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1978 Thermodynamics of Ion Association in Aqueous Solutions of Sodium Sulfate, Sodium Carbonate and Sodium Bicarbonate Yan-chun Chung Eastern Illinois University This research is a product of the graduate program in Chemistry at Eastern Illinois University. Find out more about the program.
Recommended Citation Chung, Yan-chun, "Thermodynamics of Ion Association in Aqueous Solutions of Sodium Sulfate, Sodium Carbonate and Sodium Bicarbonate" (1978). Masters Theses. 3208. https://thekeep.eiu.edu/theses/3208
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Date Author
pdm THERMODYNAMICS OF ION ASSOCIATION IN AQUEOUS SOLUTIONS OF SODIUM
SULFATE, SODIUM CARBONATE AND SODIUM BICARBONATE (TITLE)
BY
YAN-CHUN CHUNG
THESIS
SUBMITIED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
MASTER OF SCIENCE, DEPAR'IMENT OF CHEMIS'IRY
IN THE GRADUATE SCHOOL, EASTERN ILLINOIS UNIVERSITY
CHARLESTON, ILLINOIS
I ( 1978,, �'- '· �-vO:R
I HEREBY RECOMMEND THIS THESIS BE ACCEPTED AS FULFILLING
THIS PART OF THE GRADUATE DEGREE CITED ABOVE ABS'IBACT
Title of thesis: Thermodynamics of Ion Association in Aqueous Solutions
of Sodium Sulfate, Sodium Carbonate and Sodium
Bicarbonate.
Yan-chun Chung, Master of Science, 1978
Thesis directed by: David W. Ebdon, Associate Professor of Chemistry
- - Stoichiometric association constants of NaSo , Naco and NaHco 0 4 3 3 ion-uairs were determined over the temperature range 10 - 4o0c at several ionic strengths by using a sodium ion selective electrode. At zero ionic strength values for K� - are 15.8 .4, 8.5! .2 and ,NaS0 � 4
4.6 ! .3 at 10°, 25° and 4o0c, respectively; like values for K�, NaC0 - 3 are 17.1!1.4, ! .6 and 21.0: .7. Values for K� o are 19.1 ,NaHC0 3 3.35 '! .07 and 6.75 !: .02 at 25°c and 4o0c, respectively. - - The apparent association constants for NaS0 and Naco can be 4 3 I l. = 0 A 2 reproduced by the equation log K log K 4 I + BI , where A is A A the Debye-Hlickel limiting slope (equal to .51 in water at 25°c), Bis an empi-rical constant and K� is the thermodynamic association constant.
. 0 0 'l'he of log K0 - are 1.20, .93 and .67 at 10 , 25 and 400 C, va.lues A,NaS04 respectively; like values for log K� - are 1.233, 1.281 and 1.323. ,NaC0 3 - Values of B NaS0 are 1.04s ' and 2.30 at 100 , 25 0 and 40 0 C, for 4 1.b7 - respectively; like values for Naco are .69, 1.13 and 1.58. The 3 factor 4A equa.lG 2.oc, 2.04 and 2.10 at 10°, 25° and 4o0c respectively.
apparent associa,tio:n consturts fer Na.Hco ° can be reproduced by the The 3 1 c o general equation log K ' = �og , K - 2 + BI • At 25 and 400C, the - A A 2AI
366976 values of log K o are .525 and .826 and the values of B are .290 ,NaHC0 � 3 and .123, respectively.
' A linear correlation between �H (the enthalpy of formation of the ion pair from the free ions) and the ionic strength of the solu- tions ·has been obtained for each of the salts studied. The results
' show that as the ionic strength increases, AH for sodium sulfate ion association changes from negative to positive, for sodium carbonate
changes from positive to more positive and for sodium bicarbonate
changes from higher to lower positive values. These correlations can be used to calculate the apparent association constants at various
temperatures and ionic strengths. VITA
Name: Yan-chun Chung.
Permanent Address: 32-1, Wen-Hwa s. Rd., San Chung, Taiwan, R. O. c. Degree and date to be conferred: M.S., 1978 .
Date of birth: January 25, 1951 .
Place of birth: Taipei, Taiwan, Republic of China .
Secondary education: Chen Yen Junior High School, Taipei, Taiwan, Republic of China, 1963-1966. National High School of Taiwan Normal University, Taipei, Taiwan, Republic of China, 1966-1969 .
Collegiate institution attended: Date Degree Date of Degree
Taipei Institute of Technology Taipei, Taiwan, R. o. C. 1969-1972 B.S. June 1972 Eastern Illinois University Charleston, Illinois 1976-1978 M.S. May 1978
Major: Chemical Engineering, Physical Chemistry.
Positions held: Second Lieutenant, R. O. C. Army, 1972-1974. Research Assistant, Depto of Chemical Engineering, National Institute of Nuclear Energy Research, Taiwan, R. 0, C. , 1974-1976 . Graduate Assistant, Eastern Illinois University, 1976-1978. Teaching Assistant, Georgia Institute of Technology, Atlanta, Present .
iv ACKNOWLEDGMEN'IS
I would like to express my most sincere gratitude to
Dr. David W. Ebdon for his suggestion of this project, his guidance, encouragement and patience throughout this study. I would also like to thank my wife, Huei-chih, for her typing of this manuscript .
This study was partially supported by the Council on Faculty
Research of Eastern Illinois University.
I wish to dedicate this thesis to my parents,
Mr . and Mrs . Chao-da Chung.
