ANALYTICAL CHEMISTRY
1650 limitation for this particular application. The possibility exists that the conductometric measurement can be extended to the determination of traces of fluoride which can be liberated as aqueous hydrofluoric acid, as in the familiar pyrohydrolysis (5) technique. The actual ionic species in the boric acid-hydrogen fluoride solutions were not identified. However, Wamser (3, 4)has studied the system boron fluoride-water and has established that the rate of Reaction 1 is extremely rapid. He has shown further that hydrolysis of monohydroxyfluoroboric acid proceeds rapidly as HBF3OH and
+ H20Rapid +HBFs(0H)S + H F
ductances of solutions of greater concentrations were not nearly
as stable. The extreme is illustrated by the conductance of a solution of boric acid, 0.5 with respect to hydrogen fluoride, the conductance of which was reproducible for only about 1 minute before a slow drift to higher readings became evident. The effect of the hydrolysis of a very dilute concentration of monohydroxyfluoroboric acid on its conductance is apparently insignificant, in which case readings are reproducible. As the concentration increases, however, this effect becomes increasingly pronounced and erratic readings are obtained. LITERATURE CITED
(2)
(1) Hodgman, C. D., "Handbook of Chemistry and Physics," Vol. 3-1, p. 1561, Cleveland,Ohio, Chemical Rubber Publishing Co., 1952.
The reaction between hydrofluoric acid and monohydroxyfluoboric acid is extremely slow, however. HBFaOH
+ H F +HBFi + H?O
(4)
The results of this investigation substantiate the work of Wamser. The increase in conductivity of the boric acid solution upon the addition of hydrogen fluoride was noted immediately. The readings were reproducible for a t least 30 minutes for hydrogen fluoride concentration less than 1 X IO-' M , but con-
Simons, J. H., "Fluorine Chemistry," T'ol. 1 , p. 252, Kevi York, Academic Press, 1950. (3) Wamser, C. A., J . Am. Chem. SUC.7 0 , 1209-13 (1948). (2)
(4) Ibid., 73, 409-16 (1951). ( 5 ) Willard, H. H., and Winter, 0 . B., ISD.ENG.CHEM.,ANAL.ED.,
5, 7 (1933). RECEIVEDApril 30, 1953. Accepted August 10, 1953. The Oak Ridge National Laboratory is operated by the Carbide and Carbon Chemicals Co.. a Division of Union Carbide and Carbon Corp., for the Atomic Energy Commission. The work reported here was carried out under Contract No. W 7402-eng-26.
Curve for Argentometric Determination of Cyanide JOHN E . RICCI Department of C h e m i s t r y , Y e w York University, New Y o r k , iV. Y . The shape of the curve of p[Ag+]in the titration of potassium cyanide with silver nitrate has been investigated, to determine the position of the inflection point relative to the equivalent point. The relations considered include the precipitation of silver cyanide and the effect of ammonia upon the titration. Equations are derived for the calculation of the equilibrium cmnstants from the characteristics of the titration curve.
I
N AN earlier article ( 8 ) on the interrelations of the equilibrium conetants in aqueous solutions saturated with silver cyanide, the relation of the point of precipitation of silver cyanide or of silver iodide, as titration end point, to the equivalent point in the argentometric titration of cyanide was considered. The present article is concerned with the shape of the titration curve. or the curve of p[Ag+] against the quantity of silver nitrate added, in order to determine the position of the inflection point in the curve and its relation both to the equivalent point and to the pomt of first appearance of a precipitate. Two sets of conditions are considered: the titration of pure aqueous potassium cyanide with silver nitrate, in which the hydrogen ion concentration is variable during the titration; and the titration of potassium cyanide in presence of excess of ammonia, during which the value of [H+] is practically constant, fixed mainly by the ammonia. -4few of the expressions involved are identical with those of the previous discussion; two errors that crept into that paper as published are corrected here. For ease of comparison, all symbols remain as before, with the exception of the use, here, of A in place of K. for the ionization constant of hydrocyanic acid. NOMENCLATURE
el
= analytical concentration of potassium cyanide = analytical concentration of silver nitrate
z
