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A novel viscometric method for determining the stability constant of a compound of two uncharged species
J. inorg,nucl.Chem., 1969,Vol.31, pp. 159to 167. PergamonPress. PrintedinGreat Britain
A NOVEL VISCOMETRIC METHOD FOR D E T E R M I N I N G T H E S T A B I L I T Y C O N S T A N T OF A C O M P O U N D OF TWO U N C H A R G E D SPECIES H. M. N. H. I R V I N G andJ. S . S M I T H Department ot~Inorganic and Structural Chemistry, The University of Leeds, Leeds 2, England
(Received 19June 1968) A b s t r a c t - Equations are deduced relating the viscosities of dilute solutions of formally uncharged species A, B, C . . . which do not interact physically with those of admixtures of these molecular species in the same solvent. The relative viscosity, ~gret.is shown to be a simple additive function for each component i involving its mole fraction X and the respective viscosity parameters (S,)~ and (S~)~ derived from the empirical equation ~ra. = 1 + (So)X+ (S~)X 2. By introducing the condition that the species C is a chemical compound AnBm that dissociates to an extent ct into the components A and B it is shown that precise viscosity measurements can be used as the basis of a novel procedure for determining the stability constants of such dissociable compounds. The method was tested for bis-adducts of the square planar complex (diacetylbisbenzoylhydrazone)nicket(II) with pyridine, 4-methylpyridine, and isoquinoline, and was found to give stability constants in good agreement with those obtained by other methods. Limitations and advantages of the new procedure are discussed critically.
IN A PREVIOUS paper[l, 5] we showed that the relative viscosities of very dilute solutions of formally neutral coordination compounds in organic solvents may be interpreted quantitatively in terms of H I L D E B R A N D ' S Theory of Regular Solutions. The same arguments may be used to explain the viscometric behaviour of mixtures of compounds in solution. Consider a solution containing three solute species (A, B and C). We postulate that individual solutions of A, B and C in the solvent used are all regular, and that the different species do not interact. Let the mole fractions of solvent, A, B, and C be XI, Xa, X , and Xc respectively. It has been shown [ 1, 5] that for a regular solution V1 °
~ret. = ~qsot./~sot~.= ~
exp. { (AEsv - ~E1 v)/ART}
(1)
where r/ret, is the relative viscosity of the solution, V~° the molar volume of the solvent, VM the molar volume of the solution, and AEs~ and AE1~ are the molar energies of vaporisation of solution and solvent respectively. A is the constant relating activation free energy for viscous flow (AGv~sc.) to AE v for a given solvent. Specifically, AG*~c. = AE~/A.
(2)
H. M. N. H. Irving andJ. S. Smith,J. inorg, nucl. Chem. 30, 1873 (1968). L. Sacconi, G. Lombardo, and P. Paoletti, J. chem. Soc. 848 (1958). H. M. N. H. Irving and N. S. AI-Niaimi, J. inorg, nucl. Chem. 27,2231 (1965). J. H. Hildebrand and R. L. Scott, The Solubility of Non-Electrolytes, p. 432. Reinhold, New York (1950). 5. H. M. N. H. Irving and J. S. Smith, J. inorg, nucl. Chem. To be published. 159 1. 2. 3. 4.
160
H . M . N . H . IRVING and J. S. SMITH
Now AE, ~ = -- AEs~a,w + AE ~ (for the separate components) = -- A E s m m + X , AE, ~+ XAAEa v + X~AEB ~+ XcAEc ~ .
(3)
By assuming that at very low concentrations each of the solute species mixes independently with the solvent, the total energy of mixing becomes a simple sum, and the S C A T C H A R D - H I L D E B R A N D equation gives ~kEmLl~ttto,= (8, -- 8A)2XA~'rA + (8, -- 8B)2XB~-"rB"J~(8, -- 8c)2Xc~'rC where ~' represents the partial molar volume and 8 the solubility parameter of a given component. Substitution of this expression for AEm~,~ in Equation (3), followed by substitution of the resultant equations for AE~ v in Equation (1) and subsequent rearrangement of terms leads to V, 0 'Ore. = "~Mexp. {--8,' [XA (17',+ I?,) +Xn(~', + ['n) + X c ( # ' , + 17'e)] + 28, [SaXa l?a + 8BXn~'C + 8cXc~'c]}/ART.
(5)
As derived above, this equation is only valid (a) at very low concentrations (strictly, at infinite dilution), (b) when individual solutions of A, B or C are each regular, and (c) provided the species A, B and C do not interact in solution. These limitations make the equation of little use as it stands; nevertheless, a principle of practical significance can be derived from it. N o w V~, = V,0+ (V~A--V,°)X~+ (#~--V,°)XB+ (#c--V,°)Xc
(6)
so that when Xa, X8 and X c are small, and when terms involving higher powers can be neglected, V'° = 1 - ( I ? a - V'°)Xa VM
V, °
(I?B-
V,°)XB (f'c-- V,°)Xc V, °
(7)
V, °
By substituting for (V,°/VM) in Equation (5), expanding the exponential, and neglecting terms involving X, 2 and higher powers we obtain ~rel.-- 1 = {--8,2[ (~r,'~- ~'rA)XA"~ ([/'I"~- PB)XB"~ (~r,'at- ~'rc)Xc]
+ 28, [Sa ~'AXA + 8n #'nXB + 8cP'cXe] }/AR T -- (~rA-- V10)XA V, °
( V B - - VI°)XB V, °
(~rc-- V ' ° ) X c Vl °
(8)
It has been shown [ 1, 5] that for a solution of a single substance at low concentration "t]reL
-
1=
[--812X2 (|'~vl "Jr- [~2) "Jr"28182X2['~'2 "[ ART
(['~2 -- Vl°)X2 V, o
(9)
where the subscripts 1 and 2 refer to the solvent and solute respectively. Comparison of this equation with Equation (8) shows that the latter is of the f o r m ('OreL -- 1 ) mt~ture ~- (~ret. -- 1 ) A + (~rel. -- 1 )O + (~rel. -- 1 ) c
(10)
A novel viscometric m e t h o d
161
which establishes the type of function representing the viscosity of a mixture as a simple sum of the corresponding functions for the individual (non-interacting) solutes. The form of the expression for fewer or more species than the three considered above will be obvioust. N o w it has also been shown that a reasonably low concentrations[l] the relative viscosities of many solutions may be adequately represented by the empirical equation ~rel. = 1 + SoX + S1X 2.
