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The use of approximation formulae in calculations of acid-base equilibria—IV Mixtures of acid and base and titration of acid with base
Talanr~. Vol. 27. PP. 40Q to 415 0 Pcr~nmon Press Lid 1980. Rintcd
in Gmt
Brilam
THE .USE OF APPROXIMATION CALCULATIONS OF ACID-BASE MIXTURES
FORMULAE IN EQUILIBRIA-IV
OF ACID AND BASE AND TITRATION OF ACID WITH BASE HISTAKENARA%KI
Department of Chemistry, Faculty of Science. Saitama University, Shimc+Okubo. Urawa. 338. Japan (Received
17 Awgwsr 1979. Accepted 19 October 1979)
Summary-The pH of mixtures of mono- or diprotic acids and a strong base 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 results obtained are uSed to c&olate the curves for titration of mono- or diprotic acids with a strong base.
In the previous papers in this series, the solutions of the theoretically exact equations and the approximation formulae for the pH of solutions of acids,’ salts’ and their mixtures3 were compared, and the regions in which the approx~ation formulae give the pH correctly to within f0.02 were identified. This paper deals with mixtures of mono- or diprotic acids and a strong base. The calculations were done by the same method as that used in Part IL2 The results are used to allow the calculation of the curves for titration of mono- or diprotic acids with a strong base. In this paper the pH is calculated stepwise with the approxif [H’]
- {K&B
.=
When C, is less than C,, the solution will be acid, and the term [OH-J in equation (3) can be ne&cted, This provides the approximate solution [H+] rr -(Klu, + C,) + J(KA + Cal2 + 4K,(C, 2
(5) When C, is in excess of C,, the solution wiu be basic, and the term[H’J on the right of equation (3) can be neglect& This provides the approximate solution
- C,) - K,; + ,/fK,(Cr, 2G
mation formulae relevant at different titration. The quadratic formulae are approximation formulae, since they can rectly and the ranges of app&ation broad.3
stages of the used as the be solved diare relatively
- c,).
- C,) - K-)2 + 4K&Cs
Mixtures
(6)
of a weak.diprotic acid and a srrong base
For $uch a mixture in which the acid has successive dissociation constants K, and K2, material balance gives
THEORY
C, = [H,Al
Mixtures of a weak monoprotic acid and a strong base
For a mixture with an initial concentration C, and of base C,, material balance gives
of acid
C,, = [HA] + [A-]
+ wa’]
= [H’]
+ Cs = [OH-]
[H’]
+ [Na”]
(1)
+ [A-].
(2)
iD
&
CA-
cti-
EH’I + [OH-]
c, + [H’]
- [OH-]
(3)
= [OH-J
+ [HA-]
+ 2[A2-1.
(8)
These equations give the exact equation
- C,) + K,; [H’]
- C, + C,,) - K,)[H+Jz
- K,{K2(2C; -
[H+13 + (I(, + C,&H+]2 :K,(C,,
= [H* J + C,
+ (K,(K2
which yields the exact equation
-
(7)
[H+]* + (K, -I- C,,[H+]13
These equations give [H+]
+ [A’-]
and charge balance gives
and charge balance gives [H’]
f [HA-I
- KAK, = 0.
(4)
-
CB)
K,K2Kr
+
KJEH’I
= 0.
(9)
When CR is less than C,, the solution will be acid. and the terms [A’-] and [OH-] in equation; (7) and (8) can be neglected. This provides the approximate
HISATAICE NAMSAKI
410
of the base, gives
expression [H’]
= K,
‘*; :c-b;+l
a = [HA] + [A-] = &$ A
(10)
Ii
and material balance with the analytical concentration of the base 6 gives
and the solution [H’]
=
-(K,+CEi)+J(K*+CB)2+4K,(C*-Ce) 2
b-ma+]=*. Introduction
correction factor p = fraction4 t = b/a simplifies equations (18) and (19) to
Addition of the hydrogen+ contribution from the second dissociation step modifies equation (11) to3 =
-(Kt
+ CB)-I-J(K,
(12)
= K2 2;;-_cc.;$+; A
[H’]
+ ma+]
[H’] Since
K2:
+
J:(cs
6 = ut = C*pt.
