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The use of approximation formulae in calculations of acid-base equilibria—III
Talanta. Vol. 27, pp. 193 to 199 © Pergamon Press Ltd 1980. Printed in Great Britain
0039-9140/80/0201..0193502.00/0
THE USE OF APPROXIMATION FORMULAE IN CALCULATIONS OF ACID-BASE EQUILIBRIAT-III* MIXTURES O F MONO- OR DIPROTIC ACIDS AND THEIR SALTS HISATAKENARASAKI Department of Chemistry, Faculty of Science, Saitama University, Shimo-Okubo, Urawa, 338, Japan
(Received 2 May 1979. Accepted 24 June 1979)
Summary--The pH of mixtures of mono- or diprotic acids and their salts is calculated by use of approximation formulae and the theoretically exact equations. The regions for useful application of the approximation formulae (error <0.02 pH) have been identified. The regions become narrower as the concentrations of the mixtures decrease. In diprotic buffers most ranges may be calculated by the quadratic approximation formulae if K2/K~ is less than 10-4.
The previous papers in this series compared the solutions of the theoretically exact equations and the approximation formulae for the pH of solutions of acids I and salts, 2 and identified the regions in which the approximation formulae give the pH correctly within _+0.02. This paper deals with mixtures of mono- or diprotic acids and their salts. These mixtures are widely used as buffer solutions. In the earlier papers 1'2 the concentrations of the acids and salts were plotted on a logarithmic scale. However, buffer solutions are made up from approximately equimolar concentrations of an acid and its conjugate base, so here the concentrations are plotted on a decimal scale. For monoprotic acid buffers, mixtures of an acid or base and its salts are treated, whereas three couples are considered for diprotic acid buffers: H2A-NaHA, HzA-Na2A-and NaHA-Na2A. These systems appear during acid-base titrations. The calculations were done in the same way as in Part II. 2 THEORY
which yields the exact equation 3 [.H+] 3 + (K^ + Cs) [H+] 2 - (KACA + Kw)[H +] -
[H+] = - ( K ^ + Cs) + x/(K^ + Cs) z + 4K^C^ 2
(1)
and charge balance gives
[H+] _ KAC^ + Kw + x/(K^CA + Kw)2 + 4K^KwCs
2¢s
(7)
Mixtures of a weak monoprotic base and its salt For a mixture with initial concentration of base CB and that of salt Cs, material balance gives
Ca + Cs = I-B] + ['BH +] (2)
(6)
Ignoring both the terms [H +] and [ ' O H - ] on the right of equation (3) provides the simplest approximation formula, known as the Henderson equation:
[H + ] + [ N a + ] = [ H + ] + C s
=[OH-3+[A-].
(5)
When the solution is basic, the term [H +] on the right of equation (3) is ignored. This provides the approximate solution
CA ['H+-I = K A C-~s"
F o r a mixture with initial concentration of acid CA and that of salt Cs, material balance gives
(4)
When the solution is acid, the term [ O H - ] in equation (3) is ignored. This provides the approximate solution
Mixtures of a weak monoprotic acid and its salt
CA + Cs = I-HA-I + [A-'I
K^Kw = O.
and charge balance
(8)
gives
[H +] + [BH +] = [.OH-] + ['Cl-] = [ ' O H - ] + Cs.
These equations give I_H+] = KA C A [H +] + [ O H - ] Cs + [H +] - ['OH-] -
-
(3)
These equations give [ ' O H - ] = Ka Ca - [ O H - ] + [H +] Cs + [ O H - ] - [H +]
* Part II--Talanta, 1980, 27, 187 193
(9)
(10)"
194
HISATAKE NARASAKI
which yields the exact equation
and the solution
[OH-] 3 + (Ks + Cs) [OH-]2 -
(KsC. + K.,)[OH-] - K s K w = O.