( 11 )
So, which is the initial slope of the plot of relative viscosity against mole fraction, was shown to be equal to [-- 612(V1°~- ~'~2)+ 217"28,62]/ART- (I?2 - Vl°)/Vi °.
(12)
$1 is a constant that must be included to take into account the observed deviations from linearity at higher concentrations. By using the principle of additivity deduced above, and by combining Equations (10) and (1 1), it follows that
(m~,.- 1) = [(So)~X~ + (Sl)~X2] + [(So)~X~ + ( s , ) . x . 2]
+ [(S0)cXc+ (s~)cXc~].
(13)
This is a semi-empirical equation, for which it is no longer essential that .4, B and C should form regular solutions; moreover, it should be valid over the whole range of concentration used in determining the S-coefficients of the individual species involved. The only condition still retained is that A, B and C do not interact physically. We now consider the case where the component C is a compound of the species.4 and B and is formed according to the equation n,4 + m B ~ .4 nBm
If the mole fraction of the complete undissociated compound.4,Bm is denoted by X b and the actual degree of dissociation is a, we have X A = naX'c X8 = m a X b X c = (1 - ot)Xb.
Substitution in Equation (13) gives ( ~ e t . - 1) = [ (So) anaX'c + (S~) anZa2(Xb ) 2] + [ (So)BmaX'c + (S1)BmZt~z(X'c)2]
+ [(S0)c(1 - o O X b +
(S1)c(1 - ot)z(Xb)2].
(14)
t l f we write AV for the difference b e t w e e n the viscosity of a solution and that of the pure solvent (rt~eL -- 1)
"q~et.-- ~,olvent
Art
~solvent
l~solvent
Equation ( 1O) can be generalized as follows;-
A'Omt.rture~ ~ ( ATI )sotute which is the formal s t a t e m e n t of the conditions postulated above in which, inter alia, the solute species do not interact in a physical sense.
162
H.M.N.H.
I R V I N G and J. S. S M I T H
Collection of like terms in a gives ( ~ r a . - - 1) = az(xb)Z[n2(S,)a + m2(S1)B + (S1)c]
+ txX'c[n (S0)a + m (S0)B -- (So) c - - 2(S1)cX'c]
(15)
+ Xbf (So)c + (S,)cXb].
This equation may be used to determine the stability constant of compounds of the general formulaA~Bm in the following way: 1. Solutions of the individual components A and B in the chosen solvent are studied independently and the four coefficients (S0)a, (S1)A, (So)B and (SOB are calculated from the relevant viscosity measurements. 2. The S-coefficients are determined for AnBm by using sufficient of component B in the solvent to suppress dissociation. It is assumed that the presence of excess B would not materially affect the values of the S-coefficients for the pure components A and B. 3. Values of ( "Oret. 1) are then determined for a series of solutions of the compoundA~Bm of different apparent mole fractions, X~. 4. The experimental data are substituted into equation (15), which may then be solved to give the value of a at each concentration used. N o w the formation constant, -
-
(1--a)a Kn,m = ( n a a ) n ( m a a ) m
(1 - - a ) n~ . m m . a~+m . a~÷m-1
(16)
where a is the effective concentration of AnBm, i.e. assuming it to be completely undissociated (a = 0). Clearly a --- lO00Xb/Vu mol 1. -1, where I'm is the molar volume of the solution in cm 3. To examine the applicability of these principles the formation constants were studied for adducts of the square planar chelate (diacetylbisbenzoylhydrazone) nickel(II), NiD, with three different heterocyclic bases, B. Spectrophotometric measurements[2], confirmed by studies of liquid-liquid extraction[3], showed that bis-adducts occurred in each case, any mono-adducts being relatively weak. The above discussion is therefore simplified to a consideration of the equilibrium A + 2B ~ AB2 where A is equivalent to NiD, B to the heterocyclic base, andAB2 is NiD.2B. In Equations (14)-(16) n = 1 and m = 2 and Kn,m = t2 = [AB2]/ [,4] [ B ] 2 .
EXPERIMENTAL
lsoquinoline was fractionally distilled from zinc dust under reduced pressure. 4-Methylpyridine was heated under reflux with, and then fractionally distilled from, calcium hydride.
Pyridine was heated under reflux with, and then fractionally distilled from, potassium hydroxide. Benzene. AnalaR grade benzene was stood over sodium wire for one week, filtered and used without further treatment.
