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Ideal equation of state for charged monolayer at the liquid interface
Ideal Equation of State for Charged Monolayer at the Liquid Interface D. K. C H A T T O R A J 1 Chemistry Department, Jadavpur University, Calcutta-32, India
Received May 10, 1968 An attempt has been made to estimate the value of the kT-coefficient of the ideal equation of state for the charged monolayer in the presence and absence of the neutral salt. Guggenheim's concept of the surface phase is taken as the basis for such calculation. The value of the kT-coefficient has been found to be "two" both in the presence and absence of the neutral salt when integrated forms of the Boltzmann equations a r e used for the calculation of the surface concentrations of the inorganic ions. The result is in agreement with that previously suggested by Phillips and Rideal. However, if the nonintegrated forms of the Boltzmann equations are used for this calculation, p is found to be much less than two in agreement with the expectation of Davies.
INTRODUCTION I t is generally accepted t h a t the ideal equation of state for an adsorbed monolayer at the liquid interface m a y be written in the form (1) 7r = p n ~ k T
[1]
or 7rA = p k T ,
[2]
where 7r is the surface pressure equal to 7 0 - 7, n' and ~/0 being the boundary tensions of the air-water and oil-water system in the presence and absence of the surface-active solute. Here k and T are the Boltzmann constant and absolute temperature, respectively. The area A per adsorbed molecule is the reciprocal of nR, the number of solute molecules remaining in excess per unit surface area. The t e r m p, the kT-coetiicient of the equation of state, is recognized to be unity for the neutral monolayer (1). Several workers are of the opinion t h a t the forms of Eqs. [1] and [2] remain valid even
when the monolayer is charged b y ionization. Davies (2) believes t h a t p in such a case is always unity, b u t in the opinion of Phillips and Rideal (3), p is equal to " t w o . " H a y d o n (4) has a t t e m p t e d to show t h a t p --= 1 in the presence of excess neutral salt whereas in its absence, p = 2. I n the subsequent t r e a t m e n t of I t a y d o n and Taylor (5), however, the views of Davies for the/cT-coeffieient have been almost accepted b o t h in the presence and absence of the neutral salt. An a t t e m p t will be made in the present paper to derive a general expression for p for the charged monolayer from thermodynamic considerations. SURFACE AND BULK PHASES
i Present address: Department of Biochemistry, University of Iowa, Iowa City, Iowa 52240, U.S.A.
Let C~- and C be the respective concentrations of the surface-active electrolyte R N a and the neutral salt NaC1. The experimental condition is such t h a t with C kept constant, ~r for the various values of C~- is measured. Following the concept of Guggenheim (6), the system is divided with the help of two planes into surface and bulk phases. T h e physical thickness of the surface phase ~ is believed to be of the order 10-7 to 10 -6 cm. Journal of Colloid and Interface Science, Vol. 29, No. 3, March 1969
407
408
CHATTORAJ
Some experimental evidence for the surface thickness has already been presented (7). The chemical potential of each component in the two phases will be uniform and at equilibrium, ~RNa ~
~cl I-Iere ~
and ~
~l~Na
,
[3]
•
[4]
"
= .~c:
C ~ . = Ca- +C;
K(
[5]
C ~ + ' C o l - _ K~'.