(21)
+ C,,pt = [OH-]
+ [A-]. (22)
(13)
- :(cn - CA)+
(20)
= [H’]
These quations
and the solution
CH’I =
a== GP
Charge balance gives
where n is a positive integer. When Cs is in excess of C*, there are two cases. When the solution is acid for the most part, the terms [H,A] in equation (7) and [OH-] in equation (8) can be neglected. This provides the approximate expression [H’]
of a dilution
+ ug) and the stoicbiometric
VJ(V*
+ CE? + 4K,(C* - Ce.) 2
+ nK2
(19)
v, + vg
(11)
[H’]
(18)
B
-
CA)
+
give
- K,, c*pdlp;:‘,~+l;o~-l. A the
solution
(23)
will be acidic
K2i2 + 4K2(2C* - C,)
before
the
(14)
2
In the case where the solution is basic. the terms [H2A] in equation (7) and [H']in equation (8) can be neglected. This provides the approximate ex-
equivalence point, the term [OH-] in equation (23) can be negIected. This provides the approximate solution
pW3iOIl
CH’I [H’]
= K222
-;;;;z;;;
+ C*pt) + J(K,
-(K*
(15)
n-
+ C,,pt)’ + 4K*C*p(l2
t) (24) I
and the solution cH+3
=e K2W,
Cd
-
+
Kv
+
-
&2W,
c,)
+
K.,}’ + 4K2K.,(C,
- C,)
2(Cfl - C*)
Addition of the hydroxyl-ion contribution from the first dissociation step modifies equation (16) to’ LOH-I
I
-
IK2WG
-
cd
+
Kwi
+
,/{K2(2C*
At the equivalence pains C*V, = C,V,, t = 1 and if the base used is sodium hydroxide, the equivalence - C,) + KJ2
+ 4K2K,(CB
- C,) + n K, xy*
2K2
Titration
of a weak monoprotic acid with u strong hasr
If V, ml of a weak monoprotic acid with original concentration C,, are titrated with a strong base of concentration CM,material balance with the analytical concentration of the acid u. after the addition of uB ml
- IK,C,p(c - 1) - K,I [H+] = -
mixture is a solution concentration is
CH’I =
(17)
of NaA.’ The hydrogen-ion
K, + ,fK;
+ 4K*K.,C*p
2C,4P
(25)
Since the solution is basic after the equivalence point. the term [H’] on the right of equation (23) can be neglected. This provides the approximate solution
+ ./:K*C*p(t ZC*pt
- 1) - K,j2
+ 4K*K,C*pt
(26)
411
Calculations of acid-base equilibria--IV Titratiorl of a weak diprotic acid with a strong hose
Material balance of the acid gives a = CAP = [H,A]
+ [HA-]
(27)
+ [A’-]
In the case where the solution is basic, the terms [H2A] in equation (27) and [H’] in equation (28) can be neglected. This provides the approximate expression
and charge balance gives
[H’]
+ C,pt = [OH-]
[H’]
+ [HA-]
+ 2[A2-1. (28)
= K2
- t) + [OH-]
C&2
(35)
C,p(t - 1) - [OH-]
and the solution
I
CH’I =
=
- 1)
- t) + K,) + &KzC,p(2
-(KzC,p(2
Addition of the hydroxyl-ion contribution from the first dissociation step modifies equation (36) to3 - t) + K,i2
+ 4K2K,CAp(t
- I) + n K,
equations (27) and (28) can be neglected. This provides the approximate expression
CA/N - t) - &+I
At the second equivalence point, CAV, = 2CBV,, t = 2 and the equivalence mixture is a solution of Na2A.2’The hydrogen-ion concentration is
(29)
Cd + W+l
[H’]
and the solution
CH’I = -(K,
+ CApt)+ ,/(K,
+ CApt)2+4KICAp(l 2
-t)
(W Addition of the hydrogen-ion contribution from the second dissociation step modifies equation (30) to3
=
K, + ,/K;
+ 4K2K,CAp
Equation (38) can be obtained by substituting t = 2 in equation (36). Addition of the hydroxyl-ion contribution from the first dissociation step modifies equation (38) to2
[OH-]
=
-K,
+ JK:
+ 4KzK,C,p
2
+ nK2.
