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Approximate prediction of mixed-electrolyte activity coefficients
PRELIMINARY
Apfiroximate
NOTE
prediction
D-R. ROSSEINSKY
activity coefficients
and R.J. HILL
of Chemistry.
Department
of mixed-electrolyte
The
University,Exeter (England)
(Received 2nd March 1971)
Lakshmanan and Rangarajan ’ have recently suggested a useful approximate estimate of activity coefficients y in mixed electrolytes, based on the use of Gu~mheim’s specific interaction parameters2p obtained from the expression for the log 7’ of single-electrolyte solutions. For log 7’, the p are coefficients of a term in molality 112which is additional to the Debye-Htickel (DH) term, taken here’a2 to be --oz.+ lz_ i I %/( 1 + 1%) at ionic strength I (oris the Debye-Htickel constant, z + and z _ are cation and anion charge numbers, and zero superscriptsdenote single-electrolyte solutions). The innovation’ was to allow p to be molality-dependent. However, most of this dependence is due to the particularchoice of DH term, which can be made to cancel if assumed to be identical for electrolytes of similar charge type. Hence we can arriveat a much simpler method which holds with identical accuracy (the cancellation of unspecified DH terms led Guggenheim3 to favour the Akerlof-Thomas relation4
bhf~/702), = J&m
(1)
for obtaining only differences B 12 in the specific interaction coefficients). We follow the procedure of ref_ 1 exactly; but wherever u p occurs we substitute the expeninental single-electrolyte activity coefficcient 7’ (at the I of the mixture) divided by I and the appropriate stoichiometry factor, the 7’ is that which would have been used’ to calculate p_ The DH terms by assumption cancel. This then gives for the mixed 1: 1 electrolytes Rx (electrolyte 1) and NY (electrolyte 2) log(7R&
- log(rOR&
=
m2/mc~o~~“Ry)I
+
log(7°Nyl;
-
=g(Y”R& G3
*
withI=m, a21 F-
dog
+m, 7~
= m For constant I the Hamed coefficients (~~2= /dnz 1 are, from (2),
J. EZecrrti
:
dlog7&dmZ
Chem,, 30
and
(1971j.~~~7_-U) -._, “I:.’ .-_.
:
,.,/
PRELIMINARY
App.8 a21
=
c20-’
{2logboNy)J
-
log(~“NJ&
‘-
‘lo&‘Ry
NOTE
(4)
)I]
For a 1: 1 in a 2: 1 or 1.2 electrolyte the DH terms do not ali cancel, and one (but only one) f3 is still required- Thus, for RX in NY, we get (for cy12=-dIog rl IdIs)
and ~irnilarly -12 for RX in N2Y (but with replacement of suffuc. RX by NX; RY by RX, RX remaining, NX2 by RzY, NX by RY). Our 4~~~in these cases thus still depend in part on an arbitrary choice of DH term for p. Applied to the experimental data, the simple expressions (2) to (5) reproduce Tables 2 and 3 of ref. 1 so closely as to make unnecessary their repetition here. A major-part of the discrepancies, exptl, - calcd., in ref. 1, arise from the consequence of the specific-interaction assumption best shown for “common-ion” systems (R = N or X = Y), from (3) and (4), as
This is not necessarily general (Table l), and it is instructive to establish the relationship between calcd- and exptl, To distinguish, we put +,Fd
aed Now ifBlz
=o
(6a)
in (1) is truly a constant it is readily establishedr’
B12
=
exptl
Ql2
that, exactly,
exptl
-
(7)
%I
but if(as we know it sometimes to be) it is a function of m. Bzz (m), exptl
-
a12
exptl
a21
=B12
we may still write
(W
(ml
since observed departures from constancy are never gross. and the approximation satisfactory. From (6a) and (7a) we have calca.