[6]
C~-.C~+
_
C~-" C~+ and ~r
C~+" C Here C~ and Cj stand for the macroscopic volume concentrations of the j t h ions in the two phases, respectively. Keeping in mind that the surface concentration of ions of any kind per unit area divided by ¢ is its volume concentration in this phase, we can easily write Eqs. [5] and [6] in the forms r ~ - . I ' ~ + _ K1 CR-" C~+
[7]
CN~+.C
[8]
Here r j refers to the actual number and not the surface excess of the j t h ions present per unit surface area. Combining [7] and [8] leads to
"@ l~c1 - d N C l -
--
F[CR- dizl~-
[13]
where F is equal to F~/Cs, I'z and Cs being the surface and bulk concentrations of the solvent, respectively, in proper units. An extrathermodynamic assumption is that F remains constant when CR- is varied in our case. This will be true if the volumes of the surface and bulk phases remain individually unaffected by the additon of RNa in very small amount. At constant volume, the ratio of the total moles of the ions to that of solvent becomes equal to the ratio of their respective molarities. Since C is constant, on the basis of a similar argument followed before (8), we can put re1- de1- and Col- d~clto be zero, so that Eq. [13] in combination with Eq. [11] now assumes the form --
dCi~-
r ~ - - -
CR-
[14]
.[1 + r ~ + CR- 1
F~- ~
2 FdC~-
With the help of Eqs. [7], [9], [10], and [12] it can be shown that
[K,(C~- + r~-
Pc1- _ K~ C . FRC~- '
[12]
Now, following the treatment of Guggenhelm (6) again, the Gibbs equation for the present case may be written in the form
kT r ~ + . r o , - _ K2.
[11]
I'N~+ = FR- + I~ct- •
d~
and
{10]
dCN~+ = dCR- ;
represent the respective
chemical potentials of the ith component in the surface and bulk phases. The concentrations of RNa and NaC1 in the surface phase are assumed to be extremely low. RNa concentration in the bulk is also very low and since C is constant, the activity coefficient of the bulk solution is regarded as remaining constant when C~- is varied. Under this condition, Eqs. [3] and [4] will lead to the relations of the form (6)
er
K~, K2, and K3 are all constants at constant temperature. Both bulk and surface phases are individually regarded to be electroneutral so that
c)
= C~- I~,' CR- + K3 C "
[15]
[9]
Journal of Colloid and Interface Science, Vol. 29, No. 3, March 1969
Replacing Y~- in Eq. [14] by the right side of
IDEAL EQUATION OF STATE FOR CHARGED MONOLAYER [15] and rearranging further, we find }T
NCrt- q- K~ C +
ff~
-
C~- dC~-
[16]
409
their "hydrophobic head" and "hydrophilic tail" parts dipped within the nonaqueous and aqueous phases, respectively• Below this charged plane, Na + and C1- are assumed to be nonuniformly distributed following the Boltzmann law• Using the integrated forms of the Boltzmann distribution equations, we can show that
%/C7~- + (I+K~)C.Cg- +K~C ~
+ CK~
C~,+ ~00 e'~"'¢/ k T dx
PN~+
,
. fo CB'-
~¢"Ch-q-(l q-Ka)C.C~-q-K3C
2 dC~-
]
-- 2F fo CI~'- dCR- , in which is inserted the limits of the integration from 0 to ~ on the left side and zero to C~- on the right side for the bulk concentration of RNa. Equation [13] on actual integration fortunately leads to the comparatively simple relation =
+
[171
Inserting here the appropriate values of K~ and K~ from Eqs. [7] and [9], we obtain 7r = 2kT [ ( r ~ - -- FC~-) -- re~t_
~
Pc1-
,
[18] r-g:,_ c,,,o+
GOtJY MODEL OF THE DOUBLE LAYER So far we have assumed that the surface phase containing the electrolytes is behaving like a macroscopic system. The chemical potentials and the concentrations of the solute components and ions in this phase are assumed to be uniform. However, from the microscopic standpoint, the positively and negatively charged ions in the surface phase are believed to be separated in a special manner with the formation of the electrical double layer. From the standpoint of the Gofiy model, it is believed t h a t R - ions remain on the phase-boundary plane with
[19]
Col- f0 e-~'kr dx
where x is the distance from the charged plane in the normal direction and ~ stands for the absolute magnitude of the surface potential. Following Grahame's treatment again (9), the integrals may be solved to give Ccl-
re1-
I (2e~C42k~)+ log (1-k-e ~12~r) ? ~ -- log (1 -- e~¢/2~r)
+
- C x / ~ 3 ] ~ T - 2FCR-kT.