K
Kl(KzG~ + K,) K, + CAP ’
(32)
Between the first and second equivalence points, there are two cases. When the solution is acidic, the terms [HzA] in equation (27) and [OH-] in equation (28) can be neglected. This provides the approximate expression [H+] = K2
- t) - CH’I GAt - 1)+ CH’I
Gp(2
and the solution [H’]
=
-(C,p(t
1
(39)
(31)
At first equivalence point, t = 1 and if the base used is sodium hydroxide, the equivalence mixture is a solution of NaHA.2 The hydrogen-ion concentration is =
+ n K.,
where the last term is derived as described in Part IL2 After the second equivalence point, the solution is basic and equation (36) is still used.
-(K,+C,pt)+,/(K,+CApt)2+4K,CAp(l-t)
[H’]
(38)
2C*P
2K2
CH’I=
(37)
K1’
2K2
CH’I= KI
(36)
2C*p(t - 1)
Since the solution will be acidic before the first equivalence point, the terms [AZ-] and [OH-] in [OH-]
+ 4K2K,CAp(t
(K2C,,p(2 - t) + K,;’
- t) + K, +
K2C&2
- l)+
K2: +
(33)
RESULTS AND Dl!XWSlON
Mixtures of a weak monoprotic acid and a strong base
Figure 1 shows the range of applicability of equation (5) [to give results within 0.02 pH unit of the value given by equation (4)]. The ranges become narrower as CA and CB decrease. As can be seen from equation (3) in Part III and equation (3) in Part IV. if the terms C, and Cs in Part III are replaced with (C, - C,) and C,, respectively in Part IV, then equation (5) in Part III will be identical with equation (5) in Part IV. Therefore the range is the same as that shown on the right-hand side of Fig. 2 in Part IIL3 Figure 2 shows the conditions for which equation (6) give results differing by 60.02 pH unit from those obtained from equation (4). The ranges become broader as C,, and CB decrease.
!C*pO - 1) + K212 + 4K2CAp(2 - r) 2
(34)
412
HLUTAKE NMMAKI
Fig. I. The range of application of equation (5). Numbers indicate -log CA and 111corresponds to the number.
NCA 11
7
Fig. 2. The range of application of equation (6). Numbers indicate -log CA and 111corresponds to the number.
NaHA
-4
%A
14 PKI
Fig. 3. The range of application of equation (II) (--). and equation (12) with II = I (---I (-----) when CA = IO-‘M and CB < IO-‘M. Numbers indicate -log (K,/K,).
and n = 2
Calculations of acid-base equilibria-IV
413
16
Fig. 4. The range of application
of equation (14) when C, = IO-‘M indicate - log(K2/K ,).
and Cs > lo-‘M.
Numbers
PK2
Fig. 5. The range of application of equation (i4) when C, = IO-‘M and Cs > IOm2M.
Fig. 6. The range of application of equation (14) when CA = IO-‘M and Cs > IO-'M.
Fig. 7. The range of application of equation (16) (-), and equation (17) for n = 1 (---), and n = 2 (-----) when C,, = IO-*M and CB > IO-‘M.