(11)
When the solution is acid, the terms [ O H - ] on the right of equation (10) are ignored. This provides the approximate solution [OH -] = KaCB + Kw + x/(KBCB + Kw)2 +4KaKwCs 2Cs (12) When the solution is basic, the terms [H +] in equation (10) are ignored. This provides the approximate solution [ O H - ] = - ( K e + Cs)+x/(Ks + Cs) 2 +4KaCa 2
[H +] = - (KI + Ca) + x/(K, + Ca) 2 + 4 K , C A 2
Addition of the hydrogen-ion contributionfrom the second dissociation step modifies equation (20) to [H +] = - ( K , + Ca) + x/(Kl + CH)2 + 4KICA + nK2 2
(21)
where n is a positive integer. When the solution is basic, the term [H ÷] on the left of equation (16) is ignored. This provides the approximate equation CH[H+] 3 -- (KICA + Kw) [H+] 2
(13)
Ignoring both the terms [H ÷] and [ O H - ] on the right of equation (10) provides the simplest approximation formula: Ca [OH-] = Ke ~s' (14)
(20)
- K , { K 2 ( 2 C A + CH) --[-Kw}[H +] " KaK2K,, = 0.
(22)
Ignoring the term [H2A] in equation (15) further provides the approximate expression [H +] = K1 CA + [ O H - ] Ca
-
[OH-]
(23)
Mixtures o f H2A and NaHA
For the mixture with initial concentration of acid CA and that of the "acid" salt CH, material balance gives CA + 6", = [H2A] + [HA-] + [A 2-] [H+] + 1 + K[--H-~]} = [HA-] {-<:
(15)
and charge balance gives
and the solution [H +] =
K,CA + Kw + x/(K,CA + Kw)2 + 4KxKwC.
2C. (24) When the term [H ÷] on the right of equation (19) or [ O H - ] in equation (23) is ignored, both reduce to the simplest approximation formula: CA [H +] = K1 C-.H"
[H + ] + [ N a + ] = [ H + ] + C a
(25)
= I O n - ] + [HA-] + 2[A 2-] = [OH-] + [HA-]
{1 + [--H-~J" 2K21
Mixtures o f H2A and Na2A
(16)
These equations give the exact equation 3
For the mixture with initial concentration of salt Cs, material balance gives C A -I- Cs = [ H 2 A ] q- [ H A - ] + [A 2-']
[H+]" + (K, + (7.)[H+] 3
(26)
and charge balance gives
+ {Kx(K2 - CA)-- Kw}[H+] 2 [H + ] + [ N a + ] = [ H + ] + 2 C s - Ka{K2(2CA + Ca) + Kw}[H +]
= [OH-] + [HA-] + 2[A2-]. -
KxK2Kw
= 0.
(27)
(17) These equations give the exact equation
Wlaen the solution is acid, the term [ O H - ] in equation (16) is ignored. This provides the approximate equation [H+] 3 q- (Kl + CH) [H+] 2 + Kt(K2 - CA)[H +] -- K,K2(2C A + Ca) = 0.
(18)
Ignoring the terms [A 2-] in equations (15) and (16) further provides the approximate expression [ n +] = K1 CA -- [ n +]
Ca ~ [H +]
(19)
[H+] 4 + (K1 + 2Cs) [H+] 3 + {K,(K2 + Cs - CA) -- Kw} [H+] 2 - Kx(2K2CA + Kw)[H +] - K1K2Kw = O.
(28) Similar approximations to those described above give the cubic approximate equations, but they are not presented here. When the solution is acid, ignoring the terms [ O H - ] in equation (27) and [A 2-] in equations (26)
Calculations of acid-base equilibria--Ill
10
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1410 PKA Fig. 1. The range of application of equation (7). Numbers and m indicate -log(CA + Cs). and (27) provides the approximate expression [H +] = KI
CA -- Cs - [H +3 2Cs + [H +3
ignored. This provides the approximate expression (29)
[H +] = K2
Ca + [ O H - ]
(36)
Cs - [ O H - ]
and the solution [H+]
and the solution
- ( K , + 2Cs) 2
[H +] =
+ x/(K1 + 2Cs) z + 4KdCA -- Cs) 2
K z C . + Kw + 4(K2CH + K,,) 2 + 4K2KwCs
2Cs
(30)
(37)
When the solution is basic, omission of the terms [H +] in equation (27) and of [H2A] in equation (26) yields the approximate expression
Addition of the hydroxide-ion contribution from the first dissociation step modifies equation (37) to 2
2CA + [ O H - ] [H +] = K 2 C ; ~ Z_ [---~-_ ]
(31)
[ O H - ] = -(KzCH + K , ) 2K2 + x/(KzCH + Kw)2 + 4 K 2 K , Cs
and the solution
2K2
2KuCA + Kw 2(Cs - CA)
[H +] =
+ x/(2K2CA + K,,) z + 4KzK,,(Cs - CA) 2(Cs - CA) . (32)
Material balance gives (33)
and charge balance gives [H + ] + [ N a + ] = [ H + ] + C n + 2 c s = [ O H - ] + [ H A - ] + 2[A2-].