= analytical concentration of potassium iodide = [A@;+]
c
b e,
= analytical concentration of ammonia
Y 2
H W A Kb
[H+][OH-] = [ H + l [ C N - ] / [ H C N ]= 4.0 X [NH,+][OH-]/[NHa] 1.8 X [Ag+][CN-]'/[Ag(CN)2-] = 1.4 X [Ag+][NH3]'/[Ag(NHa)2+]= 6.0 X IO-* [Ag+][CN-] a t saturation with AgCN, = 1.2 X lo-'* xy a t saturation with AgCN, = P 2 / K = 1.03 X lo-'* [.4g+]/I-l a t saturation with A4gI. = 8.5 X
K K' P P' p3
n m pz # 4'
= =
r s
= 1 H/A = P' 4r P =
4 u
=
Q
y g
-drnjdpz
+
= -
v
( ( (
=
= =
)b
)p
= = value a t equivalent point = value a t inflection point
value required for precipitation activity coefficient of a univalent ion = ionic strength
= =
The italic capitals represent mass constants, and activity constants are in bold face. The e uality sign ( = ) is used only for relations which are exact by dexnition, and the symbol Z is used for all approximations, even if very accurate. c and cl represent not initial concentrations or concentrations
V O L U M E 25, NO. 1 1 , N O V E M B E R 1 9 5 3
1651
of separate solutions, but the actual analytical concentration of potassium cyanide and silver nitrate, respectively, during the titration. The volume, however, is assumed to remain constant during the titration, so that c is treated as a constant throughout while c1 is a variable, with c1 = n ( c / 2 ) = mc. The ration is significant in the usual titration, that to theequivalent point c1 = c / 2 or n = 1, in which the end point may be taken as an inflection point in the curve of px against n or as the appearance of a precipitate of either silver cyanide or silver iodide. The ratio m is used in connection with the continued precipitation of silver cyanide, starting near n = l(or m = 1/2), and carried to the equivalent point c1 = c or m = 1. TITRATION OF PURE AQUEOUS POTASSIUM C Y A 3 I D E WITH SILVER TITRATE
Relations without Precipitation of Silver Cyanide. definitions, we have c = 2y
n
(3)
+ [CX-1)
=
(5)
TITRATIOX CURVE B E F O R E EQCIVALEST P O I S T , OR TYITH 1L < 1. c1 or nc;2, Equations 1 and 4 give With y >>z and hence y
H , however, is smaller than A until near the equivalent point. For pure potassium cyanide t.lie value of H a t the start is approsimately d/WA> arid it remains, duriiig the titration, al~i-ays -~ smaller than dm. This means that r ranges from 1 1/W7/c,4, or about 1, to a maximum of less than 1 d\/T;i7/A,or about 250. The combination of the definitions, moreover, gives
- H-AZ(JV 2 K ( H + -4)- HZ) (H
Both of these expressionb hold from the very start of the titration to n g 1. ?;ear the equivalent point, then, when nK/2cx K/2cx, Equation 13 becomes v n s I + - 2x - - - uc c c __
in which u = .\/ch:/2r and = .\/uW/A TVhen (1 - 71) is still positive. but small, the neglect of the term 2xlc in Equation 14 gives, a? a quadratic. i n .rl 14, 11
.it n = 1, 1;quation 14 gives, for the equivalent point,
Since v is small compared to 1, the first approximation value of may he used €or x in v , to obtain
.T,
+ A)(W - H 2 ) H*
-
(7)
EO that H = A , and r = 2, xhen c1 E c / 2 - W / A . For (‘1 appreciably smaller than c/2, therefore, we have H < A; and a rough expression for H is sufficient in the expression for r , or the parenthesis of Equation 6. For this purpose, with r s 1 and y 2 c1, we may t,ake [CK-] c - 274 G c(l - n ) from Equation 1, so that Equation 5 becomes H e d A W / c ( l - n). This approsimate value for H in this region of the titration corresponds to the effect of potassium cyanide as a base a t the concentration e( 1 - n ) and with ionization constant W/,4. Equation 6 then becomeb
From Equation 12, then with n = 1. H e q A T 3 T K . T\.hen - n) is negative, the neglect of the last two terms of Equation 14 gives x g c ( n - 1)/2 (18)
(1
This expression holds also for n >> 1, if there is no precipitation of silver cyanide, for when el >> c/2. y then remains practically identical with c/2. and Equation 18 follows directly from Equation 2. For the slope of the curve near and through the equivalent point, Equation 14 gives 4 S 2.3 (2.72
For the shape of the titration curve of px against n in this region, we first define, as a “buffer capacitj-,” or as the reciprocal of the slope of the titration curve,
9 = - dn/dp.r
=
(12)
+
+
c1
dATl’d/aa/ncK
(2)
Also, the condition of electroneutrality means that H - T i - H - [C?;-]H/A; whence
cH4K f 9?( r$7 - HZ12
+ 1)/4c.