v TA BLE OF CONTEN'IS
Page
ABS1RACT ...... ii
VITA ...... iv
ACKNOWLEDGEMEN'IS ...... v
TA BLE OF CON'IEN'IS ...... vi
LIST OF FIGURES ...... viii
LIST OF TA BLES ...... ix
GLCBSAR Y OF SYMBOIS ...... x
IN'IRODUCTION ...... 1
II. THEORY ......
A.. Ionic Theories ...... 4
B. Ion Association ...... 7
c. Ion Selective Electrodes ...... 8
D. Determination of Association Constants ...... 10
E. Thermodynamics of Ion Association ...... 14
III. EXPERIMENTAL • •••••••••••••• 0 •••••••••••••••••••••••••• 17
A. Reagents ...... 17
B. Apparatus ...... 19
1. pH/ mv Meter ...... 19
2. Sodium Ion Selective Electrode • ••••••••0 •••••• 19
3. Waterjacketed Cell ...... 20
c. Experimental Methods ...... 20
- ° ° 1. Determination of K for NaS0 at 10 , 25 � 4 0 and 4o c ...... 20
' - 2. Determination of K for Naco at 10 0, 250 A 3 0 and 4o c ...... 22
vi I Page 0 3. Determination of K for NaHco at 25 0 A 3 0 and 4o c ••••••••••••••••••••••••••••••••••••• 23
IV • RESUL'IS AND DISCUSSION ...... 25
A. Sodium Sulfate Runs • ••••••••••••••••••••••••••••••• 25
I ° ° 1. Determination of K at 10 , 25 A,NaS0 4 0 and 40 C ••••••••••••••••••••••••••••••••••••••• 25
- 2. Thermodynamic Functions of Naso Ion 4
Association •••••••••••••••••••••••••••••••••••• 26
B. Sodium Carbonate Runs •••••••••••••••••••••••••••••• 3.3
° 1• Determination at 10 ,
0 and 40 C •••••••••••••••••••••••• ••••••••••••••• 33
- 2. Thermodynamic Functions of Naco Ion 3
Association ...... 33
C. Sodium Bicarbonate Runs ••••••••••••••• Q ••••••••••••
° 1 0 Determination of K o at 25 c A,NaHC0 3 ° and 40 c •••••••••••••••••••••••••••••••••••••••
° 2. Thermodynamic Functions of NaHco Ion 3
Association •••••••••••••••••••••••••••••••••••• 41
D. Enthalpy Changes as a Function of Ionic
Strength ••••••••••••••••••••••••••••••••••••••••••• 41
• ••••••••••••••••••••••• 0 •••• 49 v. SUGGESTIONS FOR FUTURE WO RK
REFERENCES ...... 50 LIST OF FIGURES
Figure ' � - 0 1. Plot of log K versus !2 for NaS0 at 10 , A(c) 4 0 0 25 and 4 0 C ••••• •.•••••••••• , .•••• , •••••••••••••••••••• 29
t 1 2. Plot of log K + 4A versus I for NaS0 - A(c) I2 4 0 0 0 at 1 0 , 25 and 40 C •••••••••• •••••• , ••••••••••••••••••• JO
' - Plot of ln K versus 1/T for NaSo I=O, A(c) 4 at .0967, .2544, .4923 and •7006 ••• •••••••••••••••••••••••• 31
' 4. Plot of A H versus I for NaSo - over the 4 temperature range 10° - 40°c ••••••• ••••••••••••••••••••• 32 ' � - 0 Plot of log K versus !2 for Naco at 10 , A(c) 3 0 0 25 and 4o c I I a I I I 1 1 1 I t 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I I 37
' 1 6. Plot of log K + 4AI versus for Naco - A(c) 2 I 3 o 0 0 at 10 • 25 and 40 C • , ••••••••••••••••••••••••••••••••• •
' Plot of ln K versus 1/T for Naco - at I=O, A(c) 3 .243'.r, .4879 and .6918 ...... 39
' 8. Plot of AH versus I for Naco - over the 3 o ° tempera t ure range 1o - 40 c ••••••••••••••••••••••••••• 40 ' 1 Plot of og K versus for NaHco 0 at 250 9o � A(c) !2 3
and 40 C • • •• o • o • • , • o • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • •• •••• 45 ' � 0 10. Plot of log K + versus for NaHco A(c) 2AI2 I 3 ° at 25 and 40°c •.•••••••.•••••••• •.•••.•.••••••••••••••• 46 ' 0 11. Plot of ln K versus 1/T for NaHco at I=O, A(c) 3 •2274, •Jl+-06 and •6801 •••••••••••••••••••••••••••••••••• 47 ' 12. Plot of AH versus for NaHco 0 over the I 3 0 0 4 •••••••••••••••••••••••••••• temperature range 25 - 0 C 48
viii LIST OF TABLES Ta.ble 1. Appcrent and Zero Ionic Strength Association Constants for NaSo4- at 10°, 25° and 4o0c •••••••••••••••••••••••••• 27 - 2. Comparison of Apparent Association Constants for NaS04 at 2 5°c ...... ,, ...... 28 3 - . Thermodynamic Functions of Formation for NaS04 from Free Ions at 25°c ...... 28 4. Apparent and Zero Ionic Strength Association Constants ° 0 ...•..•...... •.....•.• for Naco3- at 10 , 25° and 4o c 35 - 5. Compa.rison of Apparent Association Constants for Naco3 at 20°c and 25oc . . • • . . . • ...... • • . . • . . . . • • . . • . . • . • • • • • • • J6
6. Thermodynamic Functions of Formation for Naco3- from Free Ions at 25°c ...... 36
7. Apparent and Zero Ionic Strength Association Constants ° ° ° 0 •••• .••••••••••.•••..••• for NaHco3 at 10 , 25 and 4o c 43 8. Comparison of Apparent Association Constants for NaHco3° 0 . at 20 and 25°c ...... �
9n Thermodynamic Functions of Formation for NaHco3° from • Free Ions at 25°c ...... ft • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • �
ix GLOO SARYOF SYMBOIS
A Debye-Hiickel limiting slope (equal to .51 in water at 25°c) a distance of closest approach of ions (cm) B empirical constant b parameter in Davies equation C concentration of species indicated (mol/l) D dielectric constant of solvent E electrode response (mV) E0 standard electrode potential (mV) EM glass membrane potential (mV) EIE internal electrode potential (mV) Eref reference electrode potential (mV) E. liquid junction potential (mV) J e electronic charge (esu) t AG free energy change of formation of the ion pair from the free ions AH I enthalpy change of formation of the ion pair from the free ions 2 (t�z. I ionic strength 1 c. 1 ) KA thermodynamic association constant I KA apparent association constant 16 k Boltzmarmconstant (1.)81 x10- erg/K) m concentration of species indicated (mol/kg H20) 23 N Avogadro constant (6.023 x 10 molecules/mole)
R gas constant 1.987 (cal/(mol • K)) t AS entropy change of formation of the ion pair from the free ions T absolute temperature (K) x T activity coefficient of i i
L'± mean ionic activity coefficient (M) concentration of M ( M)I initial concentration of M ( M)T total concentration of M (M)F free ion concentration of M (M) activity of species M 2 ionic strength (t�z.1 m.1 ) I, INTRODUCTION
In the study of the thermodynamic characterization of natural water systems, knowledge of the effects of temperature, pressure and solution composition is requiredo The species in solution will not only be simple ions, organic molecules and dissolved gases, but complex ions and ion pairs. Ion association is important because it affects the viscosity, the electrical conductance, and the solubilities of minerals in seawater. To apply the ion-pair model to marine systems, reliable thermodynamic data are needed. These thermodynamic data can provide a description of the ion-pairing processes and provide some information on the hydration structure of the ion pairs.
A thermodynamic model of seawater has been developed! and refined�'J
Starting with the elemental composition of seawater, the standard free energies of the major and minor species and activity coefficients of 3 these species, Atkinson et al calculated the concentrations of the most important dissolved species by the use of a computer program which solves a system of linear equations and iteratively minimizes the total free energy of the systemo The calculated results for free ion concen trations and for prediction of precipitation agree well with those determined experimentally.
Seawater is a concentrated multicomponent electrolyte solution consisting mainly of the salts NaCl, Na 0 , Na co , NaHco , MgC1 :z8 4 2 3 3 2 and Cac1 dissolved in water. Due to the effect of carbonate and 2
1 2 sulfate on precipitation equilibria, these ions are important in sea water and other natural water systems. The concentrations of free carbonate and sulfate depend on the concentrations of all other ions in solution. Although the stability constants for NaSo -, Naco - and 4 3 NaHco ° are small, the relatively large concentration of sodium ion in 3 sea water compensates to some degree so that accurate values of these stability constants are required for computer modeling of marine systems.
Computer modeling is also very useful in characterization oil well brines from secondary recovery operations, where supply waters are pumped into the well and the oil is displaced along with produced water.
Produced water often contains high concentrations of alkali metal and alkaline earth ions with chloride, sulfate and carbonate being the principal anionic species. An undesirable side effect of this method is the production of scale, typically alkaline earth carbonates and sulfates, in the pipes and the producing formation, a porous rock.
Eventually drastic steps, such as fracturing with explosives, must be taken to maintain oil production. Accurate data and good computer models could predict when scaling would occur and also what mixtures of supply waters or what complexing agents could be used to reduce or prevent scaling in the most economically advantageous way.
