1-1 salt in M . The heavy lines are computed for infinite dilution, the light solid lines are for C, = 0.0188 monomol/l., and the thin dashed line is for Cp = 0.0342 monomol/l. The open and the filled circles are experimental data for Cp = 0.0188 and 0.0342 monomol/l., respectively.
I,
t
-3 C;”
The relationship between pH - log CY/(^ - a ) ) and the root of polymer concentration (monomol/l.) at various degrees of neutralization CY for poly(acrylic acid) at added salt concentration of 0.005 M . Open circles are experimental data3 and solid lines are the values computed theoretically.
Figure 3.
have been found. Alexandrowicz and Katchalsky used the same model as the present one, and their calculation was based on a subdivision of the potential into two parts corresponding to the close neighborhood region of the polyion and the screened-off region from the polyion, however, their skillful method included some intricate parameters. Recently, Maeda and Oosawa have presented a thermoThe Journal of Physical Chemistry. V o l 78 No 72, 1974
Katsutoshi Nitta and Shintaro Sugai
1192
I
TABLE I: Values of - ( b pK/d log C p ) ~ , s , ~ , p , ( y Evaluated from the Present Theory with a = 5.5 A and L = 2.54 A
I
a
C,/C,
0.9
0.'7
0.5
ma
0.95' 0.95 0.96 0.96 0.97 0.98
0.88 5.88 0.88 0.88 0.89 0.91
0.76 5.76 0.76 0.76 0.77 0.78
10 5 2 1 0.5
Calculated from the results of the previous paper.*
10
5
2
1
02
Figure 4. The relationship between the term - 0 . 4 3 4 ( 4 ( ~ a ) ~ ( K R and ) ) the polymer concentration with a = 5 . 5 A, L = 2.54 A , and Z = -0.9. The added salt concentrations are (from top to bottom) 0.005, 0.0'1, 0.02, 0.05, and 0.1 M . The theoretical points at the siinie C,,/Cp ratio are indicated by open circles
and joined with dashed lines. The numbers adjacent to each dashed line refer to the ratio C,/Cp.
dynamic theor,q of the dependence of pH of polyelectrolyte solution on polymer concentration. Their theory has been based on both additivity of the osmotic pressure and the counterion activity in polymers containing salts and the invariability of the osmotic or activity coefficient with respect to the polymer concentration in salt-free solution. They have obtained the following relationship between the pH of the solution and C, in terms of the present notations6
where 4p is the osmotic Coefficient of salt free-solutions. Manning and Holtzer have also developed their polyelectrolyte theory based on counterion condensation and applied it to the same problem. From their calculations, the Poisson-Boltzniann theory with point counterion a t infinite dilution hias been found to be equivalent in their special case, and the Maeda-Oosawa theory for the concentration dependence of p K to be equivalent to their theory for nansalt solutions. Also, their theory for the solution with salts give:: the following relation in terms of the present notations7
where a1 is a critical charge density and equal to 0.35 for poly(acry1ic acid). The Journalof Phyaical ChomJstry, Vol. 78, No. 72, 1974
The present calculation is based on the PoissonBoltzmann equation for the uniformly charged cylinders at finite dilution in a uniform dielectric, however, only an adjustable parameter, the radius of polymer rod including the counterion size, is used without any other assumptions such as Alexandrowicz, et al. By use of essentially the same value of the radius of polymer as that obtained in infinite dilution,2 the concentration dependence of the pH titration curve of poly(acry1ic acid) or a-helical poly(g1utamic acid) can be well explained as shown in Figures 1-3, except for the case of low degree of ionization (a = 0-0.3) of poly(acry1ic acid), which may not be assumed to have the rod-like form. To discuss the relationship between the present results and those of other authors, -0.434(4(~a) - $ ( K R ) ) ( =pK - pKo) is plotted against log C, in Figure 4 for a = 5.5 A, L = 2.54 8, and a = 0.9. Clearly, eq 10 is very well satisfied. Also, a t a glance, (a pK/d log C,)C(~)/C~,),~ is nearly independent of C,/C, and Cp in a range of C,/Cp between 10 and 0.5. When a is considerably larger than a1, the same conclusions can be obtained. In Table I, the values ! , u given for various cases. of (a pK/a log C p ) ~ ( s , , ~ ( pare The values for a = 0.9 or 0.7 agree well with ManningHoltzer's results shown in eq 12. The values for a = 0.5, however, does not agree well. Manning's limiting law cannot rigorously be applied a t near N = a l because of the oversimplification of his polyelectrolyte model. On the other hand, Maeda-Oosawa give also independence of on Cs/C, and C, as shown in eq 11 (pK/log C,)C(~~/C~,,,~ and the value should be considered to he between -1 and -0.7 for q$, of poly(acry1ic acid).14 For low values of a , polymers such as poly(acry1ic acid) must be flexible random coils and the cylindrical model is not adequate. The present treatment may not be applied to such cases. In spite of the criticism on the application of the Poisson-Bokmann e q u a t i ~ n ,the ~ present conclusion on the dependence of pH of polyelectrolyte solution on its concentration may be identical with the results from thermodynamic6 or counterion condensation the0ry.l
Acknowledgments. This work is supported in part by a Grant in Aid of the Ministry of Education in Japan. The authors are also grateful to Drs. N. Imai and H. Maeda of Nagoya University for enlightening discussions and helpful comments. References and Notes ( 1 ) L. Kotin and M . Nagasawa, J. Chem. Phys., 36,E73 (1962). (2) S. Sugai and K. Nitta, Biopolymers, 12, 1363 (1973). (3) M. Nagasawa. T. Murase, and K. Kondo, J . Phys. Chem., 69, 4005 (1965). (4) Y . Muroga, K Suzuki, Y. Kawaguchi, and M . Nagasawa, Biopolymers, 11, 137 (1972). (5) T. Takahashi, I . Noda. and M. Nagasawa, J. Phys. Chem.. 74, 1280 (1970). (6) H.Maedaand F. Oosawa, J . Phys. Chem., 7 6 . 3445 (1972)
Interaction of Iiydrated E3ectrons with the Peptide Linkage (7) G. S.Manning and A. Iioltzer, J. Phys. Chem., 77, 2206 (1973). (8) A. D. MacGillivray, J. Chem. Phys., 56, 80, 83 (1972); 57, 4071, 4075 (1972). (9) R . M. Fuoss, A . Katchalsky, and S. Lifson, Proc. Nat. Acad. Sci. U. S., 37,579 (12161). (10) S. A. Rice anti M. Nagasawa, "Polyelectrolyte Solutions," Academic Press, I-ondon, 196'1, p 233.