a12
-
cakd
a21
For constant Br2, a cakd
=
&gcd
‘Bl2
(59
(ml
is thus predicted to be exactly the mean %(cr~~pu -
w3.$le for B12. (m), it is to be obtained. Table ,I,shows the -Few measurements ion @‘able 2) For these our
is
exptl. Ql2l
h
approximately so. Where &Gpti and azfp’ have both been average of the exptl. values to be closely predicted. ha&been made on 1: 1 electrolyte mixtures without a common treatment gives (9)
PRELIMINARY
TABLE
NOTE
App-9
1
System
I/m01 kg-’
HCI-LiC15
0.5
exptl WI
_&P”
0.010 0.009 0.009 0.067
0.006 0.005 0.005 0.062
0.011 0.012 0.012 0.074
calcd
>:
.
a2
HCl-KC1
0:5
0.064 0.061 0.061
0.057 0.056
0.064 0.072
0.066
0.050
HCl-CsQ
2: 4.0 1.0
0.086
0.100
0.060
ZI: 1.0 2.0 4.0 6.0 2.0 3.0 4.0
0.077 0.073 0.04 1 0.032 0.027 0.027 0.05 1 0.05 1 0.05 2
3.0 4.0 1.0
0.016 0.014 0.028
0.099 0.073 0.021 0.008 0.003 0.003 0.032 0.025 0.024 0.013 0.013 0.013 0.027
0.046 0.041 0.047 0.044 0.044 0.044 0.067 0.075 0.080 0.013 0.011 0.014 0.029
C&l-NaCl
KCl-Lick
6
’
RI-RbI 8
2.0
NaCl-NaF
’
0.019
TABLE 2 HARNED
COEFFLClENTS”*n
0.1 0.25 0.50 0.70
* Using osmotic
FOR
SYSTEM
HCl-NaClOa
0.030 0.034
0.085 0.077
0.050 0.025
0.038
0.070
0.025
0.040 0.040 0.038
0.060 0.068 0.073
0.025 0.027 0.039
coefficients from ref. 13.
0.060 0.085 0.085 0.085 0.083 0.073
..
Expression (9) will again in general be in error, (10) again much less so, as illustrated in Table 2_ Invocation of further interaction coefficients for particular pairs of like charges appears to be the only way in which the failure of (9) can be circumvented in theories of this typCW6 _ Moving from constant -I data, we note that for trace amounts of electrolyte (mr e m2. m2 = m), eqn. (1) becomes
lo&~
)m =.% l”d~oRy)
m
+
!o&o&m
Prediction of logy, , for ml .(constant) = 0.01 mol kg-’ and m2 up to 10 mol kg-‘, is sometimes poor, but prediction of dlog rr /dm, is promising_ While for a third of the _systems estimates do lie near or vvithin the bounds of experimental error, in the former .very severe test, log rl for HBr+.NaClO, at I +_9 mol kg’! .thus is 0.78 calcd., and :
._
L’iT@&haL
Chem
..
:
.'30 (l??l~).Ap47-10_.~
Amilo 959
PRELIMINARY
expti,
1 the &rst
reproduced
discrepancy,
(Table 3)),tlioughvarying
ranges noted.
by our prediktions.
(approximately consttit sevenfold, are quite closely
The slopes, however by as much as
This agreement
compares
Skhwabe’? for the somewhat related quantity Am/Am, premises avoiding the need for singI&ion quantities.
