The use of approximation formulae in calculations of acid-base equilibria—II

Talanta. Vol. 27. pp. 187 to 191 © Pergamon Press Ltd 1980. Printed in Great Britain

0039-9140/80/0201-0187$02.00/0

THE USE OF APPROXIMATION FORMULAE IN CALCULATIONS OF ACID-BASE EQUILIBRIA--II* SALTS O F MONO- AND DIPROTIC ACIDS 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 solutions of salts of mono- and diprotic acids 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. For salts of monoprotic acids, areas are symmetrically equal to those of the acids. For salts of diprotic acids the ranges generally depend on K2/K 1.

The previous paper compared the solutions of the theoretically exact equations and the approximation formulae for the pH of solutions of mono- and diprotic acids, and identified the regions in which the approximation formulae give the correct pH (within _+0.02).' This paper deals with pure solutions of single salts of mono- and diprotic acids. These salts are produced at the equivalence points in acid-base titrations, so the regions identified in this paper can be used for the calculation of titration curves.

Ignoring the second term in equation (5) provides the simplest approximation formula:

K/2-2K. Cs "

[ H + ] = ~/

(7)

Salts of weak monoprotic bases

For the salt BHC1 of a base with dissociation constant Ks, material balance gives Cs = [ C l - ] = [BH +] + [B]

(8)'

and charge balance gives THEORY

[H +] + [BH +] = [OH-] + [el-].

Salts of weak monoprotic acids

For the salt NaA of an acid with dissociation constant KA, with analytical concentration Cs, material balance gives Cs = [Na +] = [ A - ] + [HA] (I)

(9)

These equations give [ O H - ] = Ka

[H +] - [OH-] Cs - [H +] + [OH-]

(I0)

whichyields the exact equation

and charge balance gives

[ O H - ] 3 + (Ka + C s ) [ O H - ] 2 - K w [ O H - ]

I n +] + [Na +] = [ O H - ] + [ A - ] .

(2)

These equations give

[OH-] - [H +] [H +] = KA Cs - [OH-] 4-[H +]

(3)

- KBKw = 0.

Since the solution will be acid, the term [ O H - ] on the right of equation (10) can be neglected, to give the approximate equation C s [ O H - ] z - K w [ O H - ] - KnKw = 0

which yields the exact equation2 [H+] 3 + (KA + Cs)[H+] 2 - Kw[H +] - KAK,, = O.

(4) Since the solution will be basic, the term [H +] on the right of equation (3) can he neglected, to provide the approximate equation Cs[H+] 2 - Kw[H +] - K A K , = 0

(11)

and the solution [ O H - ] = K,, + x/K~ + 4KaKwC s 2Cs

(12)

(13)

Ignoring the second term in equation (12) provides the simplest approximation formula:

(5)

[on-]

=

/KnK~.

(14)

~/ Cs

and the solution [ H + ] = rw + x / g 2 + 4 K ^ r w C s

2Cs

(6)

Salts of a weak monoprotic acid and a weak monoprotic base For the salt BHA, material balance gives

* Part I--Talanta, 1979, 26, 605.

Cs = [ a n +] + [ a ] = [ A - ] + [HA'] 187

(15)

188

HISATAKENARASAK!

and charge balance gives

gives

[H +] + [BH +] = [OH-] + [A-].

(16)

Cs = [Na +] = [H2A] + [ H A - ] + [A 2-]

These equations give

= [ H A - ] ~ "[H+] + I + t rl KsCs K ^ Cs [H +] + = [OH-] + (17) [OH-] + KB [H +] + K . and charge balance gives

which yields the exact equation

[H +] + [Na +] = [ O H - ] + [ H A - ] + 2[A 2-]

Ks[H+] 4 + {Ks(K^ + Cs) + Kw)[H+] 3 = [OH-]+[HA-]

+ Kw(KA - Ks)[H+] 2 -

(26)

[-fi-q

{

l

I + [H+] j .