1193 (11) Z. Aiexandrowicz and A. Katchalsky, J. Polymer. Sci., Part A-7, 3231 (1963). (12) P. Pierce, Ph.D. Dissertation, Department of Chemistry, Yale University, 1950, cited in ref 10, p 142. (13) M. Nagasawa and A. Hoitzer, J. Amer. Chem. Soc., 86, 531 (1964). (14) 2. Alexandrowicz, J. Polym. Sci., 56, 115 (196%).
lnteractioln of Hydrated Electrons with the Peptide Linkage P. S. Flao and E. Hayon'* Pioneering Research Laboratory, U. S. Army Natick Laboratories, Natick, Massachusetts 01670 (Received November 26, 1973) Fubiica tion costs assisted by Natich Laboratories
The reaction rate constants k of eaq- with a wide range of peptides having terminal -NH3+ and N-acetyl groups have been determined in aqueous solutions, using the technique of pulse radiolysis. These rates were shown to be markedly dependent on the state of ionization of the terminal -NH3+ groups, the number of peptide linkages, and the overall charge on the peptide molecules. Amino groups in a (3 (or 6 , y ) position are significantly less reactive than the corresponding a-amino peptides. The inductive effects on the peptide hydrogen -CONH- affect the reactivity toward eaq- , supporting previous suggestions that %q- interacts with the carbonyl group. The rate constants k have been shown to be directly proportional to barre-catalyzed rates for ionization of the peptide hydrogen of the same peptides, as determined by nmr. A similar correlation has been found between h and the ionization constants of the carboxylic acids of the corresponding peptides. Transient optical absorption spectra have been observed from the reaction of %s- with the N-acetyl derivatives of triglycine, hexaalanine, and trisarcosine. These spectra are dependent upon pH and are suggested to be formed from the interaction of ea,- with the carbonyl group of the peptide linkage. The ketyl radicals formed can undergo acid-base equilibration -C(O-)NHH+ F? --C(OH)NH-, with pK,(radical) L 12.0. These ketyl radicals have very low (negative) redox potentials and are strong reducing agents.
+
Introductnon The priiicipai linkage between the various amino acids making up a psotein is the peptide bond -CONH-. Considerable experimentall and theoretical studies have therefore been carried out on this linkage. The conductivity and electron transfer properties of proteins is also of particular importance, and hence the interest in studying the interaction of electrons with the peptide bond. Nmr studies (see ref 2 for review up to 1969) have indicated the partial double-bond character of the amide bond
0-
I
I1
As a result2, 3 OS this electronic delocalization, the linkage
is planar and long-range spin coupling from R to R1 and Rz can be expected. The peptide hydrogen (R1 = H ) can undergo both base- arid acid-catalyzed exchange reactions (see, e . g . , ref 4-6). Sheinblatt4 has determined by nmr the rates of the base-catalyzed exchange, reactions 2 and 3, and established an acidity scale for the peptide hydrogen (see more bellow)
0
0
II
+
-C-NH-
OH-
II
-C-N-
?t-
-
i-H20
(2)
or, more generally
-CONH*-
+
HOH
+=+
+
-CONH-
HOH*
(3)
The k2 rates were found4 to be strongly dependent on the nature of R and RI (when R2 = H ) . The reaction rate constants for the interaction of hydrated electrons, e,,-, with amino acids and peptides were correlated by Braam7 with the ionization constants of the terminal -NH3+ groups. This reactivity has since been correlated8-l2 with the presence and the number of carbonyl groups (i. e., the number of peptide linkages). The cooperative effect of the peptide linkage(s), the amino and carbonyl groups, and the overall charge on the molecule have been suggestedll to contribute to the reactivity toward %,-. The reaction of eaq- with simple amino acids8J3 and 01igopeptidesgJlJ~315 has been shown to result in reductive deamination eao-
+
-
~H~CH~CONHR
NH~
+
CH~CONHR (4)
The CH2CONHR radicals have been identified by pulse The Journal of Physical Chemistry, Vol. 78, NO. 12, 7974