-
• | (2e -~¢2~r) 4- log (1 q- e -~¢/2kr) |
k
-
-
log (1
-
-
e-'~nk%
[20] "
_]o
For ~ tending to zero, log (1 -- e±'~/2k~) tends to infinity so that neglecting all the small terms, we obtain I ~ + _ C~+ tim log (1 - e~¢/2kr) FclCox- ___ log (1 e-~¢/2kT)
[21]
Successive differentiation by L'Hospital's rule shows that the limiting value of Eq. [21] is unity; consequently r~a+ _ CN~+
Pc1-
[22]
Ccl-"
Putting this value in Eq. [18] and keeping in mind that r~- - FCR- is tile surface excess nR-, we find ~r = 2 nR-kT,
[231
so that p is equal to 2 with or without the presence of any amount of neutral salt. This result is in perfect agreement with the cMJournal of Colloidand InterfaceScience,Vol. 29, No. 3, March 1969
410
CI-IATTORAJ
culation of Phillips and Rideal (3) for the estimation of p. If, however, the Boltzmann equations for the calculation of FN~+ and rcl- are used without integration, r~+/I'c~- will be equal to (C~+/C) e2~lkr. If we put this result in Eq. [18] and assume r , - >> FC,-, it can be shown e ~p~/kT-
For ~e ~ 40 my, e'~SIt:r tends to unity and Pu~ will be very close to unity in aggreement with the conclusion of Davies (2). In Fig. 1, Pure has been plotted as a function of ~b~. If the double layer is partly fixed and partly diffuse on the basis of Stem's picture, then following our previous argument it can be shown
1
P I ~ = 2 [ 1 - ~ ¢ , ~ e ~ , / k 1~ + l
~r= kTn~-[2(1-(lzcl)e 2"~,/~r_1}], [24]
where the values of j in terms of C . - , C, and ~ are already given (10). In Fig. 1, P~i~. is also plotted as a function of ~ , for different concentrations of C. It must be recalled that Eqs. [24], [25], and [26] are obtained by using nonintegrated forms of the Boltzmann equations and hence their validity is extremely doubtful. In an alternative attempt made by Haydon and Taylor (5), the value of p on the
x here stands for the ratio C/C~-. The bracketed term in such a case represents the value of p. In the absence of salt (x -* 0) and p is equal to two, However, in the presence of the neutral salt p < 2. Putting x equal to infinity in Eq. [24], we obtain
[
PIi~ = 2 1
'1
e' ~ / ~ r + 1 "
] ' [26]
[25]
1
C ,mI'OM C " I X l O "I C ,i iXlO " I
'
"--- C • I X I 0 4 M MODEL
°~
i'ii:i I'|
I.oV
I
I
400 ~I)~
800 IN M V {PRAC)
] 1200
1600
FIG. I. Values of pure for various values of ~ according to equation (26) based on the Stern-Gouy model for the double layer. Journal of Colloid and In~erface Science, ¥ol. 29, No. 3, )larch 1969
I D E A L E Q U A T I O N OF S T A T E F O R C H A R G E D M O N O L A Y E R
basis of the Goiiy model of the double layer has been found to be almost unity in the presence and absence of the inorganic salt. However, their treatment is based on the equation AG
=
AGO~- eel,
[28]
C~-. C~,+ Since the surface phase is electroneutral, [29]
Combining Eqs. [10], [28], and [29] leads to
n~- = %/K'CR-(CR- ~- C).
• [1 +'~'~+ C~-dC:,.~÷]
[31]
n~- C~a+ dgR-_]" Combining again Eqs. [10], [11], [29], and [30] with Eq. [31], and putting the limits of integration in the resulting equation, we obtain
dr = kT V / ~ [c~[32] V'C~-(CR- ~- C) [ l
cR-Ca-+C ] "
Integrating this, we find
= 2nR-kT.