4 The present research is an extension of that conducted by Blum .
We have undertaken a systematic study of NaS0 -, Naco - and NaHco ° 4 3 3 ion association reactions as a function of temperature and ionic J strength. We have improved upon the method of Blum to obtain a set of reliable association constants by varying only the relative concen- trations of complexing and non-complexing ions. The apparent associ- ' ation constants KA so determined are useful in that they may be applied to stoichiometric calculations without the aid of activity coefficients and because they depend on ionic strength but not on the relative composition of the medium5. This latter characteristic, which was further substantiated for MgSo4° ion pairs5a, permitted the use in the present study of relatively simple solutions to determine the ' ' . effects of temperature on - and H o • By using KNaS04" , K NaC03 KNa COJ these data, thermodynamic parameters for ion association in sodium sulfate, carbonate and bicarbonate solutions were calculated. 4
II. THEORY
A. Ionic Theories
The first statistical theory of electrolyte solutions was developed
6 by Debye and Huckel in 1923 and its application has been remarkably successful in interpreting the behavior of very dilute solutions. The ions are regarded as point charges distributed in a continuum possessing a dielectric constant identical with that of the pure solvent, and it is also recognized that ions group themselves a little closer around an ion of opposite charge than they do around ions of like charge. From the theory, it is sought to calculate the average potential energy of a given ion in solution due to all the other ions, assuming the medium to have the dielectric constant of the pure solvent. In the argument strong electrolytes are assumed to be completely dissociated into ions, and observed deviations from this ideal behavior are then asscribed to electrical interactions between the ions.
In dilute solutions of non-electrolytes, the chemical potential of the solute G, is adeq�ately described thermodynamically by the equation:
- -o G = G + RT ln m (1) -o where G , the standard chemical potential, is the chemical potential the solute would have in a 1 molal ideal solution. Although satis- factory for non-electrolyte solutions at concentrations as high as 0.1 molal, equation (1) is not adequate for electrolyte solutions even as 3 dilute as 10- molal. As an electrolyte A B dissociates into v+ v+ v- cations and v- anions, equation (1) can be written as: 5
- -o + vRT m + vRT ln G = G ln + Y- + (2)
where- is the mean molal activity coefficient of the el ectrolyte, y+
and v = v + v , The extra free energy represented by the third term +
on the right ha nd side of equation (2) mainly reflects the energy of
interactio n of the electrical charges on the ions, We ha ve seen that
the electrostatic forces fall off rather slowly with distance between
the charges and this fact accounts for the large deviat io ns from
ideality in ionic solutions, even at high dilutions where the ions are
far apart,
In their interio nic attractio n theory, Debye and Hilckel ca lculated
the electrical contribution to the free energy, and the mea n activity
coefficient of the ions on a mole fraction scal e was given by the
limiting laws
(3)
2 where I= �c.z . is the ionic strength, and A is a constant of the t:. 1. 1. ' Debye-Hlickel theory,
1 3 3 A = (21tN)2 e /2oJOJ(l0DkT)2
The activity co efficient of a z-valent ion is:
2 1 log f. = -Az. (4) 1. 1. r2
Equatio n (4) has been well substantiated for extremely dilute solutions of elect rolytes. It is under such conditions that the experimental
uncertainties become greatestp and in an attempt to extend the theory 6 to higher concentrations, Debye and Hiickel removed the assumption of ions being point charges and considered the average distance of closest approach o� solvated cations and anions. The resulting expression for the ionic activity coefficient is
(5) 1 + B a r2 2 1 1._ 1 where B = (8�Ne )2/(10-'DkT)2 • k, T, and D are the Boltzmann constant, the absolute temperature and the dielectric constant of the solvent respectively, with e and N being the electron charge and Avogadro constant and a the mean effective diameter of the ions in the solution or the distance of closest approach o:-' the ions. Since no independent method is available for evaluating a, it is essentially an empirical parameter. A 2 in e�uation ) gives the effect of the long-range - z.J. rt (5 1 coulomb forces, while (1 + B a I2) shows how these are modified by the short-range interactions between ions. This equation agrees well with experimental results for 1-1 electrolytes to an ionic strength of abou·i:.
0.1 M. There have been many attempts to extend the Debye-Hiickel theory. 7 Guntelberg suggested the use of a formula containing no adjustable parameter:
(6) 1 + r2
Obviously such a formula cannot compete in accuracy with those con- taining param8ters, but it has the advantage that it can be extended to 8 solutions of several electrolytes. In 1938, Davies proposed the formula 1 2 I2 - log f. = Az. ( - bI ) (7) J. J. 1 + I2 7 8 It has been shown that the equation is in good agreement with the actual_values of the mean ionic activity coefficients of dilute solu- tions of 1-1, 1-2, and 2-1 electrolytes: the average deviation is about
2% in 0.1 M solution and proportionately less at lower concentrations 9 when b is taken as 0.2. Davies reassessed the accuracy of equation 10 11 (7) in the light of recent compilations of activity data • and has proposed the revised value b = 0.3 in equation (7),
B. Ion Association
For a strong electrolyte an equilibrium is established between the free ions and ion pairs. This equilibrium can be expressed by
n lll+ n- ( (8) M1* + N M_ N Ion pair) with the equilibrium expression
(9) where the parentheses represent the activities of the various species and K is the thermodynamic association constant. It is often conve- A I nient to use the apparent or stoichiometric constant K , where the ac- A tivities are replaced by the stoichiometric concentrations:
(10)
The stoichiometric equilibrium constant can be converted to the thermo- dynamic constant by multiplying by the appropriate activity coefficients to obtain: 8
= (11)
where the"( represent the activity coefficients of the species i indicated.
Since the association constants for the alkali metal sulfates and carbonates are smallp conductivity or ultrasonics may not be the best method for determining these constants. A better method of determining these constants is through ion-selective electrodes. Theoretically, the ion selective electrode responds only to the activity of free ions.
Thus, if free ions exist in equlibrium with ion pairs, the measured activity of the free ions will be less than that calculated assuming complete dissociation. If we can then attribute this decrease solely to the formation of ion pairs, we can calculate the concentration of ion pairs.
C. Ion Selective Electrodes
Membrane electrodes specific to a wide variety of ions can be categorized into three general types: (a) glass membranes (b) liquid ion-exchanger membranes (c) solid-state membranes. For a glass membrane specific ion and reference electrode in contact with a test solution, this set-up can be represented as:
Internal Internal Inner Glass Outer External Reference electrode solution hydrated membrane hydrated test electrode layer layer solution E E E IE PB1 M PB2 ref 9 where E ' E and are the potentials of the internal electrode, IE M Eref membrane and reference electrode respectively and PB , PB are phase 1 2 boundaries. The glass membrane potential is due to phase boundary potentials at each hydrated glass-solution interface and possibly a diffusion potential. Diffusion potentials arise within hydrated layers. Since ionic diffusion rates are almost equal in magnitude but opposite in direction, they will nearly cancel each other. Therefore
E EPB EM = PB1 - 2 a RT a 1,glass 2,solution = ln (12) - nF a1,solution a2,glass where a and a refer to the activities of the specific ion in the 1 2 internal and external solutions, respectively. EPB and EPB are phase 1 2 boundary potentials at each hydrated glass-solution interface. If external and internal surfaces are identical in number of exchange sites
= and if complete exchange occurs at each surface, then a 1,g 1 ass a2,glass0 Therefore
= - RT 1 RT E ln ------a a nF 2,solution (13) M nF 1,solution ln Since the internal solution is of constant composition, the activities of the ions in equation (13) will be constant and
_ ...;:;..• 3__,0 ;...__3 R _ T _ = k, _2_ log a (14) E M nF 2,solution
However, because the sides of the glass can have different environ- ments, the potential is often not zero. This has been termed the asymr.ietry potential. It is possible to write the overall voltage for the cell as: 10
2.303 RT E - E + k1- log a + E + E = IE ref nF 2 asymmetry j
- K - 2.303 RT (15) nF
K - E K K where E + k'+ E + E Since is not known, = IE ref asymmetry j"
mustbe eliminated through a calibration procedure. Therefore an ion
selective electrode is responsive to the activity of the selected ion
in the absence of any interfering ionso The same basic equation ( 5) 1
will also hold for an ion exchange or solid state membrane electrodes.