Mixtures of a weak diprotic acid and a strong base
Figure 3 shows the range of applicability of quations (I I) and (12) when the concentration of the mixture is 0.1M in acid and up to 0.1 M in base. The range spreads out as KdK, dtxmms and is modified by nK2, with n = 1 and n = 2 in equation (12X for a given ratio of K2/K, up to 10e3. As can be seen from equation (19) in Part III and equation (IO) in Part IV, if the terms C,, and Cu in Part III are replaced with (C, - C,) and C,, respectively in Part IV, then quations (20) and (21) in Part III will be id&&al with equations (11) and (12) in Pat? IV. Therefore the range is the same as shown in Fig. 6 in Part IIL3
0
2
L
When the concentration is IOmZM and IO-‘M, the range of application of equations (11) and (12) will become narrower, as can be seen from Figs. 7 and 8 in Part III.’ Figure 4 shows the range of application of equation (14) when the concentration of the mixture is 0.1 M in acid and more than 0.1 M in base. The range spreads out as KZIKI ckcmum but quatioo (14) cannot be used when Ca = 2Ck When the concentration is less than lo-‘M, the range becomes oarrower. as shown in Figs. 5 and 6. Figure 7 shows the range for use of equations (16) and (17) when the concentration of the mixture is
6
8
10
12
NaHA 1L
pK2
Fig. 8. The range of application of equation (16) (-), (-----) when C, = IO-‘&f
and equation (17) for n = and C,, z- IO-‘M.
1 (--4. and n = 2
415
Calculations of acid-base equilibria-IV
Fig. 9. The range of application of equation (16) f?). and equation (17) (-----) when CA = IO-‘A4 and Cw > IO-‘M.
O.iN in acid and more than O.iM in base. The range spreads out as K */K, decreases and is modified by nK,/K ,, with n = I and n = 2 in equation (17). for a given ratio of KJK, up to 10w3. When the concen-
tration is less than 10-2M. the range becomes narrower as shown in Figs. 8 and 9. As can he seen from equation (36) in Part III and equation (15) in Part IV. if the terms Cu and C, in Part III are replaced with (2C, - Ca) and (Ca - Ci) respectively in Part IV. then equations (37) and (38) in Part III wiil be identical with equations (16) and (17) in Part IV. Figures 7-9 are applicable to the range of the mixture of NaHA and NazA in Part III.’
for n = I h-3,
Titration of a weak ~~uprotic
and n = 2
acid with a strong base
The titration curve may be drawn from equations (24), (25) and (26) to within 0.02 pH unit except just
before the equivalence point as can be seen from Figs. 1 and 2. if activity corrections are omitted. Titration
ofa weak
diprotic acid and a strony base
The titration
curve may be drawn from equations (30) and (32) up to the first equivalence- point. Between the first and second equivalence points. if the solution is stil1 acid, the curve may be drawn from equation (34) and if the solution is basic, the curve may be drawn from equation (36). When K,jK 1 is larger, the curve may be drawn onty in the vicinity of start of titration and of the second equivalence point as can be seen from Figs. 3-9. The range spreads out as KI/Kl decreases. When this ratio is less than IO-*. the curve may be drawn over the whole area except before and after the first equivalence point, but the range becomes narrower as the concentrations of acid and base decrease. The range is modified by nK 2 in equation (31) and by nK,/K, in equation (37). with n E 1 and n = 2, for a given ratio up to 10e3. Figure 10 shows a titration curve calculated for 0.1M oxalic acid5 @K, = 1.25 and pK2 = 4.28) by use of equations (30), (32) and (34). The acid (50 ml) is titrated with O.IM strong base. REFERENCES
0’ 0
20
40
60
80
loo
In
No
V,fmi) . Fig. 10. A titration curve for O.lM oxalic acid with O.lM strong base by equations (30), (32) and (34).