(Diacetylbisbenzoylhydrazone)nickel(ll) was prepared by slowly adding a solution of nickel acetate (2-5 g) in water (150 ml) to a suspension of diacetylbisbenzoylhydrazone (K. and K. Laboratories, 2 g) in ethanol (100 ml) heated under reflux. Heating was continued for 1 hr. The product was
A novel viseometric method
163
collected by filtration of the hot mixture, and recrystallised from benzene as dark violet needles. (Found: C, 57.0; H, 4.5. Calculated for ClaH~sN402Ni: C, 57.0; H, 4.23). (Diacetylbisbenzoylhydrazone)nickel(ll)bispyridine, Pyridine (3 g) was added to the nickel complex ( 1 g) in a few ml of chloroform. The adduct, which separated as brown leaflets on the addition of light petroleum, was collected and washed with petroleum ether. (Found: C, 62.0; H, 5.0; Calculated for C1sHlsN402Ni, 2CsHsN: C, 62.6; H, 4.9%). As reported previously by Sacconi, Lombardo and Paoletti[2] the adduct loses pyridine when stored in a vacuum desiccator (H~SO4). (Diacetylbisbenzoylhydrazone)nickel(ll)bis-4-methylpyridine, when prepared similarly, formed dark brown microcrystals. (Found: C, 63.5; H, 5'4, C1sH~eN402Ni, 2 CeHTN required C, 63.7; H, 5.4%). (Diacetylbisbenzoylhydrazone)nickel(ll)bis-isoquinoline was prepared in the same way. (Found: C, 67.9; H, 4.7, C~sH~rN4OzNi, 2C9HrN requires C, 67.8; H, 4.8%). Density and viscosity measurements were made as described previously[l], in a thermostat controlled at 25-0.005°C. Density measurements were converted into molar volumes with the relationship I'M = (XIM1 + X2M2)/d where X~ and )(2 are the mole fractions of solvent and solute respectively, M1 and M2 are the corresponding molecular weights, and d is the density of the solution. Plots of VM against mole fraction for each of the pure components A, B and C were shown by least squares analysis to be linear within experimental error, and could therefore be represented by the general equation I'M = Vl°+ (V2 - VI°)XzThe results are summarised in Table 1. Table 1
Solute lsoquinoline 4-methylpyridine Pyridine NiD Isoquinoline adduct 4-methylpyridine adduct Pyridine adduct
Highest mole fraction used
I72°(ml)
tr~
25.8X 8.3X 9.5X 173.4X
0.007091 0.01712 0.04070 0.0004622
115.2 97.7 79.9 262.8
___0.002 0.001 0.001 0.001
89-937 + 394.5X
0.0005261
484-4
0-001
89.609 + 350.2X 89.025 + 299.8X
0.001098 0.0004670
439-8 388-8
0.003 zero
Equation for molar volume VM= 89.422 + 89.423 + 89.423 89.422 +
X is the mole fraction of the respective solute. O'vM is the standard deviation with respect to VM of experimental points from the best straight line derived from least squares analysis. Flow-time measurements were converted into relative viscosities for solutions of known mole fraction by using the relationship t.o,~" d,o~. ~rel. ~
~soln/~solv.
~ lsolv. " dsotv.
where t represents the flow-time and d the density. The density of each solution was calculated from the molar volume equation for pure components, or from a graph of molar volume against mole fraction
164
H.M.N.H.
I R V I N G and J. S. S M I T H
for equilibrium mixtures. F o r the pure components the results could be represented by Equation (13), and they are summarised in terms o f the calculated S-coefficients in Table 2. The final results for the calculation of formation constants from the relative viscosities of solutions of the adducts, using Equation (16), are tabulated in Tables 3, 4 and 5. Table 2 Solute Isoquinoline 4-methylpyridine Pyridine NiD Isoquinoline adduct 4-methylpyridine adduct Pyridine adduct
So
$1
1.8945 0.3456 0.4934 12.743
9-0 0.8 0.4 - 2912
(rn,,t.
8"
AE ~ (kcal)
___0.00001 zero 0-00002 0.00002
9.72 9.15 10-11 10.80
10.88 8.17 8-17 30.7
19.4246
835
0-00001
9.54
44.0
15.565 16.441
445 1042
0-00005 0.00001
9.21 9.88
37.3 38.0
* From Equation (12). O'nret" is the standard deviation with respect to */ret. o f the experimental points from the curve *?ret.= ! + S o X + S 1 X ~, determined by least squares analysis. Table 3. NiD + isoquinoline X~
a(mole 1.-1)
d. 25
Flow-time
"0tel
a
/32
log $2
zero 0.0001063 0~0003319 0.0004711 0.0007292
zero 0.001188 0.003706 0.005258 0.008128
0-87359 0.87386 0.87438 0.87473 0.87531
667.03 667.94 670-03 671.31 673.85
1.00000 1.00168 1.00541 1-00774 1.01220
-> unity 0.847 0.767 0.650
--4599 4679 4816
--3.66 3.67 3.68
Table 4. N i D + 4-methylpyridine X~
a(mole 1.-1)
zero 0-0002735 0.0008609 0-0009626
zero 0.003055 0.009594 0.01072
d425 0-87360 0-87412 0.87522 0.87546
Flow-time
~rel.