Liq_uid ion-exchange membranes are porous plastic or glass membranes
impregnated with a liq_uid ion-exchange solution which is a high mole-
cular weight organic solute with acidic9 basic or chelating functional
groups which interact strongly with ions of interest. The solid state
membranes are of three types: (a) single crystal type, (b) mixed crystal
type, and (c) precipitate type. The precipitate electrode has a membrane composed of an insoluble salt of the ion of interest pressed into a
pellet, precipitated on the electrode surface or impreganated into a
silicone or plastic matrix. A more complete treatment of these membrane
2 4 electrodes can be found in the literature.1 -1
4 D. Determination of Association Constants
The formation of an ion pair (NaSo4� for example) can be represented
by
+ 2 Na + so - � NaS0 - (Ion Pair) (16) 4 4 11
We may express the total concentration of sodium ion, lNa+) as the T concentration of free sodium ion, (Na+J F plus the concentration of the ion pair, [NaS04 -J or
(17)
Similarly,
(18) 2 Since (NajT and (so4 -JT are known, it would be possible to determine the concentration of the ion pair and to calculate the apparent associ- ation constant if the free ion concentration of either the sodium ion or the sulfate ion can be measured. We can measure the free ion concentration through the use of an ion-selective electrode.
From equation (15), we have
2.303 RT E log a • (19) cell = K - nF
The coefficient of the logarithmic term is not always equal to the
Nernstian value but varies with electrode composition and must be experimentally determined. Equation (15) then becomes
( ) E = K - S log a , 20 where S is the experimentally determined slope. The activity of the ion can be written as a ='Ye, where-yis the activity coefficient and c is the molarity of the free ion. 12
The activity coefficient should be a function of ionic strength.
Since the activity coefficient is assumed constant at constant ionic strength and the K term will be invariant, the difference between two voltage readings taken at constant ionic strength will be due to a difference in the concentration of the free ion. We can ex press the voltage at a concentration C as
E = K - S log C - S log/ (21) and for a concentration C at the same ionic strength
I I E = K - S log C - S log ( (21a)
Then the difference is given by
c E - E = S log C - S log C = S log--, (22) c
A change in voltage can then be directly related to a change in concentration of the free ion.
Two types of measurements are made. First a run is done in which the concentration of the sodium ion is varied at constant ionic strength in the absence of any significant complexing ions. This allows us to calculate S, the slope, by
E - E s = (22a) log (c/c )
Once the slope is known the results of the second run can be interpreted.
In the second run, a solution of sodium chloride and tetrarnethylarnrnonium sulfate is added to a solution of sodium chloride and tetrarnethyl- 13
' ammonium 4·,4 -biphenyldisulfonate (where all Na+ are assumed to be as the free ions). Thus both solutions have the same total sodium ion concentrat�on and ionic strength.
When aliquots of the sulfate solution are added,the voltage is lowered even though the total sodium ion concentration has not changed.
It is proposed that this decrease in electrode response and the corres- ponding decrease in the free sodium ion concentration is due to the formation of the ion pair, NaS0 - 4 . The electrode response can be compared to the known voltage and concentration of the initial sodium chloride solution to yield the free sodium ion concentration. Re- arranging equation (22) results in
(22b)
where Na is the free sodium ion concentration with an electrode [ "J F ' + response of E and (Na )1 is the initial sodium ion concentration with a voltage of E. Then using equation (22b) the free sulfate ion con- centration can be calculated.
Once the free ion and ion pair concentrations are known, it is
I possible to express the apparent equilibrium constant, KA' from equation
(10)
(23) 14
' KA can be converted to the thermodynamic constant by multiplication of the proper activity coefficient by using equation (11). The activity coefficients for the sodium sulfate ion pair and the sodium ion can be estimated at low ionic strengths u�ing the Davies equation� However, since both are of the same charge type it can be expected that in dilute solutions both will have approximately the same activity co- efficient so that equation (11) becomes
(24)
The thermodynamic association constant can be calculated by taking the logarithm of both sides of equation (24) yielding
I log KA = log K - log 2- (25) A Ys 04 Substituting log y 2- for equation (7), equation (25) becomes - 804 1. ' Iz 2 = 2 log K + A(Z 2-) ---..--- log K + A(Z 2-) b I (26) A so4 1 + r2 A so4
1. ' 2 2 ----..--I A plot of log KA + AZ versus I would have an intercept of 1 + r2 2 log KA(at zero ionic strength) and a slope of AZ b. Davies has suggested that b = 0.3 works best for most salts .
E. Thermodynamics of Ion Association
For a reaction at constant temperature, pressure and charge in which a and b moles of reactants A and B react to form c and d moles of products C and D, one may write 15
aA+bB�cC+dD (27)
F?:om the definition of activity the following equation can be written
-o aG = aRT ln a A - aGA A
-o = - bG bRT 1n a l:CB B B (28) - cG - cGo = cRT a C C ln c dG - dG0 = dRT D D 1n �
The total change in free energy for the reaction will be
o AG -4G =RT 1n (29)
where
(30) and
AG0 = cG0 + dG0 - a .G0 - bB0 (31) C D A B ll.G is the free energy change of the reaction, and .6G0 is the free energy change when all the products and reactants are in their standard states.
When the reaction is in equilibrium at constant temperature, pressure, and charge, the composition of each phase is fixed. Then from
�G VdP + d G = -SdT + 11 dn + • • • • • • •a dn -- de (32) _r-1 1 /-c c a e we know AG = 0 and as a consequence,
0 AG = - RT ln (33) 16
where K is the equilibrium constant of the reaction. In our experi eq
ment we can write
' . AG - RT ln K = A (J4)
• K where A is the association constant. From some other standard
equations,
' • - - dA G K AS = R l n + (35) ( dT ) p A
and ' . ' AG =AH - TAS (36)
One can also obtain the expression
(37)
' . Experimental values of A S and AH are often obtained from experimental ' values of KA over a temperature range by introducing the approximation I I that AS and AH are independent of temperature. Exceptionally
t I K accurate measurements of A are needed to give reliable values of AS , ' AH and AG • 17
III. EXPERIMENTAL
A. Reagents
Tetramethylammonium chloride,(cH ) NCl. 3 4 Tetraethylammonium chloride,(C H ) Ncl. 2 5 4 Stock solutions of tetramethylammonium chloride and tetraethylammonium
chloride (usually 2-3 M) were prepared. The salts were not weighed di- rectly because of their hygroscopic nature. The stock solutions were analyzed by passing a known volume of the solution through a column of
Dowex 50W-X8 cation exchanger charged with H+. The HCl produced was then titrated with a standardized base to the phenolphthalein endpoint.
Tetraethylammonium carbonate, Cc H ) N co . A Dowex 50W-X8 ( 2 5 4 J 2 3 cation exchange column in the acid form was charged with tetraethyl- ammonium ion. The effluent solution was tested with pHydrion paper until there was ·no further evidence of hydrogen ion being displaced from the column. This operation takes 3 days and requires a 15:1 excess of solu- tion. Then a small amount of tetraethylammonium hydroxide was passed through the column to eliminate any residual amounts of acid present.
A weighed amount of sodium carbonate was passed through the column and the tetraethylammonium carbon�te was collected and used. A Dowex 1-X8 anion exchange column was charged from the chloride form to the carbonate form using sodium carbonate (ca. JM). This task requires a 15;1 excess of solution and takes 3 days. The charging continued until the effluent solution contained no chloride. A known volume of a recently standar- dized tetraethylammonium chloride solution was passed through the column 18
and the tetraethylammonium carbonate was collected and used.