(34)
These equations give the exact equation a [H+] '* + (K1 + CH.+ 2Cs) [H+] a + {KI(K2 + Cs) - K , } [ H + ] 2 -
where the last term is derived as described in Part II. 2 RESULTS AND DISCUSSION Mixtures of a weak Monoprotic acid and its salt
Mixtures of NaHA and NazA
Ca + Cs -- [H2A] + [ H A - ] + EA2-]
Kw K1 (38)
+ n - - .
KI(K2CH + K,,,)[H +] - KIK~,K,, = O.
(35) Since the solution will be basic, the terms [H2A] in equation (33) and [H +] in equation (34) are
Figure I shows the range of applicability of equation (7) [to give results within 0.02 pH unit of the value given by equation (4)]. The ranges become narrower as the combined concentration (CA + Cs) decreases. Figure 2 shows the conditions for which equations (5) and (6) give results differing by 40.02 pH unit from those obtained from equation (4). The range decreases as the combined concentration decreases. When the concentration is 10-1-10-4M, the range of application of equation (5) lies to the left of the convex line for a given concentration in Fig. 2, and the range for use of equation (6) ties to the right of the corresponding concave line. Both equations (5) and (6) can be applied in the region between the convex and concave fines for a given concentration. When the total concentration is 10-SM and 10-6M, the ranges are separated as shown in Figs. 3 and 4.
196
HISATAKE NARASAKI
10'
0
8"
"2
To x/..
.6 x
2
'8
,
0
~.
2
6
8
','" 10
, 12
10 1/.
PKA Fig. 2. The range of application of equations (5) and (6) (see text for details).
Mixtures of a weak monoprotic base and its salt
Mixtures of HzA and NaHA
The considerations above apply likewise to these mixtures. The range for use of equation (14) corresponds to that for (7), of (12) to (6), and (13) to (5), in Figs. 1-4, if CA is replaced by CB and pKA by pK B.
Figure 5 shows the range of application of equation (25) when the combined concentration (CA + CH) is 10-1M. The range spreads out as K2/K 1 decreases, becoming the same as area 1 in Fig. 1 when K2/KI
10"
0
8'
u
"~
-2
¢/)
cY
2
-8
0 0
2
4
6
8
1()
12
10 1/.
PKA Fig. 3. The range of application of equation (5) on the left, and of equation (6) on the right, when CA + Cs = 10-SM.
10
0 -2
z6 O v.-
x4 (..)t o
"6
tY
-8
2 =d
o
2
/.
6
pK~
1(3
12
10 1/.
Fig. 4. The range of application of equation (5) on the left, and of equation (6) on the right, when CA + Cs = 10-6M.
Calculations of acid-base equilibria--III
197
10 .
"2
~6I C~
x/.-
d .
"8
0 0
2
10 1/.
pK~
Fig. 5. The range of application of equation (25) when CA + Ca = 10-tM. Numbers indicate - l o g (K 2/K i
).
addition of nK2 for a given ratio of K2"/KI up to 10-3. When n = 3, the range becomes much broader, but gives results more than 0.02 pH in error at some places. When the concentration is 10-2M and 10-3M, the range of application of equations (20) and (21) becomes narrower, as shown in Figs. 7 and 8. Figure 9-shows the range of application of equation
becomes 10 -6. When the concentration is less than lO-2M, the range will become narrower, as can be seen from Fig. 1. Figure 6 shows the range for use of equations (20) and (21) when the concentration is 10-IM. The range spreads out as K2/K1 decreases. When n = 1 and n = 2 in equation (21), each area is modified by the
0 -2
o
II : / .i--i 21
.
