SOLUBILITY O F SILVER CYANIDE
1375
SOME MATHEMATICAL RELATIOSS IXVOLVIKG T H E SOLUBILITY O F SILVER CYAXIDE JOHN E. RICCI Department o j Chemistry, h‘ew Y o r k University, University Heights, New Y o r k , New York Received June SO, l 9 4 l
One of the most interesting processes of analytical chemistry is the argentometric titration of cyanide, the classical Liebig method. The mathematical theory of the end point of this titration involves four equilibrium constants: IC,, the ion-product of water; K,, the ionization constant of hydrocyanic acid ; K , the instability constant of the argentocyanide complex ion, Ag(CS);; and P , the solubility product of silver cyanide. This assumes that the only species in the solution are H’, OH-, HCS, C K , Ag(CS);, and Ag+, besides water, I?, and 103,since potassium hydroxide and nitric acid are assumed t o be “strong,” or completely ionized, as base and acid, respectively. The modified, LiebigDBniges titration, in which the equivalence point is indicated by the appearance of a turbidity of silver iodide in ammoniacal solution, using potassium iodide as indicator, involves in addition three more equilibrium constants: Ka, the ionization consta,nt of ammonia; K‘, the instability constant of the silver-ammonia comples ion, Ag(NH,):; and PAgl, the solubility product of silver iodide. The usual discussion of the mathematical relations at the end points in these processes is scanty and unsatisfactory, not only, however, because of the complexity of the systems, but also because of a certain confusion frequently encountered in regard t o the definition of the solubility product of silver cyanide and in regard t o the consistency of the numerical values assigned t o the several interrelated constants pertaining t o silver cyanide. Because of the use of these equilibrium constants in the mathematical treatment of these and related analytical processes involving silver cyanide, it seems important to examine this question of the consistency of the “best” numerical values usually assigned to them in the literature. The result of this examination shows the need of clearer and more uniform definition of the solubility product, of silver cyanide and perhaps the necessity of modification of the usually quoted value for this constant, t o bring it into conformity with other independently determined quantities on the basis of which it may be calculated. Following this preliminary consideration, it xi11 be possible, with a consistent set of values of the constants, to derive at least approximate if not exact equations for various problems involving the precipitation of silver cyanide. In these formulas we shall use equilibrium constants vhich are presumably thermodynamic constants, together with symbols for the concentrations of various species in solution, assuming unity activity coefficients. The introduction of activity coefficients is then best accomplished when a final working equation, exact or approximate, is chosen for use,-as will be illustrated in an example later.
137G
JOHN E. RICCI I. ON THE SOLUBILITY PRODUCT OF SILVER CYAhTDE
A saturated pure aqueous solution of silver cyanide involves the four equilibriiini constants K , (taken as 1.O X at 25OC.),K,, K , and P. The constants K , and K may be determined in solutions unsaturated with respect t o silver cyanide, and hence without reference either to P or t o the solubility, So, of the compound in pure water. Experimentally, then, a further measurement is required, knoning K,, K a , and K , to determine either P or So, whereupon the other of these two quantities may be calculated. For K., defined as
K,
=
[H'] [ C Y ] / [ H C S ]
(1)
we shall take the value 4 X lo-", determined by Harman and Worley ( 5 ) , and chosen as the best probable value by Latimer in his Oxzdatzon Potentials (9). For the instability constant, K , defined as
K = [Ag'] [CK-]'/[Ag(CS);]
(2)
n e use the following equilibrium constant, measured in solutions saturated with respect t o silver chloride, by Randall and Halford (17) :
Ki
= [H+]2[C1-][Ag(CN)J/[HCN]' = 1.9 X IO-'