K.{KA(Ks + Cs) + Kw)[H +] -KAK

(27) z.=O.

(18)

When the solution is acid, the term [ O H - ] on the right of equation (17) is ignored, which gives the approximate equation Ks[H+] 3 + {Ks(K^ + Cs) + Kw}[H+] 2 + KAKw[H +] - KAKwCs = 0. (19) When the solution is basic, ignoring the term [H +] on the left of equation (17) gives the approximate equation K A [ O H - ] 3 + {KA(Ka + Cs) + K w ) [ O H - ] 2 + K s K w [ O H - ] - KsKwCs = 0. (20) When the second terms of both sides in equation (17) are transposed, [H+] 2 + KA[H +] - KACs [H +] + K^ =

[ O H - ] 2 + K s [ O H - ] - KsCs [ O H - ] + Ks

These equations give the exact equation 2 [H+] 4 + (KI + Cs)[H+] 3 + (KIK2 - Kw)[H+] 2

-K,(KzCs + / ~ ) [ H +] - K~K2Kw = 0.

(28)

When the solution is acid, ignoring the term [ O H - ] in equation (27) provides the approximate equation [H+] 3 + (K, + Cs)[H+] 2 + KIK2[H +] - KIK2Cs = 0. (29) When the solution is basic, if the term [H +] on the left of equation (27) is ignored, the approximate equation Cs[H+] a - K . [ H + ] 2 - KI(KzC s + Kw)[H +] -

KIK2K. = 0 (30)

is obtained. Subtraction of equation (26) from (27) gives (21)

and the terms [H+] 2 and [ O H - ] 2 are neglected, the approximate equation

{

[H +] = [OH-] + [HA-] -[H-g]

K~ J

which leads to

/K,(K2[HA-] +K.) [H+] = %/ K, + [HA-] "

Ks(K^ + Cs)[H+] 2 + Kw(K^ - Ks)[H +]

(31)

- KAKw(Ka + Cs) = 0 (22) Replacement of the term [ H A - ] in equation (31) by

is obtained, the solution being

[H+] = -Kw(K^ - Ks) + x/K~(K^ - Ks) 2 + 4KAKBKw(KA + Cs)(Ks + Cs) 2Ks(K^ + Cs) When the terms KA and Ks are almost equal, omission of the second term in equation (22) yields

K_AKw(KB + Cs) [H+] = X/ ~ . ] _ ~ 5 ~ .

(24)

When K^ and Ke are much smaller than Cs, equation (24) reduces to [H+] =

/g^Kw. V K8

(25)

"Acid" salts of weak diprotic acids For the salt NaHA of an acid with successive dissociation constants KI and K2, material balance

(23)

Cs provides

K,(KzCs + K,,) [H+] = N/ ~ Cs

(32)

When KI is smaller than Cs and K,, is neglected, equation (32) reduces to [H +] = Kx/K~K~.

(33)

Salts of weak diprotic acids For the salt Na2A. material balance gives 2Cs = [Na +] = 2{[H2A] + [ H A - ] + [A2-]}

(34)

and charge balance gives [ n +] + 2Cs = [ O H - ] + [ H A - ] + 2[A2-].

(35)

Calculations of acid-base equilibria--II

8

These equations give the exact equation 2 [H+] 4 + (K1 + 2Cs)EH+] 3

,,6

+{KI(K2 + Cs) - Kw}[H+] 2

f_)

- KIKw[H +] - KIK2Kw = 0.