.
.
i.
0
2
.
.
. . . . . . . .
,'i :
I / /I
Ol
.
, .
.
.
"8
~.'~,
'
4
d
".\'~
.
6pK 1 8
"X
",,
10
12
10 14
Fig. 6. The range of application of equation (20) ( ), and equation (21) with n = l ( - - - ) and n = 2 (. . . . . ) when CA + Cx = 10-IM. Numbers indicate -log(K2/Kl). 10
0 "~;>:. . . . . . . . . . . . . . . . .
8
x4.
-6 "~
fJ
i:
o 0
.
i"/
.
2
/*
.
.
.
. 6
.
.
...
. pK~
,o 8
10
1'2
1/*
Fig~ 7. The range of application of equations (20) and (21) when C A + CH = 10-2M-
198
HISATAKE NARASAKI
8I,/ ~ - ~ : : " ~ ti//l/l
-2
-
---
,/
"
I
o!iIiIiI( :( ', , 0
½
/.
6pK ' 8
,o 10
1'2
1/.
Fig. 8. The range of application of equations (20) and (21) when CA + C. = 10-aM. (24) when the concentration is 10-~M. The range spreads out as K2/Kt decreases but equation (24) cannot be used when C . = 0. As shown in Fig. 10, the range for use of equation (18) is very broad, but iterated computation is required to obtain the result.
Mixtures of HzA and NazA Hereafter only the ranges for use of the quadratic approximation formulae are identified, because the
formulae can be solved directly and the ranges are relatively broad, whereas the ranges for use of the simplest approximation formulae are narrow and the cubic approximation equations cannot be solved directly. Figure 11 shows the range of application of equations (30) and (32) when the combined concentration (CA + Cs) is 10-~M. The range for use of equation (30) lies under the convex lines for a given ratio of K2/K~ and the range of equation (32) is above the
10
0 5"e-
4
8"
3
-2
~ -
6-
-/.'~ b
2
0
'8
1
0
2
, /.
, 6pK 8
, 10
12
10 1/.
Fig. 9. The range of application of equation (24) when CA + Ca = 10-IM.
10
0
8-
-2
~/.o 2'
-6"~
cY
"r"
o0
-8 ,
:2
4
6pK 1 8
lo
~2
~
10
Fig. 10. The range of application of equation (18) lies under the convex lines when CA + Cs = 10-1M. Numbers indicate - log(Kz/K, ).
Calculations of acid-base equilibria--III
10 8
~
1
""1'1 0 -2
2
0 .
.
0
2
&
.
.
6' pK 1 8
199
/10
10 12
g
Fig. 11. The range of application of equation (30) lies under the convex lines and that of equation (32) above the concave lines, when C^ + Cs -- I0-1M.
$, ~"
I
.
.
.
.
I\
-
.
.
.
.
.
.
~
'/
O,
o
~
z
--
I
i/i.I"
Ix ,',,,
2" ~
I!
....... -'-,,+/i[
z
\ ',.....................Y,.'I
8
.
4
~
a
io
iz
~4
PKz Fig. 12. The range of application of equation (37) ( ), and equation (38) for n = 1 (---), and n = 2 (.....) when Ca + Cs = I0- IM.
concave lines. Both ranges spread out as K2/KI decreases. When Cs = CA, equations (30) and (32) cannot be solved. When the concentration is less than 10-2M, the range will become narrower, as can be seen from Figs. 7 and 8.
n = 2 in equation (38), for a given ratio of K2/K l up to 10 -3. When the concentration is less than 10-2M, the range will become narrower and will be shown in Part IV of this series.
Mixtures of NaHA and Na2A Figure 12 shows the range of application of equations (37) and (38) when the combined concentration (Ca + Cs) is 10-1M. The range spreads out as Kz/KI decreases and is modified by nKw/Kl, with n = 1 and
REFERENCES
1. H. Narasaki, Talanta, 1979, 26, 605. 2. Idem, ibid., 1980, 27, 187 3. J. G. Dick, Analytical Chemistry, McGraw-Hill, New York, 1973.