(3) Combined with the solubility product of silver chloride, as P.\&, = [Ag+][Cl-], equations 1, 2, and 3 give
K = KPA~cI/KI
(4
Using 1.7 x lo-'' as the value of P.*,cl, again as listed by Latimer (lo), and the Most of the analytivalue of K , already mentioned, we have K = 1.4 X to lo-''. cal texts list values of K ranging from The solubility product for silver cyanide may be defined either as
P = [Ag'] [CN-]
(5)
or as
P' = [rig+] [Ag(CN)T] Both products are constant for saturation, their relation being
P' = P 2 / K
(7) Unless otherwise noted, the "solubility product of silver cyanide" will here mean P defined as in equation 5 ; both symbols, as defined in equations 5 and 6, will be used, however, for convenience in the mathematical formulas. The soldbility prodtxt may of course be determined without reference t o So, the solubility. Thus Randall and Halford (17) found, for solutions saturated with respect t o silver cyanide, the following equilibrium constant:
K , = [H'] [Ag(CN);]/[HCN] = 3.77 X
(8)
the value calculated by Randall and Halford Latimer (10) merely quotes 3.8 X (17) through still different values of K , and P A ~ C I . 1
1377
SOLUBILITY O F SILVER CYANIDE
Combined \Tith equalions 1, 2, and 5 , this gives P equation 4,
=
KzK/K,, whence, with
Using the values already given for these constants, n-e have P = 1.3 X 10-’5.2 The solubility in pure water, Sa, v i t h which all the foregoing constants should be consistent, may presumably be determined directly and independently purely as a physical quantity. Such a Ion- solubility, however, is usually determined only indirectly, through assumptions regarding the relative concentrations of the solute ions and species in the saturated solution. The reported values, as listed by Seidell (18), are:
x 1.6 x
3.2
IO-‘ JI at l i . S o C ~by , conductivity
(1, quoting 14)
-13at 20”C., by conductivity (2)
2.1 X IO-‘ -11a t 18”C.,by potentiometry
(12)
The ratio of the solubilities calculated by Bottger (2) for silver chloride and silver cyanide (1.OG X lo-’ and 1.04 X lop6, respectively) is very closely equal to the ratio of his reported conductivities (1.33/0.19) for the tn’o substances. Since the number of ions per mole of dissolved silver cyanide tends, xith almost complete formation of the complex ion Ag(CS];, to be half the number for silver chloride, we may perhaps take as a maximum “observed” solubility, or -11. The value usually quoted is 2 X So.at 20-25”C., 3 X d l (4, 11). Such a value is evidently also understood wherever *‘P” is given as Sa g 2-4 X The implication, in such a presentation, seems to be that with such low solubility the salt in its pure saturated solution is “completely ionized”, --despite the weakness of hydrocyanic acid as an acid and the great stability of the complex ion Ag(CX)?,-into Ag+ and C Y , so that [Xg’] = [CS-j = ,So, and ‘‘KSp’’= Si. In reality, the value of P for a given value of Sodepends upon both IC, and K . P is clearly not equal to Si,since the dissolved silver, ZAg,is in two forms, Ag+ and Ag(CS);, and the dissolved cyanide, ZCs, in three forms, HCN, C S - , and Ag(CS)?. It will be shown below that a value of P , defined is in extraordinary disagreement, with values of the as [.1g+] [CS-1, of for So, for K , and lo-’’ for Ka. order of Occasionally I”, or [Ag-] [Ag(CSj;], is given, instead of P , as the “solubility product”; thus in Fales and Kenny (3), P‘ is given as 1.1 X lo-’’ (referring t o Randall and Halford ( l i ) ) , and [Ag’] = [Ag(CN),] in the pure solution. This is much more nearly correct then P lo-’’, although the concentrations of silver ion and argentocyanide ion are not, theoretically equal in the pure aqueous
-
Latimer (10) gives 7.0 X 10-“, again as calculated by Randall and Halford (17) with different values of K. and Pagcl. Kolthoff and Sandell (6): “Llgl = iCS-I = 2 4 d = 2 X 1 0 P ” a t the “equivalence point.” Willard and Furman (19) : By implication, the “solubility product” is here presented as analogous to P.4gc1 and hence as equal to Si. Pearce and Haenisch (15): “.4gCX = Agi CN-, K., = 2.2 X 10-’?.”