1. H. Narasaki, ;Ta/untu. 1979. 26. 605. 2. Idem. ibid., 19SO.27, 187.
3. IdeAs ibid., 1980,27, 193. 4. D. Betteridge and H. E. Hallam. ~~~r~ Anu/_rricaf Methods. The Chemical Society. London. 1972. 5. L. G. Sill&nand A. E. Martell. Srabiliry Consrunrs. 2nd Ed., The Chemical Society. London. 1964.
in Gmt
Brilam
THE .USE OF APPROXIMATION CALCULATIONS OF ACID-BASE MIXTURES
FORMULAE IN EQUILIBRIA-IV
OF ACID AND BASE AND TITRATION OF ACID WITH BASE HISTAKENARA%KI
Department of Chemistry, Faculty of Science. Saitama University, Shimc+Okubo. Urawa. 338. Japan (Received
17 Awgwsr 1979. Accepted 19 October 1979)
Summary-The pH of mixtures of mono- or diprotic acids and a strong base 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 results obtained are uSed to c&olate the curves for titration of mono- or diprotic acids with a strong base.
In the previous papers in this series, the solutions of the theoretically exact equations and the approximation formulae for the pH of solutions of acids,’ salts’ and their mixtures3 were compared, and the regions in which the approx~ation formulae give the pH correctly to within f0.02 were identified. This paper deals with mixtures of mono- or diprotic acids and a strong base. The calculations were done by the same method as that used in Part IL2 The results are used to allow the calculation of the curves for titration of mono- or diprotic acids with a strong base. In this paper the pH is calculated stepwise with the approxif [H’]
- {K&B
.=
When C, is less than C,, the solution will be acid, and the term [OH-J in equation (3) can be ne&cted, This provides the approximate solution [H+] rr -(Klu, + C,) + J(KA + Cal2 + 4K,(C, 2
(5) When C, is in excess of C,, the solution wiu be basic, and the term[H’J on the right of equation (3) can be neglect& This provides the approximate solution
- C,) - K,; + ,/fK,(Cr, 2G
mation formulae relevant at different titration. The quadratic formulae are approximation formulae, since they can rectly and the ranges of app&ation broad.3
stages of the used as the be solved diare relatively
- c,).
- C,) - K-)2 + 4K&Cs
Mixtures
(6)
of a weak.diprotic acid and a srrong base
For $uch a mixture in which the acid has successive dissociation constants K, and K2, material balance gives
THEORY
C, = [H,Al
Mixtures of a weak monoprotic acid and a strong base
For a mixture with an initial concentration C, and of base C,, material balance gives
of acid
C,, = [HA] + [A-]
+ wa’]
= [H’]
+ Cs = [OH-]
[H’]
+ [Na”]
(1)
+ [A-].
(2)
iD
&
CA-
cti-
EH’I + [OH-]
c, + [H’]
- [OH-]
(3)
= [OH-J
+ [HA-]
+ 2[A2-1.
(8)
These equations give the exact equation
- C,) + K,; [H’]
- C, + C,,) - K,)[H+Jz
- K,{K2(2C; -
[H+13 + (I(, + C,&H+]2 :K,(C,,
= [H* J + C,
+ (K,(K2
which yields the exact equation
-
(7)
[H+]* + (K, -I- C,,[H+]13
These equations give [H+]
+ [A’-]
and charge balance gives
and charge balance gives [H’]
f [HA-I
- KAK, = 0.
(4)
-
CB)
K,K2Kr
+
KJEH’I
= 0.
(9)
When CR is less than C,, the solution will be acid. and the terms [A’-] and [OH-] in equation; (7) and (8) can be neglected. This provides the approximate
HISATAICE NAMSAKI
410
of the base, gives
expression [H’]
= K,
‘*; :c-b;+l
a = [HA] + [A-] = &$ A
(10)
Ii
and material balance with the analytical concentration of the base 6 gives
and the solution [H’]
=
-(K,+CEi)+J(K*+CB)2+4K,(C*-Ce) 2
b-ma+]=*. Introduction
correction factor p = fraction4 t = b/a simplifies equations (18) and (19) to
Addition of the hydrogen+ contribution from the second dissociation step modifies equation (11) to3 =
-(Kt
+ CB)-I-J(K,
(12)
= K2 2;;-_cc.;$+; A
[H’]
+ ma+]
[H’] Since
K2:
+
J:(cs
6 = ut = C*pt.