0039-9140/80/0201..0193502.00/0
THE USE OF APPROXIMATION FORMULAE IN CALCULATIONS OF ACID-BASE EQUILIBRIAT-III* MIXTURES O F MONO- OR DIPROTIC ACIDS AND THEIR SALTS HISATAKENARASAKI Department of Chemistry, Faculty of Science, Saitama University, Shimo-Okubo, Urawa, 338, Japan
(Received 2 May 1979. Accepted 24 June 1979)
Summary--The pH of mixtures of mono- or diprotic acids and their salts is calculated by use of approximation formulae and the theoretically exact equations. The regions for useful application of the approximation formulae (error <0.02 pH) have been identified. The regions become narrower as the concentrations of the mixtures decrease. In diprotic buffers most ranges may be calculated by the quadratic approximation formulae if K2/K~ is less than 10-4.
The previous papers in this series compared the solutions of the theoretically exact equations and the approximation formulae for the pH of solutions of acids I and salts, 2 and identified the regions in which the approximation formulae give the pH correctly within _+0.02. This paper deals with mixtures of mono- or diprotic acids and their salts. These mixtures are widely used as buffer solutions. In the earlier papers 1'2 the concentrations of the acids and salts were plotted on a logarithmic scale. However, buffer solutions are made up from approximately equimolar concentrations of an acid and its conjugate base, so here the concentrations are plotted on a decimal scale. For monoprotic acid buffers, mixtures of an acid or base and its salts are treated, whereas three couples are considered for diprotic acid buffers: H2A-NaHA, HzA-Na2A-and NaHA-Na2A. These systems appear during acid-base titrations. The calculations were done in the same way as in Part II. 2 THEORY
which yields the exact equation 3 [.H+] 3 + (K^ + Cs) [H+] 2 - (KACA + Kw)[H +] -
[H+] = - ( K ^ + Cs) + x/(K^ + Cs) z + 4K^C^ 2
(1)
and charge balance gives
[H+] _ KAC^ + Kw + x/(K^CA + Kw)2 + 4K^KwCs
2¢s
(7)
Mixtures of a weak monoprotic base and its salt For a mixture with initial concentration of base CB and that of salt Cs, material balance gives
Ca + Cs = I-B] + ['BH +] (2)
(6)
Ignoring both the terms [H +] and [ ' O H - ] on the right of equation (3) provides the simplest approximation formula, known as the Henderson equation:
[H + ] + [ N a + ] = [ H + ] + C s
=[OH-3+[A-].
(5)
When the solution is basic, the term [H +] on the right of equation (3) is ignored. This provides the approximate solution
CA ['H+-I = K A C-~s"
F o r a mixture with initial concentration of acid CA and that of salt Cs, material balance gives
(4)
When the solution is acid, the term [ O H - ] in equation (3) is ignored. This provides the approximate solution
Mixtures of a weak monoprotic acid and its salt
CA + Cs = I-HA-I + [A-'I
K^Kw = O.
and charge balance
(8)
gives
[H +] + [BH +] = [.OH-] + ['Cl-] = [ ' O H - ] + Cs.
These equations give I_H+] = KA C A [H +] + [ O H - ] Cs + [H +] - ['OH-] -
-
(3)
These equations give [ ' O H - ] = Ka Ca - [ O H - ] + [H +] Cs + [ O H - ] - [H +]
* Part II--Talanta, 1980, 27, 187 193
(9)
(10)"
194
HISATAKE NARASAKI
which yields the exact equation
and the solution
[OH-] 3 + (Ks + Cs) [OH-]2 -
(KsC. + K.,)[OH-] - K s K w = O.