+
1378
JOHN E . RICCI
solution. Incidentally, uiing the data of Randall and Halford, and PAS, = 1.7 X lo-", we have (combining equations 4, 7 , and 9):
P'
=
1.3 X lo-''
KiP.a9~i/K1
(10)
We may list at this point, therefore, as the best arailable (but not necessarily consiatent) values of the quantities involved in the solubility of silver cyanide:
so
=2
k', = 4.
x lo-* ;M x 10-l0
K = 1.4 X lo-"
P = 1.3
x
10-l~
P' = P 2 / K = 1.3 X lo-" 11. OK THE CONSISTENCY OF T H E VALUES OF T H E COXSTANTS
so, P , K , K,
Given any three of these four quantities, the fourth may be calculated through the follonjng three equations, the first being a,statement of the principle of electroneutrality applied to pure aqueous silver cyanide:
For conyenience, ire shall use the simplified symbols N for [H'], Ti' for K w ,x for [.\g+], and y for [Xg(CS)J. Then introducing the condition for saturihtion (cquation 5 I and the cquililirium constants already defined, these equations become H - V / I I = - H P / x K , = -HPy/K,P' (14)
so= x + y = x + Pl/x = P'/y -+ y So = P / x
+ 2P'/x + HPIxK,
=
Py/P'
+ 2y + HPy/K,P'
(15) (1Gj
lye shall noir consider in turn the calculation of each of the quantities S a ,
P , Ii, and K , from the others. A. To find Sofrom P, K , and K , The two expressions for SO,of equations 1.5 and 16, are equated, to eliminate So,giving an espression for z in terms of H and constants:
+
+
' 2 = P(1 P/K H/KJ (17) When this is then equated with the expression for z obt'ainable from equation 14, also in terms of I f and constants, we obtain an equation in which the only unknown, is H , or
HS
+ H4R,(l + P / K - P/K:) - 2H3W - 2H2K,W(1 + P / K ) + HVI" + K a W Z ( l+ P / K ) = 0
(18)
1379
SOLUBILITY O F SILVER CYAZIDE
This polynomial may be solved for H by numerical approsimation, and then x may be obtained through equation 17. With x knoivn, So may then be found through equation 15 or 16. This procedure assumes no previous knoivledge of the order of magnitude of So. I n t,he actual case of silver cyanide, however, knowing that it is very insoluble and that most of the cyanide is present as Ag(CS)y rather than YS HCN despite the extreme wakness of this acid, rve may disregard equation 18 and proceed more simply by assuming that the pure saturated solution is very nearly neutral, and hence that H 2 dp. Then equation 17 gives z a t once as
and from equation 16,
Furthermore, since P K ,
>> K dw,then, less accurately, So
2 P / d / K , or 2 d P
(22)
According t o the “best” values of P and K already listed, Soshould therefore be instead of 2 X E 2.1 X which was already estimated as probably a maximum value from the data.
B. To find P from So,K , and K , The expression for .z of equation 17 is equated with that from equation 14, or x = H2P/K,(?V - H2)
(23)
whereupon we may express P in terms of H and other constants. With this result and equation 23, equation 15 then gives an expression defining H as the only unknown, in terms of So and constants:
H’(K
+ K:) + H6[Ko(K+ K % )+ S O ( K- K t ) ] - H5TV(K + 3Kt) -H’K,?V[K + 3Ki - 2SoK.I + H3(3K3V2)+ H2K:W2(3K,- SO) -H(K:’CV3) - KtW3
0 (24)
Kith H calculated from equation 24, we may then eliminate x from equations 15 and 16, and calculate P in terms of So,K , K,, and the known H. But again in the actual case of silver cyanide, we may assume H S dw, and proceed a t once as just indicated. Or, solving equation 21 now for P, we have
+ ~ P ~ K ~ F+ /P IK L~ ( W / K :- s:/K)
4p3
- S;K~V’W/K~ zo
(25)
1380
JOHX E. RICCI
With the numerical values already listed, this gives
P E +Sod/K or E 1.2 X
instead of 1.3 X
C . T o find K from So,P, and K , We eliminate K from equations 15 and 16, and introduce equation 23, obtaining
H 5 - H4(So - K ,
+ 2P/K,) - 2H3W + H'TY(S0 - 2KJ + HIT;' + K , W
=
0 (27)
With H so calculated, z is obtained from equations 15 and 16 with K eliminated; and then K from equation 15. me may again solve But if we assume, for silver cyanide, that H E equation 21, now for K, obtaining a simple quadratic:
dv,
With the numerical values listed, this gives K = 1.7 X 1.4 x as listed.