Tetramethylammonium sulfate, [(cH)4NJ2so4• A Dowex 1-X8 anion exchange column in the chloride form was charged with sodium sulfate
(ca. 3 M). This task takes 3 days and requires a 15:1 excess of solu-
tion. A known amount of tetramethylammonium chloride was passed through the column and the tetramethylammonium sulfate was collected and used.
I P 4 4 otassium , -Biphenyldisulfonate, K2(so3c6H4c6HJ303). The 4,4'- biphenyldisulfonic acid (Eastman P 4590) was treated with potassium carbonate solution to a pH 7 endpoint using a pH electrode. The neutral solution was boiled with decolorizing carbon for a moment to wet the
carbon particles and effect thorough mixing. The particles were allowed to settle, and the supernatant liquor was inspected for color. The filtrate was set in cool room for crystallization. The colorless, needle-shaped crystals were filtered in a sintered glass funnel, washed with cool water and recrystallizedfrom conductivity grade water. The crystals were then collected and used.
1 Tetramethylammonium 4,4 -Biphenyldisulfonate, �CH3)4N)2BPDS. 4 41 'I'etraethylammonium , -Biphenyldisulfonate, [(c2H5)4N)2BPDS. A Dowex 50W-X8 cation exchange column in the acid form was charged with tetramethylammonium ion (or tetraethylammonium ion) . The effluent solutions were tested with pHydrion paper until there was no further evidence of hydrogen ion being displaced from the column. Then a small amount of tetramethylammonium hydroxide (or tetraethylammonium hydroxide) 19 was passed through the column to eliminate any residual amounts of acid ' present. A weighed amount of potassium 4, 4 -biphenyldisulfonate was
' passed through the column and the tetramethylammonium 4,4 -biphenyl-
' disulfonate (or tetraethylammonium 4,4 -biphenyldisulfonate) was collected and used.
B. Apparatus
1 • pH/mV Meter
All measurements were made using an Orion 801 digital pH/mv meter.
The readings were taken on the expanded scale to a· .pJ?ecision of ± 0 .01 mV. When the meter is in the expanded scale mode the decimal point is not automatically moved but will remain in the regular place (between the third and fourth digit) . Before the start of a run, the meter was zeroed using the zero adjust on the back panel with a shorting strap across the standard and reference electrode inputs.
2. Ion Selective Electrode
Measurements on the sodium salts were made using an Orion 94-11 ion selective electrode. This is a glass electrode with a Ag-AgCl internal sodium chloride solution. The reference electrode used with the sodium ion-selective electrode was an Orion model 90-01 single junction reference electrode. The filling solution used for the sodium runs was a lithium trichloroacetate solution (Orion 90-00-19).
3. Waterjacketed Cell 20
The cell used for our experimental work was waterjacketed to main-
taln constant temperature and was fitted by a frame to a magnetic
stirrer, The cell was insulated from the stirrer with a one-inch thick-
ness of insulation. The stirrer , mV meter, and bath were placed upon
and grounded to copper sheet, The cell was fitted with a rubber stopper
which had holes drilled for the electrodes, thermometer, nitrogen inlet
and pipettes, The waterjacketed cell was connected to a 7 gal, constant
temperature bath filled with deionized water. The water was heated and
circulated with a Br ownwill heater-circulator (Brownwill Scientific )
and cooled through a copper cooling coil through which cold water was
circulated. Cool water was produced and circulated with a Forma Jr, bath
and circulator (Forma Scientific) ,
C. Experimental Methods
The solutions were made using distilled, deionized water and
calibrated volumetric glassware, Stock solutions of NaCl were prepared
by weighing dried solids to� 0,00002 g on a Mettler model H semimicro
balance, dissolving the solid and diluting to volume. Stock solutions
of tetramethylarnmonium and tetraethylarnmonium salts were prepared and
analyzed by the methods described in the reagents section.
' � 0 0 0 1. Determination of KA for NaSo4 at 10 C, 25 C and 40 C
Three solutions were prepared from stock solutions for these runs.
' Solution I contained NaCl, tetramethylarnmonium 4,4 -biphenyldisulfonate ( (((cH3)4N)2BPDS), and CH3)4NC1, Solution II contained NaCl, ( (CcH3)4NJ2so4 and cH3)4NCl, and solution III contained (ccH3)4NJ2BPL6 21 and (cH3)4NCl but no NaClo To each solution a small amount of (cH3)4NOH was added to adjust the pH of the solution to 7.5 - 8.5. Each solution had the same pH, (cH3)4NCl concentration and ionic strength. As an example, for the solution of ionic strength equal to 0.7006, the
concentration of each solute in each of the three solutions is as
follows:
Solution I Solution II Solution III
(NaCl) 1.00446 x 10-JM (NaCl] 1.00446 x 10-JM 44 (((cH3)4 NJ 2BPDSJ .0 79M (l(cH3)4N12so4) .04478M [( Ionic Strength .7006 .7005 .7006 Before a run was made the instrument was zeroed using the zero adjust control on the back panel with a shorting strap across the standard and recorder outputs and the voltage was recorded on the expanded scale • A .75 ml portion of solution I was then pipetted into the clean dry cell and the constant temperature bath was adjusted to 0 0 0 25 C ( or 10 c, 40 c ) • The electrode assembly was rinsed, wiped dry and placed in the cell. The solution was left to equilibrate until the initial reading was taken( when the electrode response did not show any drift). Then 30 ml of solution II was pipetted into the cell. Care was taken so that the pipette did not contact the solution in the cell. The corresponding new voltage was recorded when the new solution had equilibrated, Then 20 ml aliquots of solution II were pipetted into the cell and the corresponding voltages recorded until a total solution 22 volume of 145 ml (or 165 ml) was reached. The precision in the voltage readings is believed to be � 0.02 mv. The cell was emptied, rinsed and dried. A new 75 ml of solution I was pipetted into the cell and allowed to equilibrate. Then 5 ml aliquots of solution I were pipetted into the cell and the corresponding voltages recorded until a total solution volume of 100 ml was reached. This was the calibration run and was used to determine the Nerstian 0 A • slope, A mV / log (Na+) The same procedure was repeated at 10 C and 4o0c. ' - 2 K 0 2 0 4 0 . Determination of A for Naco3 at 10 C, 5 C and 0 C Three solutions were used for these runs. Solution I contained NaCl, (Cc2H5)4NJ 2BPDS and (c2H5)4 NCl. Solution II was a solution con taining NaCl, (Cc2H5)4NJ2co3 and (c2H5)4NC1, and solution III contained (Cc2H5)4NJ �PDS and (c2H5)4NC1 but no NaCl. To each solution a small amount of (c2H5)4NOH was added to adjust the pH of the solutions to 10.5 - 11.0. As an example, for the solution of ionic strength equal to .6918, the solute concentrations for each solution are as follows: Solution I Solution II Solution III (NaCl) 1.04208 x 10-JM (Nac1) 1 • 04208 x 1o-3M (�c2H5)4N] 2BPDS] .04001M ((Cc2H5)4N)2co3) .04006M (�c2H5)4N]�PDS) .04001M ( ( c2H5) 4Nc1J • 5?07M (Cc2H 5) 4Ncl) .5?07M �C2H5)4Ncl) .5716M Ionic Strength .6918 .6919 .6916 23 The sodium carbonate measurements were done in the same way as ° those described for the sodium sulfate runs at 10°c, 25 c and 4o0c. In addition, purified nitrogen was passed through water in a gas bubbler at a flow rate of about 10 ml/min. Nitrogen was used over the solution to eliminate absorption of co from the atmosphere which would add more 2 carbonate to the solution. ' . Determination of K for NaHco at 25 0C and 400 C 3 A 3 Because NaHco is a 1 1 electrolyte, the three tested solutions 3 - only contained the solutes NaCl, (cH ) NCl, and (cH ) NHCo • Solution I 3 4 3 4 3 contained NaCl and (cH3)4Ncl. Solution II was a solution containing NaCl, (cH ) NHC0 and (CH ) NC1, and solution III contained only (CH ) NC1. 3 4 3 3 4 3 4 To each solution a small amount of (ctt ) NOH was added to adjust the 3 4 pH of the solutions to 7 .5 - 8 .O. As an example, for the solution of ionic strength equal to .6801 , the solute concentrations for. each solution are as follows: Solution I Solution II Solution III 3 1.00446 x 10-3M NaCl [ ) 1.00446 x 10- M CH ) NHco .05992M � 3 4 ;1 .6?91M CH ) Ncl .6192M [( 3 4 ) Ionic Strength .6801 .6801 .6802 The sodium bicarbonate measurements were done in the same manner ° as those described for the sodium sulfate runs at 25°c and 40 c. The 24 reason we could not obtain results at 10°c was that the association constants are too small to be determined accurately by this method . 'Ihe small voltage differences produced by low degrees of association ' give rise to large uncertainies in the experimental values for KA. 25 TV, RESUL'IS AND DISCUSSION A. Sodium Sulfate Runs ' 1. Determination of KA at 100 , 25 0 and 40 0c. Data from the calibration run are used to construct a plot of E + versus log (Na ) , which should be a straight line, the slope of which is determined by the method of least squares . These experimentally observed slopes are usually lower than the Nernstian slope. The apparent association constants were calculated by using equations (22b) , (17) , (18) and (23) . For each run at 105, 125 and 145 ml readings were recorded. The reading at the 145 ml volume is usually the most precise due to the greater extent of association compared to that at the 105 and 125 ml points . The calculated apparent association constants at ° ° 0 10 , 25 and 4o c are given in Table 1. These constants were corrected ' 1 to I=O by plotting log 4A I2 versus I Figure 2 at each temperature. KA + ( ) These plots gave better straight lines than those based on the Davies equation. The thermodynamic constants obtained at 10°, 25° and 4o0c are 15.8�.4, 8 • .)t .2 and 4.6± .3 respectively. Our data are compared ' with KA values of other workers in Table 2. The values reported by 15 Fisher and Fox are much higher than the other constants at zero ionic strength . This is because conductivity tends to be insensitive to small degrees of association and �s not a specific technique but relies only on the measurement of a bulk property for determination of association constants . The measurement of ion activity by ion selective electrodes is a more sensitive method and can yield precise apparent association 26 constants even when the association constant is small . In our work, a set of reliable association constants were obtained with due attention paid to the activity factor in the tested solutions . 2. Thermodynamic Functions of Sodium Sulfate Ion Association The enthalpy changes for the formation of sodium sulfate ion pairs at each ionic strength were obtained by plotting ln K� versus 1/T (Figure J) , the slope being determined by the method of least squares . I Using the slope multiplied by the gas constant R, values of A H were f I obtained . The changes in free energy, AG , and entropy, AS , were calculated from I ' AG = RT K ln A and ' I f AS = (AH - AG )/T The calculated values for these thermodynamic functions are given in f f I Table 3, It is apparent that AG , AH and AS depend on ionic ' strength. The results show that AH for sodium sulfate ion association changes from negative to positive as the ionic strength increases and is a linear function of the ionic strength of the solution . A plot of AH versus I (Figure 4) shows a linear relationship. The equation I at based on the plot can be used to calculate values of KA, aS0 - � 4 various temperatures and ionic strengths . Values of A H also can be calculated for the range of ionic strengths covered by our measurements . 27 Table 1. Apparent and Zero Ionic Strength Association Constants for - 0 Naso4 at 10 0 , 250 and 40 c. I ' ' I {10°C) KA{c2 {1o0c2 I {22002 KA{c2 {25°c2 I {4o0c2 KA{c2 {40002 + 0 15n8 ± .4 0 8.5 - .2 0 4n6 · ! .3 .0970 4.7 +- .2 .0967 2.85 "t .06 .0962 1.74 "t .04 ,2551 2 0 93 "t .06 .25LJ4 2.10 t .04 .2532 1.49 t .04 .4936 2.03 "t .06 .4923 2.16 ! .04 .4899 2.25 "t .04 .7025 1.79 "t .06 .7006 2.44 ! .04 .6972 3.19 ! .03 The data listed above can be summarized as the following equations: 1 * log = 1.199 - 10°0: K�(c) 2.00 I2 + 1.043 I (38) 1 ' ** log = ( KA(m) 1.200 - 2.00 t'-2 + i.008 r 39) ' * 1 log = ( 25°0: KA(c) .930 - 2.04 I2 + 1.667 I 40) ' ** 1 log = KA(m) .939 - 2.04_)-l2 + 1.580_f{ (41) I * 1 log = ( 40°0: KA(c) .666 - 2.10 r2 + 2.301 I 42) I ** 1 = log _ ( KA(m) .686 2.1�2 + 2.14sr 43) * Molarity basis . ** Molality basis . 28 - Table 2. Comparison of Apparent Association Constants for NaSo at 25°c. 4 ' ' K K I -A (this work) A (o ther workers) Method Ref. 0 8.5 "t .2 12.5 :: 2 conductivity 15 6.6 conductivity 16 5.3 conductivity 17 .49 2.16 ± .04 2.5 ! .02 Na electrode 18 .61 2.26 ! .04 2.02 :t .03 pH measurements 4 .70 2.44 ! .04 - Table 3. Thermodynamic Functions of Formation for NaS04 from Free Ions at 25 0C. 1 ° ° ° 1 ° AH1 (10 C-40 c) AG (25 c AS (25 c) I (Kcal/m ole) Kcal mole (cal/mole O) 0 - 7 .18 :t .16 - 1.27 ± .02 - 19.8 ± .6 .0967 - 5.77 :t .17 .62 ! .01 - 17.3 ± .6 + .2544 - 3.96 ! .15 .44 - .01 - 11.8 ± .5 .4923 + .61 ± .05 .46 :t .01 + 306 ± .2 ,7006 + J.39 ! .04 .53 :t .01 + 13.2 :t .2 The data listed above can be summarized as the following equations: I * A H = -7.34 + 15.4 I (.54) ' ** AH = - 7.13 + 13.9_µ (55) * Molarity basis. ** Molality basis, 29 1.30 1.20 1�10 1.00 .90 ...... _.80 ..._C> - < :::.::: .70 � 0 ri .60 .50 .40 .30 .20 .10 • .8 1.1 0 .1 .2 .3 .4 ·.6 7 .9 1 .o 1 r2 - ° ° Figure 1. Plot of log K�(c) versus rt for NaS04 at 10 c, 25 c ° ° ° ° and 40 c. ( A 10 c, • 25 c, e 40 c) JO 3.00 2,80 2o60 2.40 2.20 .,..JN 2.00 H j + 1.80 ,,...... _ '-"0 - < � 1.60 bO 0 rl 1.40 1.20 1.00 .80 ,60 .40 .20 • ,7 ,9 0 .1 .2 . ) .4 5 .6 ,8 1 .o 1.1 I 1 ' 2 ° Figure 2, Plot of log + A for 10 , KA(c) 4 I versus I NaSo4- at ° 0 ° ° 0 25 and 4o c. ( A 10 c, • 25 c, e 4o c) .31 J.00 2.80 2.60 2.40 2.20 2.00 1.80 ...... -...... 