(21)
+ C,,pt = [OH-]
+ [A-]. (22)
(13)
- :(cn - CA)+
(20)
= [H’]
These quations
and the solution
CH’I =
a== GP
Charge balance gives
where n is a positive integer. When Cs is in excess of C*, there are two cases. When the solution is acid for the most part, the terms [H,A] in equation (7) and [OH-] in equation (8) can be neglected. This provides the approximate expression [H’]
of a dilution
+ ug) and the stoicbiometric
VJ(V*
+ CE? + 4K,(C* - Ce.) 2
+ nK2
(19)
v, + vg
(11)
[H’]
(18)
B
-
CA)
+
give
- K,, c*pdlp;:‘,~+l;o~-l. A the
solution
(23)
will be acidic
K2i2 + 4K2(2C* - C,)
before
the
(14)
2
In the case where the solution is basic. the terms [H2A] in equation (7) and [H']in equation (8) can be neglected. This provides the approximate ex-
equivalence point, the term [OH-] in equation (23) can be negIected. This provides the approximate solution
pW3iOIl
CH’I [H’]
= K222
-;;;;z;;;
+ C*pt) + J(K,
-(K*
(15)
n-
+ C,,pt)’ + 4K*C*p(l2
t) (24) I
and the solution cH+3
=e K2W,
Cd
-
+
Kv
+
-
&2W,
c,)
+
K.,}’ + 4K2K.,(C,
- C,)
2(Cfl - C*)
Addition of the hydroxyl-ion contribution from the first dissociation step modifies equation (16) to’ LOH-I
I
-
IK2WG
-
cd
+
Kwi
+
,/{K2(2C*
At the equivalence pains C*V, = C,V,, t = 1 and if the base used is sodium hydroxide, the equivalence - C,) + KJ2
+ 4K2K,(CB
- C,) + n K, xy*
2K2
Titration
of a weak monoprotic acid with u strong hasr
If V, ml of a weak monoprotic acid with original concentration C,, are titrated with a strong base of concentration CM,material balance with the analytical concentration of the acid u. after the addition of uB ml
- IK,C,p(c - 1) - K,I [H+] = -
mixture is a solution concentration is
CH’I =
(17)
of NaA.’ The hydrogen-ion
K, + ,fK;
+ 4K*K.,C*p
2C,4P
(25)
Since the solution is basic after the equivalence point. the term [H’] on the right of equation (23) can be neglected. This provides the approximate solution
+ ./:K*C*p(t ZC*pt
- 1) - K,j2
+ 4K*K,C*pt
(26)
411
Calculations of acid-base equilibria--IV Titratiorl of a weak diprotic acid with a strong hose
Material balance of the acid gives a = CAP = [H,A]
+ [HA-]
(27)
+ [A’-]
In the case where the solution is basic, the terms [H2A] in equation (27) and [H’] in equation (28) can be neglected. This provides the approximate expression
and charge balance gives
[H’]
+ C,pt = [OH-]
[H’]
+ [HA-]
+ 2[A2-1. (28)
= K2
- t) + [OH-]
C&2
(35)
C,p(t - 1) - [OH-]
and the solution
I
CH’I =
=
- 1)
- t) + K,) + &KzC,p(2
-(KzC,p(2
Addition of the hydroxyl-ion contribution from the first dissociation step modifies equation (36) to3 - t) + K,i2
+ 4K2K,CAp(t
- I) + n K,
equations (27) and (28) can be neglected. This provides the approximate expression
CA/N - t) - &+I
At the second equivalence point, CAV, = 2CBV,, t = 2 and the equivalence mixture is a solution of Na2A.2’The hydrogen-ion concentration is
(29)
Cd + W+l
[H’]
and the solution
CH’I = -(K,
+ CApt)+ ,/(K,
+ CApt)2+4KICAp(l 2
-t)
(W Addition of the hydrogen-ion contribution from the second dissociation step modifies equation (30) to3
=
K, + ,/K;
+ 4K2K,CAp
Equation (38) can be obtained by substituting t = 2 in equation (36). Addition of the hydroxyl-ion contribution from the first dissociation step modifies equation (38) to2
[OH-]
=
-K,
+ JK:
+ 4KzK,C,p
2
+ nK2.