instead of
D . To find K , from So,P, and K Kow z may be calculated at once from equation 15. With r known, equations 14 and 16 may then be combined to find the tn-o remaining unknowns, H and IC,. In the actual caSe of silver cyanide, the values of So, P, and K listed from the literature are seen at once to be impossible or inconsistent as a group of values, since they give no real value of z in equation 15, n-hich requires that S', be > 4P2/K. In this case then it is out of the question to evaluate K , from the given values of So and the other constants. In the sequel, then, in order to use consistent values, it seems advisable to keep as original data So,K,, and K , and to use the value of P calculated from these three constants through equation 25. Our numerical values, therefore, will be : So = 2.0
x
lo+
K , = 4.0 X lo-'' K = 1.4 X lo-'' P = 1.2
x
10-l6
P' = P a / K = 1.0 X lo-''
SOLCBILITY O F SILVER CYAXIDE
1381
111. EFFECT OFREAGENTS ON THE SOLCBILITY OF SILVER CYANIDE
We shall consider the effect of five separate reagents: A, silver nitrate; B, potassium cyanide; C, nitric acid; D, potassium hydroxide; E, hydrocyanic acid-each at the concentration c-upon the solubility, S.of a suspension of solid silver cyanide in the solution of the reagent. The calculation depends on the combination of three equations: the electroneutrality condition, the expression for the total number of equivalents of dissolved silver, ZAg, and the similar expression for dissolved cyanide, or ZCN; namely, H
- W/H
=
ZAg = [Ag’]
ZCh. = [CN-1
+ [ C Y ] + [Ag(CS);] - [Age] S + c for A; = S for B to E
[SO;]- [K’]
+ [rig((”;];
=
+ 2[Ag(CN);] + [HCN]; = S for A, C, E; = S + c for B, D
-4s in the w e of finding the solubility,
(29) (30)
(31)
Sa,of silver cyanide in pure water
from the constants, ZAg and ZCs are combined t o eliminate S , giving an equation in Z~ - ZCN, which is then combined with equation 29 t o obtain an expression for H in terms of constants and x, the [AgC]concentration. For cases A and B, i.e., for silver nitrate and potassium cyanide, this expression for H is the same as for pure saturated silver cyanide solution, or
For cases C and E (nitric acid and hydrocyanic acid)
[Sote: The symbol ( )’ under the root in such a quadratic solution Tvill represent the square of the quantity outside the root.] Equation 33 also holds for case D (potassium hydroxide), but with the sign of c reversed. Such an expression for H i s then substituted in the equation for Z,, - &, yielding an equation for x as a function of c and constants,-\vith the following results: Case
Writing Z as the sum (P‘
$PZ’
A (salver nitrate)
+ P),
- P21t’/K,] + Y’((P’)~ [P(c2 - 2) + 2cK,Z] - y 2 ( ~ ’ )[zCp 3 - K , ( ~-~z)]- y ( ~ ’ ) 4 [ 2 c~ ,PI+ (PWL = o
+ Y’P’ [ZCPZ+ KJ’
(34)
For the numerical values applying t o silver cyanide, the significant terms are the last three, giving for appreciable values of e, y
P’/c
(35)
1382
JOHN E. RICCI
Since x = P‘/y, then H , from equation 32, % hence, from equation 31:
d$.The solubility is &;
+ PIX + HPIXK, - [2P‘ + P + P 1 / w / K , ] / c % 2P’/c
S = 2y
(36) (37)
Case B (potassium cyanide) The result is the same as equation 34, with the sign of c reversed. I n this case the first three terms of the equation are the significant ones, and, approximately, y s c Then S( = Z A B )is
s=y
+ P’/y E c-
(39)
Case C (nitric acid-%Tong”, OT completely ionized, acid) In this case the polynomial corresponding to equation 34 is of the eighth de. gree, with very cumbersome coefficients. But an approximation is possible since with the actual numerical values applying to silver cyanide, and for ell but negligibly small values of c, equation 33 may be taken a8
H
cxK./(cxK.
+ P)
(40)
This leads t o
x3K,
+ x’P
- X ( C P+ K J )
- PZ
0
(41)
in which the last term, as well as K.2 in the third, are negligible. Hence
and from ZU, =x
+ P f / x s dcP/K.