satisfactorily
and is obtained
NOTE
in the
with that of from
simpler
TABLE 3 System
(25OCI
tiCl--NaC104
HBr-NaCl04 NaCl-NaCl04 NaBrTNaC104 HCI-Liao, HBr-Liclo4 HCI-HCi04 HBr-HQo&& HCl-KCl = HCl-NaQ HQ-LiCL HBr-KBr HBr-NaBr HBr-L%Br
l4
2-6 3-9 2-6 .2-10 2-6 2-6 3-10 3-10
1.5-4
1-3 1.5-4 IS-3 1.5-3 L-3
ExptL
G7ICd
0.08 0.09 0.03 0.03 0.12 0.13 0.18 0.20
o-10 O-12 0.03 0.04 0.14 0.15 0.17 0.18 0.05 0.07 0.09 0.07 0.09 0.12
Ez 0.11 0.06 0.09 0.14
REFEFiENCES
1 S. Lakshmanan and S.K. Rangamjan, J. Electruanal. C&em., 27 (1970) 170. 2 E.A. Gu~nheim,PhiL Mag, 19 <1935) 588. E.A. Guggenheim, Appt+ztions of Statistical Mechanics. Clarendon Press, Oxford, 1966, p. 177. : G. AkerlWand H.C. Thomas,J. Amer. Cirem Sot., 56 (1934) 594. _5 H.S. Ham&d and B.B. Owen, The Physical Chemistry of Electrolytic Solutions, Reinhold, New York, : 6 3rd edxi., 1958. p. 608. ILA; Robinson,J,‘Amer.. Chem Sot, 74 (1952) 6036. F2.k Robinson and C.L. Lim, nuns. FMf~day So&.49 (1953) 1144. and D.Yu. Stupin, Russ. J. Phys Chem, 35 (1961) 295_ .z L&:Makarov 9 J.N; Butler, Ama& Ctze@z, 42 <1970) 1308, 10 HS. Harr;red&nd B.B. Owen, ZXe Physical Chemistry of Electrolytic Solutions. Reinhold, New York, 3rd..edn., 1958, p_ 606. : zi. SJ. Bates arid 3-W. Urmston,J, Amer. Chem Sock, 55 (1933) 4071. 12 I.E. Me and A.J. Read,JI. ChemSoc A, (1966) 1812. 13 RA. Robinson and R.H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd edn., 1965, p. 483 14 k Fe&&and K. Schwahe, Z. Phys Chem, 230 (1965) 20. Reinhold. New York. 15 H.S. Harned and B.Bi Owen, i?TzePhysicai Chemistry of Etecrrotytic Soluti& 3rd edi,.1958, p:748. .’ 18 HS. Hamed and R_A_ Robinson. &Sulticomponenr Electrolyte Solutions, Pergamon. London, 1968, p.’ 22 !&h&&e. Elec~him slctrf.. 12 (1967) 88. -17 IL . .. .. ~-_
Apfiroximate
NOTE
prediction
D-R. ROSSEINSKY
activity coefficients
and R.J. HILL
of Chemistry.
Department
of mixed-electrolyte
The
University,Exeter (England)
(Received 2nd March 1971)
Lakshmanan and Rangarajan ’ have recently suggested a useful approximate estimate of activity coefficients y in mixed electrolytes, based on the use of Gu~mheim’s specific interaction parameters2p obtained from the expression for the log 7’ of single-electrolyte solutions. For log 7’, the p are coefficients of a term in molality 112which is additional to the Debye-Htickel (DH) term, taken here’a2 to be --oz.+ lz_ i I %/( 1 + 1%) at ionic strength I (oris the Debye-Htickel constant, z + and z _ are cation and anion charge numbers, and zero superscriptsdenote single-electrolyte solutions). The innovation’ was to allow p to be molality-dependent. However, most of this dependence is due to the particularchoice of DH term, which can be made to cancel if assumed to be identical for electrolytes of similar charge type. Hence we can arriveat a much simpler method which holds with identical accuracy (the cancellation of unspecified DH terms led Guggenheim3 to favour the Akerlof-Thomas relation4
bhf~/702), = J&m
(1)
for obtaining only differences B 12 in the specific interaction coefficients). We follow the procedure of ref_ 1 exactly; but wherever u p occurs we substitute the expeninental single-electrolyte activity coefficcient 7’ (at the I of the mixture) divided by I and the appropriate stoichiometry factor, the 7’ is that which would have been used’ to calculate p_ The DH terms by assumption cancel. This then gives for the mixed 1: 1 electrolytes Rx (electrolyte 1) and NY (electrolyte 2) log(7R&
- log(rOR&
=
m2/mc~o~~“Ry)I
+
log(7°Nyl;
-
=g(Y”R& G3
*
withI=m, a21 F-
dog
+m, 7~
= m For constant I the Hamed coefficients (~~2= /dnz 1 are, from (2),
J. EZecrrti
:
dlog7&dmZ
Chem,, 30
and
(1971j.~~~7_-U) -._, “I:.’ .-_.