The surface excess concentrations nn,c~ or nc~- for the Helmholtz model is put equal to zero so that the only adsorbed component RNa remains solely in the fixed part of the double layer. Each adsorbed R - ion is attaching specifically to a Na + ion in forming an ideal type of the fixed Helmholtz layer. Since the diffuse double layer is absent, it may be imagined that just below this fixed layer comprising the surface phase, the bulk phase exists. In this picture the difference between surface excess and actual surface concentration seems to vanish so that with the use of Eq. [5], it is possible to write,
n~- = nN~+.
dC~- kT
drr = n~- ~
7r -- 2kT%/K'C~-(CR- + C);
H E L M H O L T Z M O D E L OF T H E DOUBLE LAYER
nR-.n~,~+ _ K'.
written in the form (13)
[27]
where AGO and AG are the standard free energies of adsorption for the neutral and charged monolayers, respectively. Relation [27] has recently been found to be thermodynamically incorrect (11). Experimental results for the neutral and charged monolayers do not support this relation (12). In view of this, the conclusion reached by Haydon and Taylors for the value of p will not be very meaningful.
411
[30]
The Gibbs equation in terms of the surface excess concentrations of the ions can be
[33]
When the Gotiy layer in the surface phase is assumed to exist, a significant difference between the surface concentration and surface excess concentration will appear so that Eq. [28] for RNa and similar equations for NaC1 will not be valid. The distribution equation is related to the actual concentration and has no concern with surface excess concentration. CONCLUSION
It may be concluded, therefore, that, in agreement with the conclusion of Phillips and Rideal, the kT-eoefficient p for the charged lnonolayer of the uni-univalent organic salt is equal to "two" both in the presence or absence of any amount of the neutral salt. This value of the/cT-eoefficient is also independent of the double layer model. From the standpoint of the surface osmotic picture, the surface pressure depends on the concentration of RNA and the ionic concentrations of Na + and Rsolely obtained from its dissociation. The component NaC1 even if it enters the surface phase in excess or deficit will not be able to affect the kinetic contribution of the Journal of Colloid and Interface Science, -gal. 29, No. 3, March 1969
412
CHATTORAJ
surface pressure. H ~ y d o n ' s conclusion t h a t at a high neutral salt concentration, p = 1 can be Obtained only if the n o n i n t e g r a t e d forms of the B o l t z m a n n equations are used in the calculation. Note added'in proof: In the light of the recent experimental results on the surface pressures of the spread ionized monolayer at high area region just pulJlishe d by Robb and Alexender (J. Colloid and Interface S'c'i. 28, 1, 1968), the validity of equations (24), (25) and (26) for the adsorbed ionized monolayer may not be completely ruled out.
ACKNOWLEDGMENT The author is grateful to his colleague Dr. S. Khamrui for many helpful discussions. 4
REFERENCES 1. DAWES, J. T., AND RIDEAI,, E. K., "Interfacial Phenomena." Academic Press~ New York, 1961.
Journal of Colloid and In,trace Science, Vol. 29, No. 3, ~aroh 1969
2. DAVIES, J. T., Proc. Roy. Soc. (London) A208, 224 (1951). 3. PHILLIPS, J. ~/[., AND RIDEAL, E. K., Proc.
Roy. Soc. (London) A232, 159 (1954). 4. HAYDON,D. A., J. Colloid Sci. 13, 159 (1958). 5. HAYDON, D. A., AND TAYLOR, F. H., Phil. Trans. Roy. Soc. (London) A252, 225 (1960). 6. GVOGEN~EIU, E. A., "Thermodynamics." North Holland Publishing Co., Amsterdam, 1957. 7. CttATTORAJ , D. K., AND CttATTERJEE, A. K., J. Colloid and Interface Sci. 21, 159 (1966). 8. C~tATTO~AZ, D. K., J. Colloid and Interface Sci. 26, 379 (1968). 9. GRAmAME, D. C., Chem. Rev. 41, 481 (1947). 10. CmATTORAZ,D. K., J. Colloid and Interface. Sci. Submitted. 11. LEVINE, S., BELL, C. M., AND PETHICA, B. A., J. Chem. Phys. 40, 2304 (1964). 12. CHATT~RJEE, A. K., AND CHATTOaAJ, D. K., J. Colloid and Interface Sci. 26, 1 (1968). 13. C~ATTORAZ, D. K., J. Phys. Chem. 70, 2687 (1966).