K
Kl(KzG~ + K,) K, + CAP ’
(32)
Between the first and second equivalence points, there are two cases. When the solution is acidic, the terms [HzA] in equation (27) and [OH-] in equation (28) can be neglected. This provides the approximate expression [H+] = K2
- t) - CH’I GAt - 1)+ CH’I
Gp(2
and the solution [H’]
=
-(C,p(t
1
(39)
(31)
At first equivalence point, t = 1 and if the base used is sodium hydroxide, the equivalence mixture is a solution of NaHA.2 The hydrogen-ion concentration is =
+ n K.,
where the last term is derived as described in Part IL2 After the second equivalence point, the solution is basic and equation (36) is still used.
-(K,+C,pt)+,/(K,+CApt)2+4K,CAp(l-t)
[H’]
(38)
2C*P
2K2
CH’I=
(37)
K1’
2K2
CH’I= KI
(36)
2C*p(t - 1)
Since the solution will be acidic before the first equivalence point, the terms [AZ-] and [OH-] in [OH-]
+ 4K2K,CAp(t
(K2C,,p(2 - t) + K,;’
- t) + K, +
K2C&2
- l)+
K2: +
(33)
RESULTS AND Dl!XWSlON
Mixtures of a weak monoprotic acid and a strong base
Figure 1 shows the range of applicability of equation (5) [to give results within 0.02 pH unit of the value given by equation (4)]. The ranges become narrower as CA and CB decrease. As can be seen from equation (3) in Part III and equation (3) in Part IV. if the terms C, and Cs in Part III are replaced with (C, - C,) and C,, respectively in Part IV, then equation (5) in Part III will be identical with equation (5) in Part IV. Therefore the range is the same as that shown on the right-hand side of Fig. 2 in Part IIL3 Figure 2 shows the conditions for which equation (6) give results differing by 60.02 pH unit from those obtained from equation (4). The ranges become broader as C,, and CB decrease.
!C*pO - 1) + K212 + 4K2CAp(2 - r) 2
(34)
412
HLUTAKE NMMAKI
Fig. I. The range of application of equation (5). Numbers indicate -log CA and 111corresponds to the number.
NCA 11
7
Fig. 2. The range of application of equation (6). Numbers indicate -log CA and 111corresponds to the number.
NaHA
-4
%A
14 PKI
Fig. 3. The range of application of equation (II) (--). and equation (12) with II = I (---I (-----) when CA = IO-‘M and CB < IO-‘M. Numbers indicate -log (K,/K,).
and n = 2
Calculations of acid-base equilibria-IV
413
16
Fig. 4. The range of application
of equation (14) when C, = IO-‘M indicate - log(K2/K ,).
and Cs > lo-‘M.
Numbers
PK2
Fig. 5. The range of application of equation (i4) when C, = IO-‘M and Cs > IOm2M.
Fig. 6. The range of application of equation (14) when CA = IO-‘M and Cs > IO-'M.
Fig. 7. The range of application of equation (16) (-), and equation (17) for n = 1 (---), and n = 2 (-----) when C,, = IO-*M and CB > IO-‘M.