(43)
The effect of strong acid, then, upon the solubility of a substance like silver cyanide is strictly a function of the four equilibrium constants K,, K, P , and W . The effect of W has already dropped out in the approximation of equation 41; in the final simple formula, equation 43,the effect of P‘ (or therefore of K ) is apparently also absent, but it is implicit in the interconnection of P and the solubility itself. In Kolthoff and Sandell ( 7 ) ,me read that strong acid does not dissolve silver cyanide because it “is a salt of the strong acid HAg(CN)z.” Aside from the fact that we do hot know the ionization constant of such a hypothetical “acid,” we see here that strong acid does affect the solubility despite the assumption that iiHAg(CN)z” is a “strong acid.” The final approximate formula, equation 43, shows a small but definite effect; in pure water, x S y C
SOLUBILITY O F SILVER CPAKIDE
1383
d p ;Tyith nitric acid present, x rises t o dcmawhile y falls to P ’ / d h x P . The relation P‘ = P 2 / K must be kept in mind in interpreting these expressions. This effect may moreover be compared with that in the absence of complex ion formation. I n each case, H is again given by equation 33, and if this is assumed similarly to be simplifiable to equation 40, then x3Ko
+ xZP - X P ( C+ IC,)
- P2 2 0
(44)
which may also be written directly from equation 41 on setting P’ = 0 in Z. ?;om if K , and P are comparable in magnitude, the last term may be neglected, and again, as in equations 42 and 43,
For a given set of values of P and K., then, the effect is the same as in the case of silver cyanide (equation 43). But if we compare the salts of two acids of the \ / M 3 1 and , of the same valuc of K , (in both cases, as in silver cyanide, v i t h K, cP2/2P', or > 85c. ( 1 ) If Q > cPc/2P', the first precipitate is silver chloride. It appears when, with Zag = c1 and ZCN = c, H is given by equation 57. But with x = Pz/cr and [ C Y ] = d K y / z , N
H E
7 d c t z
Then from PcN,
For an approximate solution, v-e assume that we may set 1~ = c / 2 in tlic paren. thesis, in which we may furthermore neglect the term 1 since c must here bc < 10-2c2,obtaining:
-
Hence the appearance of silver chloride occurs (since z = P ~ / c zwhen ) c1
= y
+ P*/c, E c/2 - 8.9 x
lo-B(ccz)~~
(74)
1387
SOLUBILITY O F SILVER CYANIDE
The second precipitate, silver cyanide, will appear, mixed with the silver chloride, when H , ascording t o equation 32, has the value
H E .\/xK,W/P since z is expected t o be very small. From CL
neglecting P as
- 2(P’
(75)
ZCN,
or
+ P ) = HP/R.
(76)
> c. For example, if cp = 1 and c = silver chloride precipitates (alone) nhen (from equation 74) c1 = 4.98 X and silver cyanide appears as a second precipitate when (from equation 80) c1 = 0.9155; if c? = 1 and c = lo-’, the corresponding values of c1 are 4.997 X and 0.155. (2) If Q < cP2/2P’, the first precipitate, silver cyanide, appears with the conditions discussed in equations 66-68. For the value of CI when silver chloride begins t o precipitate, mixed with the silver cyanide, this occurs when
[CY-] = [Cl-]P/P, = c2P/P2
(81)
Hence, since H is still given by equation 57,
Also, ~1
-c
x - [ C Y ] - 2/ - [HCS]
(83)
and since x = P/[CS-1, y = P’/x and HCN = H [ C X ] / K , ,then, exactly,
(3) Finally, when the solution is saturated with both precipitates and CI =
1388 c
JOHN E. RICCI
+
c2, we have a suspension of silver chloride and silver cyanide in aqueous potassium nitrate, with H again given by equation 32. But since ZAg
=
ZCN
+ [Cl-]
(85)
and
x
+ P ’ / x = P ~ / +x P / x + ~ P ’ / +x [HCN]
(86)
- Z = HP/K,
(87)
then X’
in which Z = PI
+ P + P’.