Received May 10, 1968 An attempt has been made to estimate the value of the kT-coefficient of the ideal equation of state for the charged monolayer in the presence and absence of the neutral salt. Guggenheim's concept of the surface phase is taken as the basis for such calculation. The value of the kT-coefficient has been found to be "two" both in the presence and absence of the neutral salt when integrated forms of the Boltzmann equations a r e used for the calculation of the surface concentrations of the inorganic ions. The result is in agreement with that previously suggested by Phillips and Rideal. However, if the nonintegrated forms of the Boltzmann equations are used for this calculation, p is found to be much less than two in agreement with the expectation of Davies.
INTRODUCTION I t is generally accepted t h a t the ideal equation of state for an adsorbed monolayer at the liquid interface m a y be written in the form (1) 7r = p n ~ k T
[1]
or 7rA = p k T ,
[2]
where 7r is the surface pressure equal to 7 0 - 7, n' and ~/0 being the boundary tensions of the air-water and oil-water system in the presence and absence of the surface-active solute. Here k and T are the Boltzmann constant and absolute temperature, respectively. The area A per adsorbed molecule is the reciprocal of nR, the number of solute molecules remaining in excess per unit surface area. The t e r m p, the kT-coetiicient of the equation of state, is recognized to be unity for the neutral monolayer (1). Several workers are of the opinion t h a t the forms of Eqs. [1] and [2] remain valid even
when the monolayer is charged b y ionization. Davies (2) believes t h a t p in such a case is always unity, b u t in the opinion of Phillips and Rideal (3), p is equal to " t w o . " H a y d o n (4) has a t t e m p t e d to show t h a t p --= 1 in the presence of excess neutral salt whereas in its absence, p = 2. I n the subsequent t r e a t m e n t of I t a y d o n and Taylor (5), however, the views of Davies for the/cT-coeffieient have been almost accepted b o t h in the presence and absence of the neutral salt. An a t t e m p t will be made in the present paper to derive a general expression for p for the charged monolayer from thermodynamic considerations. SURFACE AND BULK PHASES
i Present address: Department of Biochemistry, University of Iowa, Iowa City, Iowa 52240, U.S.A.
Let C~- and C be the respective concentrations of the surface-active electrolyte R N a and the neutral salt NaC1. The experimental condition is such t h a t with C kept constant, ~r for the various values of C~- is measured. Following the concept of Guggenheim (6), the system is divided with the help of two planes into surface and bulk phases. T h e physical thickness of the surface phase ~ is believed to be of the order 10-7 to 10 -6 cm. Journal of Colloid and Interface Science, Vol. 29, No. 3, March 1969
407
408
CHATTORAJ
Some experimental evidence for the surface thickness has already been presented (7). The chemical potential of each component in the two phases will be uniform and at equilibrium, ~RNa ~
~cl I-Iere ~
and ~
~l~Na
,
[3]
•
[4]
"
= .~c:
C ~ . = Ca- +C;
K(
[5]
C ~ + ' C o l - _ K~'.
[6]
C~-.C~+
_
C~-" C~+ and ~r
C~+" C Here C~ and Cj stand for the macroscopic volume concentrations of the j t h ions in the two phases, respectively. Keeping in mind that the surface concentration of ions of any kind per unit area divided by ¢ is its volume concentration in this phase, we can easily write Eqs. [5] and [6] in the forms r ~ - . I ' ~ + _ K1 CR-" C~+
[7]
CN~+.C
[8]
Here r j refers to the actual number and not the surface excess of the j t h ions present per unit surface area. Combining [7] and [8] leads to
"@ l~c1 - d N C l -
--
F[CR- dizl~-
[13]
where F is equal to F~/Cs, I'z and Cs being the surface and bulk concentrations of the solvent, respectively, in proper units. An extrathermodynamic assumption is that F remains constant when CR- is varied in our case. This will be true if the volumes of the surface and bulk phases remain individually unaffected by the additon of RNa in very small amount. At constant volume, the ratio of the total moles of the ions to that of solvent becomes equal to the ratio of their respective molarities. Since C is constant, on the basis of a similar argument followed before (8), we can put re1- de1- and Col- d~clto be zero, so that Eq. [13] in combination with Eq. [11] now assumes the form --
dCi~-
r ~ - - -
CR-
[14]
.[1 + r ~ + CR- 1
F~- ~
2 FdC~-
With the help of Eqs. [7], [9], [10], and [12] it can be shown that
[K,(C~- + r~-
Pc1- _ K~ C . FRC~- '
[12]
Now, following the treatment of Guggenhelm (6) again, the Gibbs equation for the present case may be written in the form
kT r ~ + . r o , - _ K2.