(36) I

Since the solution will be basic, the term [H +] in equation (35) can be ignored, giving the approximate equation 2Cs[H+] 3 + (KICs - K . ) [ H + ] 2 -- KIKw[H +] -

2

.(11111 X'IllJll A[ll)[ll

0

K2Kw

= 0

(38)

and the solution [H +] = Kw + x/K~ + 4K2KwCs 2Cs

(39)

Ignoring the second term in equation (38) provides the simplest approximation formula: Cs "

(40)

Addition of the hydroxide-ion contribution from the first dissociation step modifies equations (39) and (40), in terms of [ O H - ] , to give

[ O H - ] = - K w + x/K~ + 4K2KwCs n K. 2K 2 + ~

(41)

[OH-] =

(42)

+ n-X/ /(2 KI where n is a positive integer.*

Calculations Calculations were first done by the same method as described in Part I. 1 Then two calculators were used, the first programmed to solve the approximation formula for pH and the other set to see whether a given * The dissociation and hydrolysis reactions of Na2A are Na2A ~ 2Na + + A 2A 2- + H 2 0 ~ H A - + O H HA- + H20 ~- H2A + OH-.

(i) (ii) (iii)

The contribution of [OH-] from reaction (iii) is approximately given by [H2A]. The equilibrium equation of reaction

(iii) can

be written as Kw KI

A

2

/.

[I II

6

8

10

12

l&

KIK2Kw = O. (37)

Omission of the terms [H2A] in equation (34) and [H +] in equation (35) yields the approximate expression C s [ H + ] 2 - K w [ H +] -

[ I

/I

411 ~-illl

0

[H+] = ~/

189

['H2A] [ O H - ] [HA-]

(iv)

From reaction (ii), [ H A - ] ~ [:OH-], so [:H2AI ~ K__~~ KI the [ O H - ] contribution from reaction (i//~ The factor n provides a simple means of assessing the effect of a fairly wide variation in the contribution of this correction term.

Fig. 1. The range of application of (equation (7) (area A, horizontal hatching) and equation (6) (area B, vertical hatching). pH satisfied the exact equation. The pH-value obtained from the first at a certain concentration was put into the second, and if correct would produce a result of zero. If the result was positive, 0.02 was added to the pH value put in, or if it was negative; 0.02 was subtracted from the value, and this procedure was repeated until the sign of the result changed; in this way the correct solution to the exact equation could be obtained within +0.02 pH unit. The same evaluation was applied for other concentrations and for various values of the constants. Thus the regions in which the approximation formulae give the pH correctly within + 0.02 were delineated correct to one decimal place with respect to both the pK and - log Cs values. RESULTS AND DISCUSSION

Salts of weak monoprotic acids F i g u r e 1 shows the conditions for which equations (6) and (7) give results differing by ~<0.02 pH unit from those obtained from equation (4). Area (A) applies to the range for use of equation (7) and area (B) to that for equation (6). Area (B) increases with increasing pKA and the boundary reaches a plateau at pKA ~> 8.

.Salts of weak monoprotic bases The considerations above apply equally to salts of weak monoprotic bases if KA is replaced by KB in Fig. 1. Area (A) applies to the range for use of equation 0 4 ) and area (B) to that for equation (13), because the term [:H +] in equations (3)-(7) is replaced by I O H - ] in equations (10)-(14), and KA by KB.

Salts of a weak monoprotic acid and a weak monoprotic base Figure 2 shows the range of applicability of equation (25) if KB/K ^ <~ 10-1 [to give results within 0.02 pH unit of the value given by equation (18)]. The range depends on Ka/KA. As this ratio decreases, the range becomes smaller and shifts to the left. If K~/KB <<,10- l, Fig. 2 still applies if K^ is replaced by KB in the abscissa.

190

HISATAKE NARASAKI

10

.

8"

u~6T /.20

0 0

2

/.

6

pKA

8

10

12

1/.

Fig. 2. The range of application of equation (25). Numbers inc~cate-log(Ks/K^). The range covers the area bounded by the curve to the right of the number. As shown in Fig. 3, the range of application of equation (24) becomes smaller as Kn/KA decreases. Figure 4 shows that equation (23) gives a similar result, As shown in Fig. 5, the range of application of equation (19) spreads as KB/K^ decreases, becoming the same as area (B) in Fig. 1 of Part 11 when Kn/KA becomes 10-s. This indicates that the salts effectively behave, as monoprotic acids if KB/K^ <~10-s. These

2

4

6

1:

PKA

8

10

12

1/.