:
,.,/
PRELIMINARY
App.8 a21
=
c20-’
{2logboNy)J
-
log(~“NJ&
‘-
‘lo&‘Ry
NOTE
(4)
)I]
For a 1: 1 in a 2: 1 or 1.2 electrolyte the DH terms do not ali cancel, and one (but only one) f3 is still required- Thus, for RX in NY, we get (for cy12=-dIog rl IdIs)
and ~irnilarly -12 for RX in N2Y (but with replacement of suffuc. RX by NX; RY by RX, RX remaining, NX2 by RzY, NX by RY). Our 4~~~in these cases thus still depend in part on an arbitrary choice of DH term for p. Applied to the experimental data, the simple expressions (2) to (5) reproduce Tables 2 and 3 of ref. 1 so closely as to make unnecessary their repetition here. A major-part of the discrepancies, exptl, - calcd., in ref. 1, arise from the consequence of the specific-interaction assumption best shown for “common-ion” systems (R = N or X = Y), from (3) and (4), as
This is not necessarily general (Table l), and it is instructive to establish the relationship between calcd- and exptl, To distinguish, we put +,Fd
aed Now ifBlz
=o
(6a)
in (1) is truly a constant it is readily establishedr’
B12
=
exptl
Ql2
that, exactly,
exptl
-
(7)
%I
but if(as we know it sometimes to be) it is a function of m. Bzz (m), exptl
-
a12
exptl
a21
=B12
we may still write
(W
(ml
since observed departures from constancy are never gross. and the approximation satisfactory. From (6a) and (7a) we have calca.
a12
-
cakd
a21
For constant Br2, a cakd
=
&gcd
‘Bl2
(59
(ml
is thus predicted to be exactly the mean %(cr~~pu -
w3.$le for B12. (m), it is to be obtained. Table ,I,shows the -Few measurements ion @‘able 2) For these our
is
exptl. Ql2l
h
approximately so. Where &Gpti and azfp’ have both been average of the exptl. values to be closely predicted. ha&been made on 1: 1 electrolyte mixtures without a common treatment gives (9)
PRELIMINARY
TABLE
NOTE
App-9
1
System
I/m01 kg-’
HCI-LiC15
0.5
exptl WI
_&P”
0.010 0.009 0.009 0.067
0.006 0.005 0.005 0.062
0.011 0.012 0.012 0.074
calcd
>:
.
a2
HCl-KC1
0:5
0.064 0.061 0.061
0.057 0.056
0.064 0.072
0.066
0.050
HCl-CsQ
2: 4.0 1.0
0.086
0.100
0.060
ZI: 1.0 2.0 4.0 6.0 2.0 3.0 4.0
0.077 0.073 0.04 1 0.032 0.027 0.027 0.05 1 0.05 1 0.05 2
3.0 4.0 1.0
0.016 0.014 0.028
0.099 0.073 0.021 0.008 0.003 0.003 0.032 0.025 0.024 0.013 0.013 0.013 0.027
0.046 0.041 0.047 0.044 0.044 0.044 0.067 0.075 0.080 0.013 0.011 0.014 0.029
C&l-NaCl
KCl-Lick
6
’
RI-RbI 8
2.0
NaCl-NaF
’
0.019
TABLE 2 HARNED
COEFFLClENTS”*n
0.1 0.25 0.50 0.70
* Using osmotic
FOR
SYSTEM
HCl-NaClOa
0.030 0.034
0.085 0.077
0.050 0.025
0.038
0.070
0.025
0.040 0.040 0.038
0.060 0.068 0.073
0.025 0.027 0.039
coefficients from ref. 13.
0.060 0.085 0.085 0.085 0.083 0.073
..