[11]
I'N~+ = FR- + I~ct- •
d~
and
{10]
dCN~+ = dCR- ;
represent the respective
chemical potentials of the ith component in the surface and bulk phases. The concentrations of RNa and NaC1 in the surface phase are assumed to be extremely low. RNa concentration in the bulk is also very low and since C is constant, the activity coefficient of the bulk solution is regarded as remaining constant when C~- is varied. Under this condition, Eqs. [3] and [4] will lead to the relations of the form (6)
er
K~, K2, and K3 are all constants at constant temperature. Both bulk and surface phases are individually regarded to be electroneutral so that
c)
= C~- I~,' CR- + K3 C "
[15]
[9]
Journal of Colloid and Interface Science, Vol. 29, No. 3, March 1969
Replacing Y~- in Eq. [14] by the right side of
IDEAL EQUATION OF STATE FOR CHARGED MONOLAYER [15] and rearranging further, we find }T
NCrt- q- K~ C +
ff~
-
C~- dC~-
[16]
409
their "hydrophobic head" and "hydrophilic tail" parts dipped within the nonaqueous and aqueous phases, respectively• Below this charged plane, Na + and C1- are assumed to be nonuniformly distributed following the Boltzmann law• Using the integrated forms of the Boltzmann distribution equations, we can show that
%/C7~- + (I+K~)C.Cg- +K~C ~
+ CK~
C~,+ ~00 e'~"'¢/ k T dx
PN~+
,
. fo CB'-
~¢"Ch-q-(l q-Ka)C.C~-q-K3C
2 dC~-
]
-- 2F fo CI~'- dCR- , in which is inserted the limits of the integration from 0 to ~ on the left side and zero to C~- on the right side for the bulk concentration of RNa. Equation [13] on actual integration fortunately leads to the comparatively simple relation =
+
[171
Inserting here the appropriate values of K~ and K~ from Eqs. [7] and [9], we obtain 7r = 2kT [ ( r ~ - -- FC~-) -- re~t_
~
Pc1-
,
[18] r-g:,_ c,,,o+
GOtJY MODEL OF THE DOUBLE LAYER So far we have assumed that the surface phase containing the electrolytes is behaving like a macroscopic system. The chemical potentials and the concentrations of the solute components and ions in this phase are assumed to be uniform. However, from the microscopic standpoint, the positively and negatively charged ions in the surface phase are believed to be separated in a special manner with the formation of the electrical double layer. From the standpoint of the Gofiy model, it is believed t h a t R - ions remain on the phase-boundary plane with
[19]
Col- f0 e-~'kr dx
where x is the distance from the charged plane in the normal direction and ~ stands for the absolute magnitude of the surface potential. Following Grahame's treatment again (9), the integrals may be solved to give Ccl-
re1-
I (2e~C42k~)+ log (1-k-e ~12~r) ? ~ -- log (1 -- e~¢/2~r)
+
- C x / ~ 3 ] ~ T - 2FCR-kT.