(1)
c:2
H = dAW/(A
K ( 42
so that Equation 3 beconies
+ [ C S - ] + [ H C S ] = 2y + r [ C W ]
= x 4 IJ =
H
FIom the
+ 5 - (r/2)[CS-] = c1/(c/2) = 1 + 2 x / c - ( r / ~ ) [ c K - ]
c1
with x
1. CURVE THROUGH EQUIVALENT P O I N T , OR FOR n Even up to and just through the equivalent point, [CK-] remains greater than A . At n = 1, we have [ C N - ] = 2x/r from Equation 3 , and [ C S - I z S c K / 2 x from Equation 4. so that 85,3 g cr,*K. But H remains always smaller than d F v , so that even mith re g d W / A in x,, the value of [CX-le>or d w e remains , much greater than A , even down to c = lo-?. From the start to just through the equivalent point, therefore, Equation 5 becomes H t’AW/[ chT], while a t the same time y c1 or nc/2. Hence TlTR.4TIOK
(log, 10) z(dn/dr)
+ u/2 + v/4)/c
d+/dpx ?3( 2 . 3 ) ’ ( 2 ~- u/4 - ~ / 1 6 ) ~
(19)
(20)
The “end-point” inflection or the vertical type of inflection corresponding to a maximum value of Idpx/di~I,therefore occurs at
(9)
Equation (6) then gives
or as under Equations (16) and (17):
1, I l + n cH3 o=(2.3)[mtAW(H+&]
iVith the relatively small second term neglected, differentiation of Equation 10 shows that there will be a horizontal type of inflection point in the curve, corresponding to a maximum of 9,at
n
(J2
-
I ) , or 1/(J.2
+ I)
(11)
Since x i < xe, from Equation 17, the inflection point occurs before the equivalent point; but the titration error, with the inflection point as end point, would be very small. [Actually, however,
1652
ANALYTICAL CHEMISTRY
precipitation of silver cyanide occurs even before the inflection point, as seen below, so that both .cj and x, here represent metastable relations for the supersaturated solution.] Combination of Equations 14, 20, and 21 gives
+
For the slope a t the inflection point, Equations 19, 20; and 21 give
Relations Involving Precipitation of Silver Cyanide. During saturation with silver cyanide, we have y = P ’ / x , [CN-] = P / x , and hence
H = 1/AW/(A
+ P/x)
(25)
The first appearance of the precipitate, causing a break in the titration curve, occurs, under equilibrium conditions, when, from r P ) / c . Therefore the value of n a t Equation 1, zP = ( 2 P ‘ the first appearance of the precipitate is, from Equation 3,
Furthermore, since d H / d x = HaP/2xlAW, from Equation 25, the exact Equation 29 gives
The first two terms of Equation 31 are the largest, and since, near the equivalent point, x s 1/a and H G d W / ( 1 -
c2
rP
Practically, since r is always smaller than
x p E 2P’/c or 2P2/cK
+ rP)
-
(PP’
250, we have (27)
and as long as c >>2 A P / K , or c >> 7 X 10-6, then, from Equation 25, HP S 4 2 A P W / c K . K i t h this expression for H in r in Equation 26 we see that np is less than 1 if c * > 2(2 P’ t P ) z / r P , and hence for c greater than about 2 x 10-6. .4t the same time, Equation 26 becomes
+
CJC X/C
=
1
E (2.3) 2 d / P ‘ / c , or (2.3) 2P/c 1. For the curve during precipitation of silver cyanide, Equation 29, with H as in Equation 25, gives directly m for specified 2; rearrangement of the equation may be used to calculate x for specified in:
With P’ >> P ( l H / A ) , a first approximation value of x is easily used to see if a second approximation is needed Prac-
0 (Curve B *)
n -2 m = l
n = l in = 0.5
1
+
- Q/CX
(29)
Here Q is defined as Q = P‘ rP = P‘ P H P / A , with H as in Equation 25. At the equivalent point for the precipitation, then, or when m = 1, Equation 29 gives x, = Q < i Z dp or P / f l K .
+
+’,,i
+
-
fixing the titration error, if the first appearance of the precipitate ( a t equilibrium) is taken as the end point. ( T h e negative sign in Equation 28 appears 94 incorrectly as a positive sign in Equations 67 and 68 of 2 . ) For the titration curve during precipitation, in which the change of p x is considered with respect to the ratio rn (not n), the number of moles of precipitate formed per liter of solution is c1 2 - 21 = c - 22( rP/x, so that
m =
(32)
From Equation 29, then, tni S‘ 1 Le., x i 61 or P/ 1. The left half of Figure 1 shows the titration curve for pure potassium cyanide a t c = 0.1. The solution is unsaturated up to point 1, where silver cyanide first begins to precipitate. The dotted section, 1-2-3, etc., rcprcseiits supersaturated solution.