(11)
When the solution is acid, the terms [ O H - ] on the right of equation (10) are ignored. This provides the approximate solution [OH -] = KaCB + Kw + x/(KBCB + Kw)2 +4KaKwCs 2Cs (12) When the solution is basic, the terms [H +] in equation (10) are ignored. This provides the approximate solution [ O H - ] = - ( K e + Cs)+x/(Ks + Cs) 2 +4KaCa 2
[H +] = - (KI + Ca) + x/(K, + Ca) 2 + 4 K , C A 2
Addition of the hydrogen-ion contributionfrom the second dissociation step modifies equation (20) to [H +] = - ( K , + Ca) + x/(Kl + CH)2 + 4KICA + nK2 2
(21)
where n is a positive integer. When the solution is basic, the term [H ÷] on the left of equation (16) is ignored. This provides the approximate equation CH[H+] 3 -- (KICA + Kw) [H+] 2
(13)
Ignoring both the terms [H ÷] and [ O H - ] on the right of equation (10) provides the simplest approximation formula: Ca [OH-] = Ke ~s' (14)
(20)
- K , { K 2 ( 2 C A + CH) --[-Kw}[H +] " KaK2K,, = 0.
(22)
Ignoring the term [H2A] in equation (15) further provides the approximate expression [H +] = K1 CA + [ O H - ] Ca
-
[OH-]
(23)
Mixtures o f H2A and NaHA
For the mixture with initial concentration of acid CA and that of the "acid" salt CH, material balance gives CA + 6", = [H2A] + [HA-] + [A 2-] [H+] + 1 + K[--H-~]} = [HA-] {-<:
(15)
and charge balance gives
and the solution [H +] =
K,CA + Kw + x/(K,CA + Kw)2 + 4KxKwC.
2C. (24) When the term [H ÷] on the right of equation (19) or [ O H - ] in equation (23) is ignored, both reduce to the simplest approximation formula: CA [H +] = K1 C-.H"
[H + ] + [ N a + ] = [ H + ] + C a
(25)
= I O n - ] + [HA-] + 2[A 2-] = [OH-] + [HA-]
{1 + [--H-~J" 2K21
Mixtures o f H2A and Na2A
(16)
These equations give the exact equation 3
For the mixture with initial concentration of salt Cs, material balance gives C A -I- Cs = [ H 2 A ] q- [ H A - ] + [A 2-']
[H+]" + (K, + (7.)[H+] 3
(26)
and charge balance gives
+ {Kx(K2 - CA)-- Kw}[H+] 2 [H + ] + [ N a + ] = [ H + ] + 2 C s - Ka{K2(2CA + Ca) + Kw}[H +]
= [OH-] + [HA-] + 2[A2-]. -
KxK2Kw
= 0.
(27)
(17) These equations give the exact equation
Wlaen the solution is acid, the term [ O H - ] in equation (16) is ignored. This provides the approximate equation [H+] 3 q- (Kl + CH) [H+] 2 + Kt(K2 - CA)[H +] -- K,K2(2C A + Ca) = 0.
(18)
Ignoring the terms [A 2-] in equations (15) and (16) further provides the approximate expression [ n +] = K1 CA -- [ n +]
Ca ~ [H +]
(19)
[H+] 4 + (K1 + 2Cs) [H+] 3 + {K,(K2 + Cs - CA) -- Kw} [H+] 2 - Kx(2K2CA + Kw)[H +] - K1K2Kw = O.
(28) Similar approximations to those described above give the cubic approximate equations, but they are not presented here. When the solution is acid, ignoring the terms [ O H - ] in equation (27) and [A 2-] in equations (26)
Calculations of acid-base equilibria--Ill
10
195
0
-~r- 8
-2_ I--g
E
"4 _E
v--
2
1
x
3
3
2
b
1
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o
2,
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-8
2
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fi
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1'0 12
1410 PKA Fig. 1. The range of application of equation (7). Numbers and m indicate -log(CA + Cs). and (27) provides the approximate expression [H +] = KI
CA -- Cs - [H +3 2Cs + [H +3
ignored. This provides the approximate expression (29)
[H +] = K2
Ca + [ O H - ]
(36)
Cs - [ O H - ]
and the solution [H+]
and the solution
- ( K , + 2Cs) 2
[H +] =
+ x/(K1 + 2Cs) z + 4KdCA -- Cs) 2
K z C . + Kw + 4(K2CH + K,,) 2 + 4K2KwCs
2Cs
(30)
(37)
When the solution is basic, omission of the terms [H +] in equation (27) and of [H2A] in equation (26) yields the approximate expression
Addition of the hydroxide-ion contribution from the first dissociation step modifies equation (37) to 2
2CA + [ O H - ] [H +] = K 2 C ; ~ Z_ [---~-_ ]
(31)
[ O H - ] = -(KzCH + K , ) 2K2 + x/(KzCH + Kw)2 + 4 K 2 K , Cs
and the solution
2K2
2KuCA + Kw 2(Cs - CA)
[H +] =
+ x/(2K2CA + K,,) z + 4KzK,,(Cs - CA) 2(Cs - CA) . (32)
Material balance gives (33)
and charge balance gives [H + ] + [ N a + ] = [ H + ] + C n + 2 c s = [ O H - ] + [ H A - ] + 2[A2-].