a
t2
log fl~
667.44 669.40 674.41 675.27
1-00000 1-00353 1.01232 1.01389
-0-927 0-430 0.398
-2457 19520 20838
-(3.41) 4.29 4.32
a
fl~
log/32
-0.971 0.929 0.919 0.880
-2490 5227 5700 3630
-3.40 3.72 3.76 3.56
Table 5. NiD + pyridine Xb
a(mole 1.-1
d4~5
Flow-time
zero 0.0001593 0.0001835 0.0001914 0.0003122
zero 0.0017804 0.0020507 0.0021389 0.0034874
0.87354 0.87388 0-87393 0-87395 0.87420
667.50 668.66 668.85 668.90 669.78
~,~l. 1.00000 1.00213 1.00247 1.00258 1.00417
A novel viscometric method
165
DISCUSSION
Formation constants of the bis-adducts of N i D with isoquinoline, 4-methylpyridine and pyridine determined as above may be compared with the values calculated from spectrophotometric measurements by Sacconi et al.[2], by Irving and AI-Niaimi [3] (Table 6). The agreement between the two sets of values is very Table 6. Formation constants of adducts of NiD with heterocyclic bases in benzene at 25°C
Base lsoquinoline 4-Methylpyridine Pyridine
Log ,Sz (25°C) (a) (b) 3.67 4.3 3.6 +- 0.2
3.78 _+0.02 4.45 ___0-02 3-78 _+0.02
(c) 3.73 4.54 --
(a) Present work. (b) Values from [2]. (c) Calculated from the values K = (8.15_+0.39) × 103 and (5-53_+0-13)×104 for 20_2°C[3], using the entropy values A S = - - 3 0 . 1 and - 3 2 . 9 c a l deg. -1 respectively from [2].
satisfactory, and we suggest that our method may therefore find applications where other methods are not suitable for the determination of dissociation constants. A study of Fig. 1 will assist in understanding the choice of conditions for obtaining the most reliable answer by this method. The figure represents the general shape of the two curves that would be obtained experimentally; their separation is a function of the extent of dissociation of the compound A,,Bm and it is the ability to measure this separation accurately that determines the accuracy of the method. As required by theory, the two curves converge at high mole fraction. The scale along the mole fraction axis is compressed for a strong adduct and extended for a weak adduct. The following considerations may limit the usefulness of this method in certain systems. (a) Fig. 1 shows that the separation between the curves (and hence the accuracy of results) increases initially with the concentration of the dissolved species. If the solutes involved have low solubilities, the experimental points will be crowded near the origin, leading to low precision. This is exemplified in our own results, since N i D and its adducts (in particular the species NiD, 2 pyridine) have a rather low solubility in benzene. (b) Equation (17) shows that in the absence of added base the dissociation of the adduct AB2 will be I> 50% when log/3~ ~< 4 for 4 for a = 0.01 M (mole fraction 0.001 for Vu = 100) and when log/32 ~< 6 for a = 0-001 M (X = 0.0001). Since an appreciable dissociation of the adduct is necessary to get a meaningful separation of the curves (Fig. 1), this suggests 1og/32 = 6 as an upper limit for this method.
166
H.M.N.H.
I R V I N G and J. S. SMITH
IL
mole f r a o t i o n o f adduot Fig. 1. Upper eurve-undissociated AB2; lower curve-equilibrium mixtures, A +2B.
AB2
(c) The determination of the S-coefficients for the undissociated compound depends on being able to inhibit its dissociation with a concentration of one component which is small enough not to change the characteristics of the solvent. If b is the additional concentration of base B which must be added to the adduct at concentration a = l A B s ] such that the fraction dissociated is no greater than a, it is easy to show that b = { ( 1 - ct)/afl~} 1/2
__
Table 7 gives calculated values of b for
2t~a.
a = 0-01 M
(17) and a =
0-005. Since the
Table 7. Concentrations of heterocyclic base needed to suppress dissociation of 0.01 M solutions of adduct in benzene to a = 0-005 log K
1-0
2"0
3-0
4.0
5'0
6"0
7"0
Conc. of additional B (mole I.-')
4.5
1.4
0-47
0.16
0.065
0-034
0.024
A novel viscometricmethod
167
second term in Equation (17) is very small compared to the first for values of log fl~ < l01°, the estimates of b are equally valid for a = lABs] from 0.1 to 0.001 M. Practical considerations limit the permissible upper concentration range, and the need to keep the concentration of added B as low as possible would seem to fix a lower limit of about log fl~ = 2.5. As it stands the proposed procedure demands very careful measurements of viscosity involving long flow-times, in order to realise significant differences between solvent and dilute solutions. Quite apart from the experimental difficulties encountered in such work, e.g. the avoidance of dust particles, the accurate measurement of flow-times, precise temperature control etc., the measurements are intrinsically time-consuming and the procedure would not compete with, for example, spectrophotometric methods should these be applicable. However, where no alternative is available, the new procedure offers considerable advantages, and the construction of some form of direct-reading viscometer giving rapid but accurate results for low-viscosity liquids would do much to make it acceptable. Under optimum conditions it would be possible to simplify the new procedure considerably by dispensing with the need to determine experimentally the values for the viscosity parameters (So)c and (SOc for the compound A,Bm. For being given experimental data in (~/re~, X) to form a sufficient number of equations of the form (12), a solution by successive approximations can be applied in which arbitrary values of K are varied until the standard deviations in the calculated average values of (So)c and (SOc are both minimised. A few other points deserve comment. The value of 82 = 10.1 for pyridine obtained in the course of this work agrees very well with values of 10.7, 10.4, 9-7 and I0.5 obtained by quite independent methods [4]. The values of the viscosity parameters So and $1 show interesting trends. So is of much the same order for the heterocyclic bases, is higher for the complex NiD and higher still for the three adducts. S~ for the bases is again small, but large for the three adducts. The very large negative value of $1 = - 2912 for the complex NiD appears unexpected. However it is noteworthy that whereas values of So and $1 are normally positive, negative values have been reported for chromium(lII) acetylacetonate in methanol and ethanol[l], where there is a known degree of non-regularity which has been attributed to solvation.