-
• | (2e -~¢2~r) 4- log (1 q- e -~¢/2kr) |
k
-
-
log (1
-
-
e-'~nk%
[20] "
_]o
For ~ tending to zero, log (1 -- e±'~/2k~) tends to infinity so that neglecting all the small terms, we obtain I ~ + _ C~+ tim log (1 - e~¢/2kr) FclCox- ___ log (1 e-~¢/2kT)
[21]
Successive differentiation by L'Hospital's rule shows that the limiting value of Eq. [21] is unity; consequently r~a+ _ CN~+
Pc1-
[22]
Ccl-"
Putting this value in Eq. [18] and keeping in mind that r~- - FCR- is tile surface excess nR-, we find ~r = 2 nR-kT,
[231
so that p is equal to 2 with or without the presence of any amount of neutral salt. This result is in perfect agreement with the cMJournal of Colloidand InterfaceScience,Vol. 29, No. 3, March 1969
410
CI-IATTORAJ
culation of Phillips and Rideal (3) for the estimation of p. If, however, the Boltzmann equations for the calculation of FN~+ and rcl- are used without integration, r~+/I'c~- will be equal to (C~+/C) e2~lkr. If we put this result in Eq. [18] and assume r , - >> FC,-, it can be shown e ~p~/kT-
For ~e ~ 40 my, e'~SIt:r tends to unity and Pu~ will be very close to unity in aggreement with the conclusion of Davies (2). In Fig. 1, Pure has been plotted as a function of ~b~. If the double layer is partly fixed and partly diffuse on the basis of Stem's picture, then following our previous argument it can be shown
1
P I ~ = 2 [ 1 - ~ ¢ , ~ e ~ , / k 1~ + l
~r= kTn~-[2(1-(lzcl)e 2"~,/~r_1}], [24]
where the values of j in terms of C . - , C, and ~ are already given (10). In Fig. 1, P~i~. is also plotted as a function of ~ , for different concentrations of C. It must be recalled that Eqs. [24], [25], and [26] are obtained by using nonintegrated forms of the Boltzmann equations and hence their validity is extremely doubtful. In an alternative attempt made by Haydon and Taylor (5), the value of p on the
x here stands for the ratio C/C~-. The bracketed term in such a case represents the value of p. In the absence of salt (x -* 0) and p is equal to two, However, in the presence of the neutral salt p < 2. Putting x equal to infinity in Eq. [24], we obtain
[
PIi~ = 2 1
'1
e' ~ / ~ r + 1 "
] ' [26]
[25]
1
C ,mI'OM C " I X l O "I C ,i iXlO " I
'
"--- C • I X I 0 4 M MODEL
°~
i'ii:i I'|
I.oV
I
I
400 ~I)~
800 IN M V {PRAC)
] 1200
1600
FIG. I. Values of pure for various values of ~ according to equation (26) based on the Stern-Gouy model for the double layer. Journal of Colloid and In~erface Science, ¥ol. 29, No. 3, )larch 1969
I D E A L E Q U A T I O N OF S T A T E F O R C H A R G E D M O N O L A Y E R
basis of the Goiiy model of the double layer has been found to be almost unity in the presence and absence of the inorganic salt. However, their treatment is based on the equation AG
=
AGO~- eel,
[28]
C~-. C~,+ Since the surface phase is electroneutral, [29]
Combining Eqs. [10], [28], and [29] leads to
n~- = %/K'CR-(CR- ~- C).
• [1 +'~'~+ C~-dC:,.~÷]
[31]
n~- C~a+ dgR-_]" Combining again Eqs. [10], [11], [29], and [30] with Eq. [31], and putting the limits of integration in the resulting equation, we obtain
dr = kT V / ~ [c~[32] V'C~-(CR- ~- C) [ l
cR-Ca-+C ] "
Integrating this, we find
= 2nR-kT.