0 1.60 - < :::.::: � rl 1.40 1.20 1.00 .80 .60 .40 .20 J.O .3.1 J.2 .3• .3 .3.4 1/T x 10-.3 Figure o, .3· Plot of ln K�(c) versus 1/T for NaS04, at I = = • O, & .0967, .2544, .492.3 and .7006 ( • I= I .0967, 4 • • I = .2544, e I = . 92.3 , t I = 7006) 32 6.oo 4.oo 3.00 2.00 1.00 '"' Q) '6 0 �r-i ct! {) � -1.00 :i:: � -2.00 -J.00 -4.00 -5.00 -6.oo -7.00 -8.00 0 .1 .2 .4 .6 .7 • 8 .9 1.0 1.1 I ' Figure L�. Plot of � H versus I for NaSo4- JJ B. Sodium Carbonate Runs ' 0 1. Determination of K at 100 , 25 0 and 40 c. A The data for the sodium carbonate runs were treated in the same 0 way as those described for the sodium sulfate runs at 10 , 25 0 and 400 c. The calculated apparent association constants at 10° , 25° and 40°c are given in Table 4 . These constants were extrapolated to zero ionic ' .!. strength by plotting log K 4AI2 versus I (Figure 6) at each tempera- A + ture. Just as in the sodium sulfate runs , these plots gave better straight lines than those based on the Davies equation . The thermo 0 dynamic constants obtained at 10°, 25° and 4o c are 17.1 : 1.4, 19.1 � .6 and 21 .0 : .7 respectively. Our data are compared with ' K ' - values of other workers in Table 5. The value for K - A,Naco A,NaC0 3 3 agrees well with the value calculated from Garrels and Thompson 's results . By using equation (56), the apparent association constant of sodium carbonate at 20°c was calculated . Our result agrees with the 0 value of Lin and Atkinson at 20 C and I = .19. It is believed that our work yields the most precise values due to the fact that it utilizes a "first order" method to measure the constant " Ion-selective electrodes are directly sensitive to the free sodium ion concentration, which can be directly related to the concentration of the sodium carbonate ion pair. 2. Thermodynamic Functions of Sodium Carbonate Ion Association The enthalpy changes for the formation of sodium carbonate ion I pairs at each ionic strength were obtained by plotting K versus ln A 1/T (Figure 7) . The data for these runs were treated in the same manner as those described for the sodium sulfate runs . The calculated ' ' values for these thermodynamic functions , AH , AG and AS are given ' in Table 6, A plot of AH versus I (Figure 8) shows a linear relation- ' ship and that AH changes from positive to more positive as the ionic strength increases. As described for the sodium sulfate runs , the I equation based on this plot can be used to calculated values of - KA,NaC0 3 at various temperatures and ionic strengths . c. Sodium Bicarbonate Runs I 1. Determination of at 25 0 C and 400 c. KA The data for the sodium bicarbonate runs were treated in the same 0 way as those described for the sodium sulfate runs at 25°c and 4o c. The calculated apparent association constants at 25°c and 4o 0c are given in Table 7. These constants were extrapolated to zero ionic I .!. 2AI2 ° ° strength by plotting log KA + versus I (Figure 10) at 25 and 40 c. These plots gave better straight lines than those based on the Davies equation . The thermodynamic constants obtained at 25°c and 4o0c are 3.35 .07 and 6.75 -: .02. Our data are compared with o � KA,NaHC0 3 values of other workers in Table 8. Our values are higher than the values reported by other workers . By using equation (58) , the apparent association constant of sodium bicarbonate at 20°c was calculated . Again , we believe our results to be the most accurate be�ause we are using a method which unambiguously yields the free sodium ion concen- tration . 35 Table 4. Apparent and Zero Ionic Strength Association Constants for - 0 Naco3 at 10 , 250 and 40 0C. ' ' ' I (1 0°C) KA(c) (1 o0c) I (25°c) KA(c) (25°c) I (4o0c) KA(c) (40°c) 0 17 .1 ± 1.4 0 19.1 -: .6 0 21 .0 -: .7 .2440 2.64 ! .06 .2434 3.58 : .04 02422 4.74 -: .04 + .4892 1.43 -: .04 .4879 2.52 ! .04 .4855 4.18 - .04 .6936 1.14 ! .04 .6918 2.35 ! .02 .6884 4o72 "!:: .04 The data listed above can be summarized as the following equations: ' * .!. 10°C: log KA(c) = 1.233 - 2.00 I2 + .693 I (44) ' ** ..!. log KA(m) = 1.229 - 2.00).£-2 + .684jJ- (45) ' * ..!. 25°C: log KA(c) = 1.281 - 2.04 I2 + 1.131 I (46) ' ** .!. = log KA(m) 1.293 - 2.04.)4 2 + 1.058.Jl (47) ' * ..!. 4o0c: log KA(cJ = 1.323 - 2.10 I2 + 1.583 I (48) ' ** .!. log KA(m) = 1.)48 - 2.10_fl2 + 1.448)"' (49) * Molarity basis. ** Molality basis. J6 Table Comparison of Apparent Association Constants for - 5. Naco3 o at 20 C and 250 C. t t K K I A (this work) A (other workers) Method Ref. 0 19.1 't .6 18.6 pH measurements 1 3.5 + .1 from Kd of H2co3 19 2.2 Na electrode, 20 Amalgam electrode .19 3.70 ± .03(20°c) 4.2 ± .08(20°c) Na electrode 22 .50 2.54 ± .04 1.38 : .5 Na electrode, 20 Amalgam electrode .72 2.31 't .02 4.25 ± .3 pH measurements 21 - Table 6. Thermodynamic Functions of Formation for Naco3 from Free Ions at 250 C. ' t ' AH (10°C-4-0°C) AG (25°C AS (25°c) I (Kcal/mole) Kcal mole (cal/mole O) 0 + 1.22 ± .02 - 1.75 ± .02 + 9.97 ± .13 .24.34 + 3n4J ± .02 .76 ± .01 + 14.1 : .1 .4879 + 6.28 t .02 .55 : .01 + 22.9 't .1 .6918 + 8.33 : .16 .51 ± .01 + 29.7 t .6 The data listed above can be stunmarized as the following equations: ' * AH = 1.11 + 10 .4 I (56) t ** = A H 1 .48 + 8 .8_fi- (57) * Molarity basis. ** Molality basis . 37 1.30 1.20 1.10 1.00 .80 ,_...._ () - • ';:<< . 70 tiD 0 r-l .60 .50 .40 .30 .20 .10 0 .1 .2 .3 . 4 .5 .6 .7 .8 .9 1.0 1.1 1 r2 I � 0 ° Figure 5. Plot of log versus for Naco - at 10 , 25 KA(c) I2 3 0 ° ° ° and 40 c. ( .A 10 C, • 25 C, 0 40 C) 3.60 .3.40 J.20 J.00 2.80 2.60 .-!ll\I H � 2.40 +-... () ...... - < :::..:: 2.20 bD 0 ...-1 2.00 1.80 1.60 1.40 1.20 1.00 .80 .4 .9 1 .o 0 .1 .2 • .3 .6 .8 1 .1 I - ' 1 0 of log A versus I for Naco at 10 , Figure 6. Plot KA + 4 I2 3 0 (c) 0 ° ° 25 and 4o c. ( A 10 c, • 25°c, • 4o c) 39 J.20 3.00 2.80 2.60 2.40 - 2.20 2.00 1.80 ';' 1.60 '-" • < � � . 1.40 1.20 1.00 .80 .60 .40 .20 0 3.0 3.1 3.2 3.3 3.4 3.5 . 3 1/T x 1o- Figure 7. Plot of for Naco at I = ln K�(c) versus 1/T 3 O, .2434, • = II I = = �4H79 and .6918 (A I O, .24J4, GI ,Ll-879, 9 I = .6918) 40 14o0 13.0 12.0 11.0 10.0 9.0 ...... ,. (l) ,..... 