A NOVEL VISCOMETRIC METHOD FOR D E T E R M I N I N G T H E S T A B I L I T Y C O N S T A N T OF A C O M P O U N D OF TWO U N C H A R G E D SPECIES H. M. N. H. I R V I N G andJ. S . S M I T H Department ot~Inorganic and Structural Chemistry, The University of Leeds, Leeds 2, England
(Received 19June 1968) A b s t r a c t - Equations are deduced relating the viscosities of dilute solutions of formally uncharged species A, B, C . . . which do not interact physically with those of admixtures of these molecular species in the same solvent. The relative viscosity, ~gret.is shown to be a simple additive function for each component i involving its mole fraction X and the respective viscosity parameters (S,)~ and (S~)~ derived from the empirical equation ~ra. = 1 + (So)X+ (S~)X 2. By introducing the condition that the species C is a chemical compound AnBm that dissociates to an extent ct into the components A and B it is shown that precise viscosity measurements can be used as the basis of a novel procedure for determining the stability constants of such dissociable compounds. The method was tested for bis-adducts of the square planar complex (diacetylbisbenzoylhydrazone)nicket(II) with pyridine, 4-methylpyridine, and isoquinoline, and was found to give stability constants in good agreement with those obtained by other methods. Limitations and advantages of the new procedure are discussed critically.
IN A PREVIOUS paper[l, 5] we showed that the relative viscosities of very dilute solutions of formally neutral coordination compounds in organic solvents may be interpreted quantitatively in terms of H I L D E B R A N D ' S Theory of Regular Solutions. The same arguments may be used to explain the viscometric behaviour of mixtures of compounds in solution. Consider a solution containing three solute species (A, B and C). We postulate that individual solutions of A, B and C in the solvent used are all regular, and that the different species do not interact. Let the mole fractions of solvent, A, B, and C be XI, Xa, X , and Xc respectively. It has been shown [ 1, 5] that for a regular solution V1 °
~ret. = ~qsot./~sot~.= ~
exp. { (AEsv - ~E1 v)/ART}
(1)
where r/ret, is the relative viscosity of the solution, V~° the molar volume of the solvent, VM the molar volume of the solution, and AEs~ and AE1~ are the molar energies of vaporisation of solution and solvent respectively. A is the constant relating activation free energy for viscous flow (AGv~sc.) to AE v for a given solvent. Specifically, AG*~c. = AE~/A.
(2)
H. M. N. H. Irving andJ. S. Smith,J. inorg, nucl. Chem. 30, 1873 (1968). L. Sacconi, G. Lombardo, and P. Paoletti, J. chem. Soc. 848 (1958). H. M. N. H. Irving and N. S. AI-Niaimi, J. inorg, nucl. Chem. 27,2231 (1965). J. H. Hildebrand and R. L. Scott, The Solubility of Non-Electrolytes, p. 432. Reinhold, New York (1950). 5. H. M. N. H. Irving and J. S. Smith, J. inorg, nucl. Chem. To be published. 159 1. 2. 3. 4.
160
H . M . N . H . IRVING and J. S. SMITH
Now AE, ~ = -- AEs~a,w + AE ~ (for the separate components) = -- A E s m m + X , AE, ~+ XAAEa v + X~AEB ~+ XcAEc ~ .
(3)
By assuming that at very low concentrations each of the solute species mixes independently with the solvent, the total energy of mixing becomes a simple sum, and the S C A T C H A R D - H I L D E B R A N D equation gives ~kEmLl~ttto,= (8, -- 8A)2XA~'rA + (8, -- 8B)2XB~-"rB"J~(8, -- 8c)2Xc~'rC where ~' represents the partial molar volume and 8 the solubility parameter of a given component. Substitution of this expression for AEm~,~ in Equation (3), followed by substitution of the resultant equations for AE~ v in Equation (1) and subsequent rearrangement of terms leads to V, 0 'Ore. = "~Mexp. {--8,' [XA (17',+ I?,) +Xn(~', + ['n) + X c ( # ' , + 17'e)] + 28, [SaXa l?a + 8BXn~'C + 8cXc~'c]}/ART.
(5)
As derived above, this equation is only valid (a) at very low concentrations (strictly, at infinite dilution), (b) when individual solutions of A, B or C are each regular, and (c) provided the species A, B and C do not interact in solution. These limitations make the equation of little use as it stands; nevertheless, a principle of practical significance can be derived from it. N o w V~, = V,0+ (V~A--V,°)X~+ (#~--V,°)XB+ (#c--V,°)Xc
(6)
so that when Xa, X8 and X c are small, and when terms involving higher powers can be neglected, V'° = 1 - ( I ? a - V'°)Xa VM
V, °
(I?B-
V,°)XB (f'c-- V,°)Xc V, °
(7)
V, °
By substituting for (V,°/VM) in Equation (5), expanding the exponential, and neglecting terms involving X, 2 and higher powers we obtain ~rel.-- 1 = {--8,2[ (~r,'~- ~'rA)XA"~ ([/'I"~- PB)XB"~ (~r,'at- ~'rc)Xc]
+ 28, [Sa ~'AXA + 8n #'nXB + 8cP'cXe] }/AR T -- (~rA-- V10)XA V, °
( V B - - VI°)XB V, °
(~rc-- V ' ° ) X c Vl °
(8)
It has been shown [ 1, 5] that for a solution of a single substance at low concentration "t]reL
-
1=
[--812X2 (|'~vl "Jr- [~2) "Jr"28182X2['~'2 "[ ART
(['~2 -- Vl°)X2 V, o
(9)
where the subscripts 1 and 2 refer to the solvent and solute respectively. Comparison of this equation with Equation (8) shows that the latter is of the f o r m ('OreL -- 1 ) mt~ture ~- (~ret. -- 1 ) A + (~rel. -- 1 )O + (~rel. -- 1 ) c
(10)
A novel viscometric m e t h o d
161
which establishes the type of function representing the viscosity of a mixture as a simple sum of the corresponding functions for the individual (non-interacting) solutes. The form of the expression for fewer or more species than the three considered above will be obvioust. N o w it has also been shown that a reasonably low concentrations[l] the relative viscosities of many solutions may be adequately represented by the empirical equation ~rel. = 1 + SoX + S1X 2.