Hence
+ x4P - 2x3K,Z: - 2x2PZ + x(K,Za - Fw/K,)+ PZ’ = 0
x5Ka
Approximately,
x4
(88)
- 2x2 + Z3 E 0
and 5
EdP2
+ P + P’
Equation 88 is interesting since it may be modified directly according t o variations in the system. In the absence of complex ion formation, we simply drop P’ from the sum Z. If the acid “ H C S ” 11-ere strong, though still forming the complex ion, we would divide through by K , and set K , = =, thereby obtaining equation 85, and hence equation 90, as an exact equation. C. Effect o j ammonia (at analytical concentration b) o n solubiliiy oj silver chloride or silver cyanide
Before treating the Liebig-DBnighs titration of potassium cyanide, we shall first consider the relations inrolred in the effect of ammonia on the solubility of precipitates such as silver chloride and silver cyanide. We here introduce two other equilibrium constants: Kb
ISH:] [OH-]/[NHJ
=
1.8 X lo-’
and
I(’ = [Ag’] [SH3]?/[hg(NH3):] = 6.0 X lo-* the values again taken from Latimer (10). (1) For silver ch!oride and aqueous ammonia: Let z represent [Ag(SH,):], and S = [CI-] = the solubility of silver chloride, or S = x + z
For saturation, then, z =
s - P2/S
(‘31)
SOLUBILITY OF SILVER CYANIDE
1389
Then
whence [OH-] = d K a Y
+W
(95)
But
+ [SH:] + 22
b = [”I]
(96)
or
b = 2[S - P,/Sl
+ Y+K,Y/.\/Km
(97)
Kith Y defined by equation 93. This result gives b explicitly and exactly for any specified value of the solubility S . For appreciable values of S, the concentration of ammonia required is approximately
(a) Equation 97 will now be used to illustrate the introduction of actiaity coejkienk. Since the equation is based on the principle of electroneutrality, it holds strictly in terms of concentrations. Hence if the equilibrium constants involved in it are thermodynamic constants, they must be replaced with “mass” constants holding in the ionic atmosphere of the particular solution. The usual simplifying assumption made in such transformation is that if the ionic strength is not too high the activity coefficient of an uncharged species may be taken as unity while the activity coefficients of all univalent ions (the only ones involved in the whole of this discussion), whether positive or negative, are equal to each other. On this basis, equation 97 becomes
with
Y
=
z/K’rqS? + P2/y2) P?
(100)
If the solution contains only the ammonia and silver chloride, then the ionic strength is p =
[OH-]
+ [Cl-]
(101)
with [OH-] given explicitly by equation 95 and [Cl-] = S. Hence if the DebyeHuckel limiting law is assumed t o apply, so that log y = - 0 . 5 0 5 z / i , equation 99 may be used for a complete calculation. For appreciable S, the approximation 98 becomes b GZ S(2
+ 1/K’y2/P2)
(102)
1390
JOHN E. RICCI
and p E S. If the solution also contains a n indifferent salt such ss potassium nitrate at concentration cat then P =
[OH-]
+S +
(103)
C,
(b) For an approximate solution for S in terms of b, with appreciable 6 :
S >> P'/S; Y E S 4 K T z or E SZ (with Z defined as 4 K ' l p 2 ) ; [OH-] > [H'], or [OH-] 4 K b Y g 4 K X S Z . Then equation 97 becomes b % 2s
.