The surface excess concentrations nn,c~ or nc~- for the Helmholtz model is put equal to zero so that the only adsorbed component RNa remains solely in the fixed part of the double layer. Each adsorbed R - ion is attaching specifically to a Na + ion in forming an ideal type of the fixed Helmholtz layer. Since the diffuse double layer is absent, it may be imagined that just below this fixed layer comprising the surface phase, the bulk phase exists. In this picture the difference between surface excess and actual surface concentration seems to vanish so that with the use of Eq. [5], it is possible to write,
n~- = nN~+.
dC~- kT
drr = n~- ~
7r -- 2kT%/K'C~-(CR- + C);
H E L M H O L T Z M O D E L OF T H E DOUBLE LAYER
nR-.n~,~+ _ K'.
written in the form (13)
[27]
where AGO and AG are the standard free energies of adsorption for the neutral and charged monolayers, respectively. Relation [27] has recently been found to be thermodynamically incorrect (11). Experimental results for the neutral and charged monolayers do not support this relation (12). In view of this, the conclusion reached by Haydon and Taylors for the value of p will not be very meaningful.
411
[30]
The Gibbs equation in terms of the surface excess concentrations of the ions can be
[33]
When the Gotiy layer in the surface phase is assumed to exist, a significant difference between the surface concentration and surface excess concentration will appear so that Eq. [28] for RNa and similar equations for NaC1 will not be valid. The distribution equation is related to the actual concentration and has no concern with surface excess concentration. CONCLUSION
It may be concluded, therefore, that, in agreement with the conclusion of Phillips and Rideal, the kT-eoefficient p for the charged lnonolayer of the uni-univalent organic salt is equal to "two" both in the presence or absence of any amount of the neutral salt. This value of the/cT-eoefficient is also independent of the double layer model. From the standpoint of the surface osmotic picture, the surface pressure depends on the concentration of RNA and the ionic concentrations of Na + and Rsolely obtained from its dissociation. The component NaC1 even if it enters the surface phase in excess or deficit will not be able to affect the kinetic contribution of the Journal of Colloid and Interface Science, -gal. 29, No. 3, March 1969
412
CHATTORAJ
surface pressure. H ~ y d o n ' s conclusion t h a t at a high neutral salt concentration, p = 1 can be Obtained only if the n o n i n t e g r a t e d forms of the B o l t z m a n n equations are used in the calculation. Note added'in proof: In the light of the recent experimental results on the surface pressures of the spread ionized monolayer at high area region just pulJlishe d by Robb and Alexender (J. Colloid and Interface S'c'i. 28, 1, 1968), the validity of equations (24), (25) and (26) for the adsorbed ionized monolayer may not be completely ruled out.
ACKNOWLEDGMENT The author is grateful to his colleague Dr. S. Khamrui for many helpful discussions. 4
REFERENCES 1. DAWES, J. T., AND RIDEAI,, E. K., "Interfacial Phenomena." Academic Press~ New York, 1961.
Journal of Colloid and In,trace Science, Vol. 29, No. 3, ~aroh 1969
2. DAVIES, J. T., Proc. Roy. Soc. (London) A208, 224 (1951). 3. PHILLIPS, J. ~/[., AND RIDEAL, E. K., Proc.
Roy. Soc. (London) A232, 159 (1954). 4. HAYDON,D. A., J. Colloid Sci. 13, 159 (1958). 5. HAYDON, D. A., AND TAYLOR, F. H., Phil. Trans. Roy. Soc. (London) A252, 225 (1960). 6. GVOGEN~EIU, E. A., "Thermodynamics." North Holland Publishing Co., Amsterdam, 1957. 7. CttATTORAJ , D. K., AND CttATTERJEE, A. K., J. Colloid and Interface Sci. 21, 159 (1966). 8. C~tATTO~AZ, D. K., J. Colloid and Interface Sci. 26, 379 (1968). 9. GRAmAME, D. C., Chem. Rev. 41, 481 (1947). 10. CmATTORAZ,D. K., J. Colloid and Interface. Sci. Submitted. 11. LEVINE, S., BELL, C. M., AND PETHICA, B. A., J. Chem. Phys. 40, 2304 (1964). 12. CHATT~RJEE, A. K., AND CHATTOaAJ, D. K., J. Colloid and Interface Sci. 26, 1 (1968). 13. C~ATTORAZ, D. K., J. Phys. Chem. 70, 2687 (1966).