( 11 )
So, which is the initial slope of the plot of relative viscosity against mole fraction, was shown to be equal to [-- 612(V1°~- ~'~2)+ 217"28,62]/ART- (I?2 - Vl°)/Vi °.
(12)
$1 is a constant that must be included to take into account the observed deviations from linearity at higher concentrations. By using the principle of additivity deduced above, and by combining Equations (10) and (1 1), it follows that
(m~,.- 1) = [(So)~X~ + (Sl)~X2] + [(So)~X~ + ( s , ) . x . 2]
+ [(S0)cXc+ (s~)cXc~].
(13)
This is a semi-empirical equation, for which it is no longer essential that .4, B and C should form regular solutions; moreover, it should be valid over the whole range of concentration used in determining the S-coefficients of the individual species involved. The only condition still retained is that A, B and C do not interact physically. We now consider the case where the component C is a compound of the species.4 and B and is formed according to the equation n,4 + m B ~ .4 nBm
If the mole fraction of the complete undissociated compound.4,Bm is denoted by X b and the actual degree of dissociation is a, we have X A = naX'c X8 = m a X b X c = (1 - ot)Xb.
Substitution in Equation (13) gives ( ~ e t . - 1) = [ (So) anaX'c + (S~) anZa2(Xb ) 2] + [ (So)BmaX'c + (S1)BmZt~z(X'c)2]
+ [(S0)c(1 - o O X b +
(S1)c(1 - ot)z(Xb)2].
(14)
t l f we write AV for the difference b e t w e e n the viscosity of a solution and that of the pure solvent (rt~eL -- 1)
"q~et.-- ~,olvent
Art
~solvent
l~solvent
Equation ( 1O) can be generalized as follows;-
A'Omt.rture~ ~ ( ATI )sotute which is the formal s t a t e m e n t of the conditions postulated above in which, inter alia, the solute species do not interact in a physical sense.
162
H.M.N.H.
I R V I N G and J. S. S M I T H
Collection of like terms in a gives ( ~ r a . - - 1) = az(xb)Z[n2(S,)a + m2(S1)B + (S1)c]
+ txX'c[n (S0)a + m (S0)B -- (So) c - - 2(S1)cX'c]
(15)
+ Xbf (So)c + (S,)cXb].
This equation may be used to determine the stability constant of compounds of the general formulaA~Bm in the following way: 1. Solutions of the individual components A and B in the chosen solvent are studied independently and the four coefficients (S0)a, (S1)A, (So)B and (SOB are calculated from the relevant viscosity measurements. 2. The S-coefficients are determined for AnBm by using sufficient of component B in the solvent to suppress dissociation. It is assumed that the presence of excess B would not materially affect the values of the S-coefficients for the pure components A and B. 3. Values of ( "Oret. 1) are then determined for a series of solutions of the compoundA~Bm of different apparent mole fractions, X~. 4. The experimental data are substituted into equation (15), which may then be solved to give the value of a at each concentration used. N o w the formation constant, -
-
(1--a)a Kn,m = ( n a a ) n ( m a a ) m
(1 - - a ) n~ . m m . a~+m . a~÷m-1
(16)
where a is the effective concentration of AnBm, i.e. assuming it to be completely undissociated (a = 0). Clearly a --- lO00Xb/Vu mol 1. -1, where I'm is the molar volume of the solution in cm 3. To examine the applicability of these principles the formation constants were studied for adducts of the square planar chelate (diacetylbisbenzoylhydrazone) nickel(II), NiD, with three different heterocyclic bases, B. Spectrophotometric measurements[2], confirmed by studies of liquid-liquid extraction[3], showed that bis-adducts occurred in each case, any mono-adducts being relatively weak. The above discussion is therefore simplified to a consideration of the equilibrium A + 2B ~ AB2 where A is equivalent to NiD, B to the heterocyclic base, andAB2 is NiD.2B. In Equations (14)-(16) n = 1 and m = 2 and Kn,m = t2 = [AB2]/ [,4] [ B ] 2 .
EXPERIMENTAL
lsoquinoline was fractionally distilled from zinc dust under reduced pressure. 4-Methylpyridine was heated under reflux with, and then fractionally distilled from, calcium hydride.
Pyridine was heated under reflux with, and then fractionally distilled from, potassium hydroxide. Benzene. AnalaR grade benzene was stood over sodium wire for one week, filtered and used without further treatment.