+ SZ + 4 K b Z
(104)
whence
But with b >> Kb, this becomes
as from equation 98. For silver bromide and silver iodide, where K'
>> Ps,
S E bdP2/K' (107) but for silver chloride this final approximation would give S = b/18.8, whereas equation 106 gives S = b/20.8, an appreciable difference. (c) A reagent' containing silver nitrate (c,), potassium nitrate, and ammonia ( b ) is sometimes used t o dissolve silver chloride, and t o separate it from silver bromide and silver iodide. To find the solubility of silver chloride in such a g b, then with z = z[XH3la/K'g solution: if b is appreciable, so that ["a] xb2/K', and x = P2/[C1-] = P 2 / S , b2P2 s = [a-l= z + 2 - E P? - +-s SR' s + SCl - P2 &) K' + b2 E O c1
c1
(1C%
(2) For silver cyanide and aqueous ammonia we shall assume, for the sake of an approximate formula, that the concentration b is large enough so that [OH-] Z/bKb. From XCN = s, then,
+ P + HP/K,)/S G 2P'/S g S / 2 ; and since S = x + y + z , then x
Also, y = P'/x
=
(2P' 2
= s - s / 2 - 2P'/S g s / 2
But b = [NHJ 4
Known as Miller's reagent (13)
+ [NH:] + 22
(111) (112) (113)
SOLUBILITY OF SILVER CYAKIDE
and = 4 K T xE
&
Hence
Rearranging,
b3 - 2b2(Q
+ S)+ b ( Q + S)' - Q ' K a
0
(117)
with Q = (S/2)4/K'/P'. For any appreciable value of S, (2 >> S ; hence
b g Q + S s S
(,'y +
1) g S(123)
~
Converse!y, then,
D. The Liebig-Dknigds tifration This is the titration of potassium cyanide Tvith silver nitrate, in ammoniacal solution, with potassium iodide as indicator. The concentrations are c1 for silver nitrat,e, c for potassium cyanide, CQ for potassium iodide, and b 'For "3. S o w P, = solubility product of silver iodide = 8.5 X lo-'' (10); the other constants are as already defined. We shall assume again that b is large enough so that [OH-] s l / b x . Disregarding temporarily the presence of the iodide, Tve shall first consider the effect of ammonia on the usual appearance of a precipitate of silver cyanide as an end point. For the value of [Ag'], or I, when silver cyanide would begin t o precipitate, the expression for ZCNgives
x
1
- (2P'
+ P +HP/K,)
S
-1 (2P' + PTV/K,.\/bTb)
(120)
Then from Zag, c1
= I
+y +z
= I
+ P'/x + z[SHJ2/K'
But with b large enough, we shall assume [NHJ b. Hence, since from equation 120, and y = P ' / x and hence >> x , CI
2/
+ 2b2P'/cK'
(121) I
P'/c, (122)
1392
JOHN E. RICCI
Also,
Therefore
or
(
c,E! I---
2
3.4
x
10-7
3.3
46-) +
x
10-52,‘
i125)
c
With c = 0.1 and 2, = dfi, c1 = 0.05003. The ammonia therefore would cause a slight delay in the appearance of the end point, whereas in the absence of ammonia we found (equations 13-68) that silver cyanide n.ould precipitate just before the equivalence point. However, x i t h potaisium iodide also present, the precipitate will not he silver cyanide a t all, but silver iodide, precipitating again slightly before the equivalence point. To find the value of c1 when the solution becomes saturated with respect t o silver iodide: From &, c = 2y
+ [CS-](l + H/K,)
( 126)
It-riting CL ( = degree of ionization of hydrocyanic acid) for K , / ( K , using relations already derived, c =
2y
whence, introducing the condition
+ H ) , and
+ dhTx (I/cy) .T
(1 2 i )
= Pa/ca,
‘This is esact and genera‘, but CY invo‘ves the unknown hydrogen-ion concen) if h tration. If the so’ution is sufficiently a’ka’ine (ie.,if [OH-] % dbxaand is appreciab’e, then CY S 1. Also, if z ( = [Bg(.UH,):]), which equals X[NH,]~/K’, is assumed E xb2/k” and hence E b* P3/c3K‘, then we may calculate c1 from Van. or c1 = x y z . as
+ +
the lust three terms all being negligible. early, with
The end point therefore occurs slightly
-
The conditions usually recommended (8, 16, 20) are b z 0.3 and cg c 0.01 for c g 0.06; and then according t o equation 130 the error would be -0.4 per
SOLCBILITY O F SILVER C Y A N D E
1393
cent, lyhich is considerable. If the error is actually not so great, then something would seem t o be wrong n.ith the values of the equilibrium constants appearing in equation 128 and implicit in equation 129. We may finally evaluate the theoretical value of cg (for given values of c and b) for zero error in the titration, or the value required so that c1 = c2/2 when silver iodide appears as a precipitate. With no precipitation at the equivalence point, when c1 = 4 2 , 2BAg
=
z
+y +z
Seglecting [HCX] (i.e,, assuming g xb2,/K’ this means
&N 01
=
[CY]
+ 2y + [HCN
(131)
E I ) , and !since [CS-] = d K y / x , while
z
4 K y x E 2x
+ 2xb2/K’ E 2xb2/K1
(132)
Hence y
4x3l~~/li(I