(34)
These equations give the exact equation a [H+] '* + (K1 + CH.+ 2Cs) [H+] a + {KI(K2 + Cs) - K , } [ H + ] 2 -
where the last term is derived as described in Part II. 2 RESULTS AND DISCUSSION Mixtures of a weak Monoprotic acid and its salt
Mixtures of NaHA and NazA
Ca + Cs -- [H2A] + [ H A - ] + EA2-]
Kw K1 (38)
+ n - - .
KI(K2CH + K,,,)[H +] - KIK~,K,, = O.
(35) Since the solution will be basic, the terms [H2A] in equation (33) and [H +] in equation (34) are
Figure I shows the range of applicability of equation (7) [to give results within 0.02 pH unit of the value given by equation (4)]. The ranges become narrower as the combined concentration (CA + Cs) decreases. Figure 2 shows the conditions for which equations (5) and (6) give results differing by 40.02 pH unit from those obtained from equation (4). The range decreases as the combined concentration decreases. When the concentration is 10-1-10-4M, the range of application of equation (5) lies to the left of the convex line for a given concentration in Fig. 2, and the range for use of equation (6) ties to the right of the corresponding concave line. Both equations (5) and (6) can be applied in the region between the convex and concave fines for a given concentration. When the total concentration is 10-SM and 10-6M, the ranges are separated as shown in Figs. 3 and 4.
196
HISATAKE NARASAKI
10'
0
8"
"2
To x/..
.6 x
2
'8
,
0
~.
2
6
8
','" 10
, 12
10 1/.
PKA Fig. 2. The range of application of equations (5) and (6) (see text for details).
Mixtures of a weak monoprotic base and its salt
Mixtures of HzA and NaHA
The considerations above apply likewise to these mixtures. The range for use of equation (14) corresponds to that for (7), of (12) to (6), and (13) to (5), in Figs. 1-4, if CA is replaced by CB and pKA by pK B.
Figure 5 shows the range of application of equation (25) when the combined concentration (CA + CH) is 10-1M. The range spreads out as K2/K 1 decreases, becoming the same as area 1 in Fig. 1 when K2/KI
10"
0
8'
u
"~
-2
¢/)
cY
2
-8
0 0
2
4
6
8
1()
12
10 1/.
PKA Fig. 3. The range of application of equation (5) on the left, and of equation (6) on the right, when CA + Cs = 10-SM.
10
0 -2
z6 O v.-
x4 (..)t o
"6
tY
-8
2 =d
o
2
/.
6
pK~
1(3
12
10 1/.
Fig. 4. The range of application of equation (5) on the left, and of equation (6) on the right, when CA + Cs = 10-6M.
Calculations of acid-base equilibria--III
197
10 .
"2
~6I C~
x/.-
d .
"8
0 0
2
10 1/.
pK~
Fig. 5. The range of application of equation (25) when CA + Ca = 10-tM. Numbers indicate - l o g (K 2/K i
).
addition of nK2 for a given ratio of K2"/KI up to 10-3. When n = 3, the range becomes much broader, but gives results more than 0.02 pH in error at some places. When the concentration is 10-2M and 10-3M, the range of application of equations (20) and (21) becomes narrower, as shown in Figs. 7 and 8. Figure 9-shows the range of application of equation
becomes 10 -6. When the concentration is less than lO-2M, the range will become narrower, as can be seen from Fig. 1. Figure 6 shows the range for use of equations (20) and (21) when the concentration is 10-IM. The range spreads out as K2/K1 decreases. When n = 1 and n = 2 in equation (21), each area is modified by the
0 -2
o
II : / .i--i 21
.