(Diacetylbisbenzoylhydrazone)nickel(ll) was prepared by slowly adding a solution of nickel acetate (2-5 g) in water (150 ml) to a suspension of diacetylbisbenzoylhydrazone (K. and K. Laboratories, 2 g) in ethanol (100 ml) heated under reflux. Heating was continued for 1 hr. The product was
A novel viseometric method
163
collected by filtration of the hot mixture, and recrystallised from benzene as dark violet needles. (Found: C, 57.0; H, 4.5. Calculated for ClaH~sN402Ni: C, 57.0; H, 4.23). (Diacetylbisbenzoylhydrazone)nickel(ll)bispyridine, Pyridine (3 g) was added to the nickel complex ( 1 g) in a few ml of chloroform. The adduct, which separated as brown leaflets on the addition of light petroleum, was collected and washed with petroleum ether. (Found: C, 62.0; H, 5.0; Calculated for C1sHlsN402Ni, 2CsHsN: C, 62.6; H, 4.9%). As reported previously by Sacconi, Lombardo and Paoletti[2] the adduct loses pyridine when stored in a vacuum desiccator (H~SO4). (Diacetylbisbenzoylhydrazone)nickel(ll)bis-4-methylpyridine, when prepared similarly, formed dark brown microcrystals. (Found: C, 63.5; H, 5'4, C1sH~eN402Ni, 2 CeHTN required C, 63.7; H, 5.4%). (Diacetylbisbenzoylhydrazone)nickel(ll)bis-isoquinoline was prepared in the same way. (Found: C, 67.9; H, 4.7, C~sH~rN4OzNi, 2C9HrN requires C, 67.8; H, 4.8%). Density and viscosity measurements were made as described previously[l], in a thermostat controlled at 25-0.005°C. Density measurements were converted into molar volumes with the relationship I'M = (XIM1 + X2M2)/d where X~ and )(2 are the mole fractions of solvent and solute respectively, M1 and M2 are the corresponding molecular weights, and d is the density of the solution. Plots of VM against mole fraction for each of the pure components A, B and C were shown by least squares analysis to be linear within experimental error, and could therefore be represented by the general equation I'M = Vl°+ (V2 - VI°)XzThe results are summarised in Table 1. Table 1
Solute lsoquinoline 4-methylpyridine Pyridine NiD Isoquinoline adduct 4-methylpyridine adduct Pyridine adduct
Highest mole fraction used
I72°(ml)
tr~
25.8X 8.3X 9.5X 173.4X
0.007091 0.01712 0.04070 0.0004622
115.2 97.7 79.9 262.8
___0.002 0.001 0.001 0.001
89-937 + 394.5X
0.0005261
484-4
0-001
89.609 + 350.2X 89.025 + 299.8X
0.001098 0.0004670
439-8 388-8
0.003 zero
Equation for molar volume VM= 89.422 + 89.423 + 89.423 89.422 +
X is the mole fraction of the respective solute. O'vM is the standard deviation with respect to VM of experimental points from the best straight line derived from least squares analysis. Flow-time measurements were converted into relative viscosities for solutions of known mole fraction by using the relationship t.o,~" d,o~. ~rel. ~
~soln/~solv.
~ lsolv. " dsotv.
where t represents the flow-time and d the density. The density of each solution was calculated from the molar volume equation for pure components, or from a graph of molar volume against mole fraction
164
H.M.N.H.
I R V I N G and J. S. S M I T H
for equilibrium mixtures. F o r the pure components the results could be represented by Equation (13), and they are summarised in terms o f the calculated S-coefficients in Table 2. The final results for the calculation of formation constants from the relative viscosities of solutions of the adducts, using Equation (16), are tabulated in Tables 3, 4 and 5. Table 2 Solute Isoquinoline 4-methylpyridine Pyridine NiD Isoquinoline adduct 4-methylpyridine adduct Pyridine adduct
So
$1
1.8945 0.3456 0.4934 12.743
9-0 0.8 0.4 - 2912
(rn,,t.
8"
AE ~ (kcal)
___0.00001 zero 0-00002 0.00002
9.72 9.15 10-11 10.80
10.88 8.17 8-17 30.7
19.4246
835
0-00001
9.54
44.0
15.565 16.441
445 1042
0-00005 0.00001
9.21 9.88
37.3 38.0
* From Equation (12). O'nret" is the standard deviation with respect to */ret. o f the experimental points from the curve *?ret.= ! + S o X + S 1 X ~, determined by least squares analysis. Table 3. NiD + isoquinoline X~
a(mole 1.-1)
d. 25
Flow-time
"0tel
a
/32
log $2
zero 0.0001063 0~0003319 0.0004711 0.0007292
zero 0.001188 0.003706 0.005258 0.008128
0-87359 0.87386 0.87438 0.87473 0.87531
667.03 667.94 670-03 671.31 673.85
1.00000 1.00168 1.00541 1-00774 1.01220
-> unity 0.847 0.767 0.650
--4599 4679 4816
--3.66 3.67 3.68
Table 4. N i D + 4-methylpyridine X~
a(mole 1.-1)
zero 0-0002735 0.0008609 0-0009626
zero 0.003055 0.009594 0.01072
d425 0-87360 0-87412 0.87522 0.87546
Flow-time
~rel.