Fig. 5. The range of application of equation (19).

0; 0

2

/.

(5 8 pl~

10

12

1l.

Fig. 6. The range of application of equation (33). Numbers indicate -log(K2/K~). The ranges are the funnel shapes bounded by the line to the right of the number, and the corresponding image. considerations apply equally to equation (20) if KA is replaced by KB in Fig. 5.

"Acic~' salts of weak diprotic acids 2

0 0

2

/,

6

PKA

8

10

12

14

Figure 6 shows that the range of application of equation (33) becomes smaller and shifts to the left as K2/KI decreases. On the other hand, the range for

Fig. 3. The range of application of equation (24). Designation as in Fig. 2.

,o

Tz.

.

0

2. 0

0

2

4

6

pK,

8 "

10

12

14

Fig. 4. The range of application of equation (23). Designation as in Fig. 2.

2

6

pK~

8

10

12

Fig. 7. The range of application of equation (32). The numbers indicate -log(K2/Kl). The range for K2/Kt = 0.1 is given by the funnel shape symmetric with respect to the number 1. For other ratios the range is bounded by the line to the left of the number and by the broken line on the right.

Calculations of acid-base equilibria--II .

.

~'

1.3

(.3

I

191

¢..3

o~/," _9.0 I

2 0

0

2" '

2

i

Z,

6

pK,

8

10

12

14

Fig. 8. The range of application of equation (29) lies under the convex lines to the right of the number which corresponds to -log(K2/K 1). The range of application of equation (30) lies under the upper of the two right-hand lines for K2/Kx = 0.1 and under the lower line for other ratios.

0

i

J

i

i

t

6

8

10

12

14 PK2 Fig. 10. The range of application of equation (39). The numbers are -log(K~/Kl) and the range is bounded by 2

4

the curve to the right of the number.

.

6.

i"

o

2

I

2

0

0

0 0 2

/.

6

8

10

12

14

PK2 Fig. 9. The range of application of equation (40).

use of equation (32) spreads out as K2/K 1 decreases, as shown in Fig. 7. O n the left-hand side of F i g 8 is shown the range of application of equation (29), which decreases as Kz/KI decreases. The range for use of equation (30) is described on the right-hand side of Fig. 8. The range is not affected by K2/K1 when the ratio is 10-2 or less.

Salts of weak diprotic acids Figure 9 shows the range of application of equation (40). The range depends on K2/Kt. When this ratio is 10-1 the range covers both areas (A) and (B), but if this ratio is less than 10 -2 only area (B) applies and is the same as area (A) in Fig. 1, derived from equation (7). As shown in Fig. 10, the range of application of equation (39) spreads out as Kz/K1 decreases, becoming the same as area (B) in Fig. 1 when K2/K1 becomes 10 -s. This indicates that salts of diprotic acids effectively behave as those of monoprotic acids if K2/K1 <~ 10 -s.

"rAL.27/2--H

2

4

6

8

10

12

14

PK2 Fig. 11. The range of application of equation (41) when n = I. Designation as for Fig. 10.

When n = 1, the range of application of equation (42) is described by area (B) in Fig. 9, regardless of K,/K1. When n = 2, the same range as for n = 1 applies if K2/K1 <~ 10 -2, but the range becomes slightly narrower at the right-hand side of area (B) if Kz/KI = 10 -1. Figure 11 shows that when n = 1, equation (41) gives a slightly wider range than equation (39). When n = 2, the range for use of equation (41) is almost the same as the areas in Fig. 10 for a given ratio. The range of application of equation (37) is not affected by the ratio Kz/KI and is the same as the total area delineated in Fig. 10.

REFERENCES

I. H. Narasaki, Talanta, 1979, 26, 605. 2. J. G. Dick, Analytical Chemistry, McGraw-Hill, New York, 1973.