0 8.0 �,..... � ::.:: ...... 7.0 . :z: 5.0 4.o 3.0 2.0 1.0 I 0 .1 .2 .3 .4 .5 .6 .7 .8 ..9 1 .o 1.1 I ' Figure Plot of t::.. H versus for Naco - 8. I 3 41 2. Thermodynamic Functions of Sodium Bicarbonate Ion Association The enthalpy changes for the formation of sodium bicarbonate ion ' pairs at each ionic strength were obtained by plotting lnKA versus 1/T (Figure 11) . The data for these runs were treated in the same way as those described for the sodium sulfate runs. The calculated I I I values for these thermodynamic functions, AH , AG and AS are ' given in Table 9. By plotting AH versus I (Figure 12) , a linear equation was also demonstrated. By using this equation, values of AH can be calculated for the range of ionic strengths covered by I our measurements and values of o at various temperatures and �aHC03 ionic strengths can be obtained. D. EnthalpyChan ges as a Function of Ionic Strength - At zero ionic strength we observe that for NaSo4 the association - constant decreases with increasing temperature while for Naco3 and NaHco3° the opposite trend is observed. Increasing temperature produces two effects: increasing thermal energy and a decreasing dielectric con- stant leading to an increase in potential energy between the ions. For - NaS04 the former effect is determining; for Naco3- and NaHco3 the latter effect is predominant. Trends in AH as a function of ionic strength are more dif:ficult to rationalize. Increasing ionic strength tends to decrease the dielectric constant in most cases and has an effect on the structure of the solvent - - and its ability to coordinate to solute ions . For NaS 04 and Naco3 ' the increase in A H with increasing ionic strength can be explained by 42 + 2- 2 assuming an increased potential energy between Na and or co - so4 3 0 due to a decrease in effective dielectric constant. For NaHCO the J opposite trend is observed. Presumably an increased ionic strength results in a marked change in the water structure that produces a lowering - of stability for the ion pair. This effect is not observed for Naco 3 or NaS04- where the potential energy is much stronger due to the doubly charged anion . 43 Table 7. Apparent and Zero Ionic Strength Association Constants for 0 250 0 NaHco3 at and 40 C. ' ' I (2,2o C2 KA (25°C) I (4o0c2 KA (40°C2 0 3.35 .07 0 6.75 .02 ! ! .06 n2263 2.28 .04· .2274 1.27 � � .04 .3389 1.82 .04 .)406 1.07 't ! .6801 .04 .6768 1 n12 .04 .76 't '± The data listed above can be summarized as the following equations: ' 1 25 * .525 - 1.02 r2 a290 I (50) °C : log KA(c) = + I 1 ** .521 - 1.0 2 n278J'- (51 log K � + ) A(m) = I ° * .829 -1.05 r21 .123 (52) 40 C: log KA(c) = + I I ** .826 - 1.05�21 .122jk (53) log K + A(m) = * Molarity basis. ** f1olality basis. 44 Table 8. Comparison of Apparent Association Constants for NaHco30 at 20°c and 25°c. ' ' I KA (this work2 KA (other workers2 Methods Ref. 0 3.35 ± .01 1.45 from Kd of H2co3 19 .56 pH measurements 1 ° ° .36 .86 ± 003 (20 C) o5 t .1 (20 C) Na electrode 22 .5 .89 ± .03 .39 ! .15 Na electrode 20 Amalgam electrode .66 .77 ± .04 .26 pH measurements 1 .72 .74 ± .04 .280 ! .001 pH measurements 21 Table 9. Thermodynamic Functions of Formation for NaHco3° from Free Ions at 250 C. ' ° ' ' AH (10 C-40°C) .t:.G (2S0c) AS (25°C) I (Kca1Lm ole2 (Kca1Lmole2 (calLmole 02 0 + 8.66 - .72 ! .01 + 31 o5 ± �1 .2274 + 7.22 - .14 ± 003 + 24.7 ! .1 .3406 + 6.56 .04 ! n02 + 22.1 � .1 .6801 + 4.79 + .16 ! 003 + 15.5 ± .1 The data listed above can be summarized as the following equations: ' * A H = 8 • 57 - 5.66 I (54) ' ** .t:.H 8 .57 - 5-37!-"' (55) * Molarity basis. ** Molality basiso 45 1.00 .90 .80 .70 .60 .50 .40 ,...._ (.) ...... - < ::.:: .JO l:lD rl0 .20 .10 0 -,10 -.20 -.JO -.40 . • 1.1 0 .1 .2 .J .4 .5 1 .6 7 8 .9 1.0 r2 ' i 0 0 Figure versus z at 25 9. Plot of log KA(c) r for NaHCOJ and and 4o0c . ( • 25°c, o 4o0c) 46 1.20 1 .15 1.10 1.05 1.00 .95 ..-ilN H < .90 C\I + ,-.... (.) .85 ...... � < :::.::: tll) 0 .80 r-i .75 .70 .65 .60 .55 .50 I - l I f 0 .1 .2 .J .4 .5 .6 .8 • 9 1 .o 1.1 I � I 0 0 Figure 10. Plot of log K�(c) + 2AI2 versus for NaHco3 at 25 ° ° and 4o0c. ( • 25 , • 40 c) 47 2.40 2.20 2.00 1.80 1.60 1.40 1.20 ...... - < � 1.00 s:: r-1 .80 .60 .40 .20 0 -.20 - .40 3.0 3.1 3.2 3.3 3.4 1/T x 10-3 I versus I = O, Figl!.re 11. Plot of ln KA(c) 1/T for NaHCOJ at .2274, .)406, .6801 • ( A I = 0, • I = .2274, e I = .4879, �I = .6801) 48 11 .0 10.0 "'"' 8.0 Q.l ,.; 0 �,.; ro () � '-' 7.0 :::r: 6.o 4.0 0 .1 .2 .3 .4 .6 .7 .8 1.0 1.1 I I 0 Figure 12. Plot of AH versus I for NaHco3 49 V. SUGGES TIONS FOR FUWRE WORK Future studies should be made to further the understanding of ion association in natural water systems . Measurements of appar ent associ ation constants of other alkali metal ions with sulfate, carbonate and bicarbonate could be made using methods described in this thesis ft It is also important to study ion association and solubility equilibria involving alkaline earth ions and sulfate, carbonate and bicarbonate. Each time a comparison method should be used in which a model electrolyte such as ((cH3)4NJ 2BPOO , is replaced by ((cH)4N)2so4, for exampleo The decrease in cation activity can then be directly related to the formation of ion pairs . Another interesting task is determination of apparent association constants for sodium ion pairs with ortho, meta and para-benz ene disulfonate ions . Comparison of these three association constants with the charge separation of the sulfonate groups in ortho, meta and para benzenedisulfonate ions will provide useful information regarding the relationship of structure and ion association. Work with the sulfate and carbonate electrodes could also lead to interesting results . Data from this method will tend to be less precise because the slope is only half that of the sodium ion-selective electrode. Also these electrodes tend to be less stable than the sodium electrode. Lack of a reliabl e cation-selective electrode may necessitate use of carbonate or sulfate-selective electrodes , however . .50 REFERENCES 1. Garrels, R. M. and Thompson, M. E., Am. J. Sci., 260, 57 (1962). 2. Whitfield, M.,Marine Chemistry, J, 197 (1975) . 3. Atkinson, G., Dayhoff, M. 0. and Ebdon, D. W., Marine Electro chemistry, Electrochemical Society, Princeton, N. J. , 1973. 4. Blum, F. D., Stability Constants of Sodium Carbonate, Sodium Sulfate and Potassium Carbonate in Water at 25oc, Master's Thesis, Charleston, Illinois, Eastern Illinois University, 1977. 5. Kester, D. R. and Pytkowicz, R. M., Limnol. Oceanogr., �14, 686-92 (1969) . _5a.Kester, D. 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