a
t2
log fl~
667.44 669.40 674.41 675.27
1-00000 1-00353 1.01232 1.01389
-0-927 0-430 0.398
-2457 19520 20838
-(3.41) 4.29 4.32
a
fl~
log/32
-0.971 0.929 0.919 0.880
-2490 5227 5700 3630
-3.40 3.72 3.76 3.56
Table 5. NiD + pyridine Xb
a(mole 1.-1
d4~5
Flow-time
zero 0.0001593 0.0001835 0.0001914 0.0003122
zero 0.0017804 0.0020507 0.0021389 0.0034874
0.87354 0.87388 0-87393 0-87395 0.87420
667.50 668.66 668.85 668.90 669.78
~,~l. 1.00000 1.00213 1.00247 1.00258 1.00417
A novel viscometric method
165
DISCUSSION
Formation constants of the bis-adducts of N i D with isoquinoline, 4-methylpyridine and pyridine determined as above may be compared with the values calculated from spectrophotometric measurements by Sacconi et al.[2], by Irving and AI-Niaimi [3] (Table 6). The agreement between the two sets of values is very Table 6. Formation constants of adducts of NiD with heterocyclic bases in benzene at 25°C
Base lsoquinoline 4-Methylpyridine Pyridine
Log ,Sz (25°C) (a) (b) 3.67 4.3 3.6 +- 0.2
3.78 _+0.02 4.45 ___0-02 3-78 _+0.02
(c) 3.73 4.54 --
(a) Present work. (b) Values from [2]. (c) Calculated from the values K = (8.15_+0.39) × 103 and (5-53_+0-13)×104 for 20_2°C[3], using the entropy values A S = - - 3 0 . 1 and - 3 2 . 9 c a l deg. -1 respectively from [2].
satisfactory, and we suggest that our method may therefore find applications where other methods are not suitable for the determination of dissociation constants. A study of Fig. 1 will assist in understanding the choice of conditions for obtaining the most reliable answer by this method. The figure represents the general shape of the two curves that would be obtained experimentally; their separation is a function of the extent of dissociation of the compound A,,Bm and it is the ability to measure this separation accurately that determines the accuracy of the method. As required by theory, the two curves converge at high mole fraction. The scale along the mole fraction axis is compressed for a strong adduct and extended for a weak adduct. The following considerations may limit the usefulness of this method in certain systems. (a) Fig. 1 shows that the separation between the curves (and hence the accuracy of results) increases initially with the concentration of the dissolved species. If the solutes involved have low solubilities, the experimental points will be crowded near the origin, leading to low precision. This is exemplified in our own results, since N i D and its adducts (in particular the species NiD, 2 pyridine) have a rather low solubility in benzene. (b) Equation (17) shows that in the absence of added base the dissociation of the adduct AB2 will be I> 50% when log/3~ ~< 4 for 4 for a = 0.01 M (mole fraction 0.001 for Vu = 100) and when log/32 ~< 6 for a = 0-001 M (X = 0.0001). Since an appreciable dissociation of the adduct is necessary to get a meaningful separation of the curves (Fig. 1), this suggests 1og/32 = 6 as an upper limit for this method.
166
H.M.N.H.
I R V I N G and J. S. SMITH
IL
mole f r a o t i o n o f adduot Fig. 1. Upper eurve-undissociated AB2; lower curve-equilibrium mixtures, A +2B.
AB2
(c) The determination of the S-coefficients for the undissociated compound depends on being able to inhibit its dissociation with a concentration of one component which is small enough not to change the characteristics of the solvent. If b is the additional concentration of base B which must be added to the adduct at concentration a = l A B s ] such that the fraction dissociated is no greater than a, it is easy to show that b = { ( 1 - ct)/afl~} 1/2
__
Table 7 gives calculated values of b for
2t~a.
a = 0-01 M
(17) and a =
0-005. Since the
Table 7. Concentrations of heterocyclic base needed to suppress dissociation of 0.01 M solutions of adduct in benzene to a = 0-005 log K
1-0
2"0
3-0
4.0
5'0
6"0
7"0
Conc. of additional B (mole I.-')
4.5
1.4
0-47
0.16
0.065
0-034
0.024
A novel viscometricmethod
167
second term in Equation (17) is very small compared to the first for values of log fl~ < l01°, the estimates of b are equally valid for a = lABs] from 0.1 to 0.001 M. Practical considerations limit the permissible upper concentration range, and the need to keep the concentration of added B as low as possible would seem to fix a lower limit of about log fl~ = 2.5. As it stands the proposed procedure demands very careful measurements of viscosity involving long flow-times, in order to realise significant differences between solvent and dilute solutions. Quite apart from the experimental difficulties encountered in such work, e.g. the avoidance of dust particles, the accurate measurement of flow-times, precise temperature control etc., the measurements are intrinsically time-consuming and the procedure would not compete with, for example, spectrophotometric methods should these be applicable. However, where no alternative is available, the new procedure offers considerable advantages, and the construction of some form of direct-reading viscometer giving rapid but accurate results for low-viscosity liquids would do much to make it acceptable. Under optimum conditions it would be possible to simplify the new procedure considerably by dispensing with the need to determine experimentally the values for the viscosity parameters (So)c and (SOc for the compound A,Bm. For being given experimental data in (~/re~, X) to form a sufficient number of equations of the form (12), a solution by successive approximations can be applied in which arbitrary values of K are varied until the standard deviations in the calculated average values of (So)c and (SOc are both minimised. A few other points deserve comment. The value of 82 = 10.1 for pyridine obtained in the course of this work agrees very well with values of 10.7, 10.4, 9-7 and I0.5 obtained by quite independent methods [4]. The values of the viscosity parameters So and $1 show interesting trends. So is of much the same order for the heterocyclic bases, is higher for the complex NiD and higher still for the three adducts. S~ for the bases is again small, but large for the three adducts. The very large negative value of $1 = - 2912 for the complex NiD appears unexpected. However it is noteworthy that whereas values of So and $1 are normally positive, negative values have been reported for chromium(lII) acetylacetonate in methanol and ethanol[l], where there is a known degree of non-regularity which has been attributed to solvation.