Mixtures of a weak diprotic acid and a strong base
Figure 3 shows the range of applicability of quations (I I) and (12) when the concentration of the mixture is 0.1M in acid and up to 0.1 M in base. The range spreads out as KdK, dtxmms and is modified by nK2, with n = 1 and n = 2 in equation (12X for a given ratio of K2/K, up to 10e3. As can be seen from equation (19) in Part III and equation (IO) in Part IV, if the terms C,, and Cu in Part III are replaced with (C, - C,) and C,, respectively in Part IV, then quations (20) and (21) in Part III will be id&&al with equations (11) and (12) in Pat? IV. Therefore the range is the same as shown in Fig. 6 in Part IIL3
0
2
L
When the concentration is IOmZM and IO-‘M, the range of application of equations (11) and (12) will become narrower, as can be seen from Figs. 7 and 8 in Part III.’ Figure 4 shows the range of application of equation (14) when the concentration of the mixture is 0.1 M in acid and more than 0.1 M in base. The range spreads out as KZIKI ckcmum but quatioo (14) cannot be used when Ca = 2Ck When the concentration is less than lo-‘M, the range becomes oarrower. as shown in Figs. 5 and 6. Figure 7 shows the range for use of equations (16) and (17) when the concentration of the mixture is
6
8
10
12
NaHA 1L
pK2
Fig. 8. The range of application of equation (16) (-), (-----) when C, = IO-‘&f
and equation (17) for n = and C,, z- IO-‘M.
1 (--4. and n = 2
415
Calculations of acid-base equilibria-IV
Fig. 9. The range of application of equation (16) f?). and equation (17) (-----) when CA = IO-‘A4 and Cw > IO-‘M.
O.iN in acid and more than O.iM in base. The range spreads out as K */K, decreases and is modified by nK,/K ,, with n = I and n = 2 in equation (17). for a given ratio of KJK, up to 10w3. When the concen-
tration is less than 10-2M. the range becomes narrower as shown in Figs. 8 and 9. As can he seen from equation (36) in Part III and equation (15) in Part IV. if the terms Cu and C, in Part III are replaced with (2C, - Ca) and (Ca - Ci) respectively in Part IV. then equations (37) and (38) in Part III wiil be identical with equations (16) and (17) in Part IV. Figures 7-9 are applicable to the range of the mixture of NaHA and NazA in Part III.’
for n = I h-3,
Titration of a weak ~~uprotic
and n = 2
acid with a strong base
The titration curve may be drawn from equations (24), (25) and (26) to within 0.02 pH unit except just
before the equivalence point as can be seen from Figs. 1 and 2. if activity corrections are omitted. Titration
ofa weak
diprotic acid and a strony base
The titration
curve may be drawn from equations (30) and (32) up to the first equivalence- point. Between the first and second equivalence points. if the solution is stil1 acid, the curve may be drawn from equation (34) and if the solution is basic, the curve may be drawn from equation (36). When K,jK 1 is larger, the curve may be drawn onty in the vicinity of start of titration and of the second equivalence point as can be seen from Figs. 3-9. The range spreads out as KI/Kl decreases. When this ratio is less than IO-*. the curve may be drawn over the whole area except before and after the first equivalence point, but the range becomes narrower as the concentrations of acid and base decrease. The range is modified by nK 2 in equation (31) and by nK,/K, in equation (37). with n E 1 and n = 2, for a given ratio up to 10e3. Figure 10 shows a titration curve calculated for 0.1M oxalic acid5 @K, = 1.25 and pK2 = 4.28) by use of equations (30), (32) and (34). The acid (50 ml) is titrated with O.IM strong base. REFERENCES
0’ 0
20
40
60
80
loo
In
No
V,fmi) . Fig. 10. A titration curve for O.lM oxalic acid with O.lM strong base by equations (30), (32) and (34).
1. H. Narasaki, ;Ta/untu. 1979. 26. 605. 2. Idem. ibid., 19SO.27, 187.
3. IdeAs ibid., 1980,27, 193. 4. D. Betteridge and H. E. Hallam. ~~~r~ Anu/_rricaf Methods. The Chemical Society. London. 1972. 5. L. G. Sill&nand A. E. Martell. Srabiliry Consrunrs. 2nd Ed., The Chemical Society. London. 1964.