.
.
i.
0
2
.
.
. . . . . . . .
,'i :
I / /I
Ol
.
, .
.
.
"8
~.'~,
'
4
d
".\'~
.
6pK 1 8
"X
",,
10
12
10 14
Fig. 6. The range of application of equation (20) ( ), and equation (21) with n = l ( - - - ) and n = 2 (. . . . . ) when CA + Cx = 10-IM. Numbers indicate -log(K2/Kl). 10
0 "~;>:. . . . . . . . . . . . . . . . .
8
x4.
-6 "~
fJ
i:
o 0
.
i"/
.
2
/*
.
.
.
. 6
.
.
...
. pK~
,o 8
10
1'2
1/*
Fig~ 7. The range of application of equations (20) and (21) when C A + CH = 10-2M-
198
HISATAKE NARASAKI
8I,/ ~ - ~ : : " ~ ti//l/l
-2
-
---
,/
"
I
o!iIiIiI( :( ', , 0
½
/.
6pK ' 8
,o 10
1'2
1/.
Fig. 8. The range of application of equations (20) and (21) when CA + C. = 10-aM. (24) when the concentration is 10-~M. The range spreads out as K2/Kt decreases but equation (24) cannot be used when C . = 0. As shown in Fig. 10, the range for use of equation (18) is very broad, but iterated computation is required to obtain the result.
Mixtures of HzA and NazA Hereafter only the ranges for use of the quadratic approximation formulae are identified, because the
formulae can be solved directly and the ranges are relatively broad, whereas the ranges for use of the simplest approximation formulae are narrow and the cubic approximation equations cannot be solved directly. Figure 11 shows the range of application of equations (30) and (32) when the combined concentration (CA + Cs) is 10-~M. The range for use of equation (30) lies under the convex lines for a given ratio of K2/K~ and the range of equation (32) is above the
10
0 5"e-
4
8"
3
-2
~ -
6-
-/.'~ b
2
0
'8
1
0
2
, /.
, 6pK 8
, 10
12
10 1/.
Fig. 9. The range of application of equation (24) when CA + Ca = 10-IM.
10
0
8-
-2
~/.o 2'
-6"~
cY
"r"
o0
-8 ,
:2
4
6pK 1 8
lo
~2
~
10
Fig. 10. The range of application of equation (18) lies under the convex lines when CA + Cs = 10-1M. Numbers indicate - log(Kz/K, ).
Calculations of acid-base equilibria--III
10 8
~
1
""1'1 0 -2
2
0 .
.
0
2
&
.
.
6' pK 1 8
199
/10
10 12
g
Fig. 11. The range of application of equation (30) lies under the convex lines and that of equation (32) above the concave lines, when C^ + Cs -- I0-1M.
$, ~"
I
.
.
.
.
I\
-
.
.
.
.
.
.
~
'/
O,
o
~
z
--
I
i/i.I"
Ix ,',,,
2" ~
I!
....... -'-,,+/i[
z
\ ',.....................Y,.'I
8
.
4
~
a
io
iz
~4
PKz Fig. 12. The range of application of equation (37) ( ), and equation (38) for n = 1 (---), and n = 2 (.....) when Ca + Cs = I0- IM.
concave lines. Both ranges spread out as K2/KI decreases. When Cs = CA, equations (30) and (32) cannot be solved. When the concentration is less than 10-2M, the range will become narrower, as can be seen from Figs. 7 and 8.
n = 2 in equation (38), for a given ratio of K2/K l up to 10 -3. When the concentration is less than 10-2M, the range will become narrower and will be shown in Part IV of this series.
Mixtures of NaHA and Na2A Figure 12 shows the range of application of equations (37) and (38) when the combined concentration (Ca + Cs) is 10-1M. The range spreads out as Kz/KI decreases and is modified by nKw/Kl, with n = 1 and
REFERENCES
1. H. Narasaki, Talanta, 1979, 26, 605. 2. Idem, ibid., 1980, 27, 187 3. J. G. Dick, Analytical Chemistry, McGraw-Hill, New York, 1973.