Expression (9) will again in general be in error, (10) again much less so, as illustrated in Table 2_ Invocation of further interaction coefficients for particular pairs of like charges appears to be the only way in which the failure of (9) can be circumvented in theories of this typCW6 _ Moving from constant -I data, we note that for trace amounts of electrolyte (mr e m2. m2 = m), eqn. (1) becomes
lo&~
)m =.% l”d~oRy)
m
+
!o&o&m
Prediction of logy, , for ml .(constant) = 0.01 mol kg-’ and m2 up to 10 mol kg-‘, is sometimes poor, but prediction of dlog rr /dm, is promising_ While for a third of the _systems estimates do lie near or vvithin the bounds of experimental error, in the former .very severe test, log rl for HBr+.NaClO, at I +_9 mol kg’! .thus is 0.78 calcd., and :
._
L’iT@&haL
Chem
..
:
.'30 (l??l~).Ap47-10_.~
Amilo 959
PRELIMINARY
expti,
1 the &rst
reproduced
discrepancy,
(Table 3)),tlioughvarying
ranges noted.
by our prediktions.
(approximately consttit sevenfold, are quite closely
The slopes, however by as much as
This agreement
compares
Skhwabe’? for the somewhat related quantity Am/Am, premises avoiding the need for singI&ion quantities.
satisfactorily
and is obtained
NOTE
in the
with that of from
simpler
TABLE 3 System
(25OCI
tiCl--NaC104
HBr-NaCl04 NaCl-NaCl04 NaBrTNaC104 HCI-Liao, HBr-Liclo4 HCI-HCi04 HBr-HQo&& HCl-KCl = HCl-NaQ HQ-LiCL HBr-KBr HBr-NaBr HBr-L%Br
l4
2-6 3-9 2-6 .2-10 2-6 2-6 3-10 3-10
1.5-4
1-3 1.5-4 IS-3 1.5-3 L-3
ExptL
G7ICd
0.08 0.09 0.03 0.03 0.12 0.13 0.18 0.20
o-10 O-12 0.03 0.04 0.14 0.15 0.17 0.18 0.05 0.07 0.09 0.07 0.09 0.12
Ez 0.11 0.06 0.09 0.14
REFEFiENCES
1 S. Lakshmanan and S.K. Rangamjan, J. Electruanal. C&em., 27 (1970) 170. 2 E.A. Gu~nheim,PhiL Mag, 19 <1935) 588. E.A. Guggenheim, Appt+ztions of Statistical Mechanics. Clarendon Press, Oxford, 1966, p. 177. : G. AkerlWand H.C. Thomas,J. Amer. Cirem Sot., 56 (1934) 594. _5 H.S. Ham&d and B.B. Owen, The Physical Chemistry of Electrolytic Solutions, Reinhold, New York, : 6 3rd edxi., 1958. p. 608. ILA; Robinson,J,‘Amer.. Chem Sot, 74 (1952) 6036. F2.k Robinson and C.L. Lim, nuns. FMf~day So&.49 (1953) 1144. and D.Yu. Stupin, Russ. J. Phys Chem, 35 (1961) 295_ .z L&:Makarov 9 J.N; Butler, Ama& Ctze@z, 42 <1970) 1308, 10 HS. Harr;red&nd B.B. Owen, ZXe Physical Chemistry of Electrolytic Solutions. Reinhold, New York, 3rd..edn., 1958, p_ 606. : zi. SJ. Bates arid 3-W. Urmston,J, Amer. Chem Sock, 55 (1933) 4071. 12 I.E. Me and A.J. Read,JI. ChemSoc A, (1966) 1812. 13 RA. Robinson and R.H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd edn., 1965, p. 483 14 k Fe&&and K. Schwahe, Z. Phys Chem, 230 (1965) 20. Reinhold. New York. 15 H.S. Harned and B.Bi Owen, i?TzePhysicai Chemistry of Etecrrotytic Soluti& 3rd edi,.1958, p:748. .’ 18 HS. Hamed and R_A_ Robinson. &Sulticomponenr Electrolyte Solutions, Pergamon. London, 1968, p.’ 22 !&h&&e. Elec~him slctrf.. 12 (1967) 88. -17 IL . .. .. ~-_