I¶
1653 Table I.
Titration of Pure Aqueous Potassium Cyanide with Silver Nitrate
[Values of pz for precipitation of AgCN (point I ) , a t inflection point (point 2), and a t equivalent point (point 3)l Percentage Error C Point PZ 100(n-1) 1 1 11.686 -9.64 X 10-8 2 6.958 -2.54 x 10-6 3 6.093 0 -1.79 x 10-4 10 -1 1 10.686 2 7.195 -1.60 X 1 0 - 8 3 6.300 0 10 - 2 1 9.686 -4.40 X lo-: 2 7.422 -1.00 x io3 6.504 0 10 -3 1 8.686 -1.26 X 10-1 2 7,642 -6.34 X 10-2 6.707 0 3
various ways: from xp and c through Equation 27; from x at the end point of precipitation through Equation 32; and from the slope of the px curve and the value of c, a t the end point of precipitation, through Equation 33. In each case, the thermodynamic constant is calculated from the mass constant so obtained, through the relation log P’= log P’ 2 log y, while for the value of y we have 2p E [K+] [NOS-] y E c CI q E 2c, since y c/2 a t CI = c / 2 and y 0 a t c, = c.
t 2 8
+ +
+
+ +
4
-02 100(n
Figure 2.
-
t L
0 1) +
Titration Curves near n of Ammonia
=
1, in -4bsence
tx 6
c = l c = 0.1 C. c = 0.01 d. c = 0.001 Horizontal branch of each curve starts at point 1 of Figure 1, A . for precipitation of AgCN. a.
b.
a.
4
Point 2 is the inflection point, and 3 is the equivalent point a t n = 1, both being metastable. Point 4 is the inflection and equivalent point for stoichiometric precipitation, a t rn = 1. The co-
ordinates of points 1, 2. and 3 for various values of c are listed in Table I, in ~ h i c h“% error” = 100 (n - 1). The detailed course of the curves near n = 1 and near m = 1 is shown in Figures 2 and 3. These curves apply, strictly, for the addition of silver nitrate solution of such high concentration that the volume remains constant, so that c is a constant during the variation of ratio n or ni. The approximate formulas for the relations near the equivalent points, however, hold regardless of the relatively negligible volume changes near these points, provided that c represents the final, not the initial, analytic4 concentration of the potassium cyanide. K i t h the inflection point taken as the end point, therefore, the error TT o d d be negative but practically negligible for the whole range of c considered. The precipitation of silver cyanide, however, would precede the inflection point which, in absence of ammonia, would occur only in the supersaturated solution. But even the precipitation of silver cyanide, if it should become visible when the solution first becomes theoretically saturated, would give only a negligible (negative) error as an end point. Determination of Constants from px Curve. The two equilibrium constants in question are K , the instability constant of the complex ion Ag(CN)?-, and the solubility product of silver cyanide, which may be taken either as P = [Ag+][CN-] or as P‘ = P 2 / K [Ag+][Ag(CN)Z-]. The solubility product, P’, may be obtained, theoretically, in
P -1
100(m
Figure 3.
+I
-0 1 )
Titration Curves near-m = 1, in Absence of Ammonia c c c d. c a.
b.
c.
= l
= 0.1 = 0.01 = 0.002
The instability conetant, K , may be obtained from corresponding values of x and n for given c, before precipitation, through Equation 13 as
K =m-2n” ( - ~
~
+
J
~
+
1
- 22)‘ n +
,(35)
Roughly, for n appreciably smaller than 1, this becomes
K
2cx (1 - n ) 2 / n
(36)
as given directly by Equation 6 with T E 1 . Again, log K = log K 2 log y, while 2p E [K+l [NOS-] y [CN-I c 2cI c ( l - n ) = 2c. K may also be calculated from the value of c and the slope of the px curve a t the first appearance of the precipitate of silver cyanide, combined with the observed value of xp, when Equation 19, with its first term neglected, gives
+ +
+
+ +
+
ANALYTICAL CHEMISTRY
1654
If the value of P' is already known from the later portions of the curve, the theoretical value of 2P'/c may be used for xp in this expression. In connection with Equation 37, moreover, p c.
H as in Equation 44. Since the second term of Equation 10 is again negligible, we have the relations in Equation 11, the slope and position of the titration curve in this region being independent of the ammonia. At the equivalent point, when n = 1 and [CN-] d/cK/22, Equation 47 gives
TITRATION IN PRESENCE OF AMMONIA
General Relations. For the relation between [Ag(NH&+] or z , and [-4g+]or x, we have, with b as the total or analytical concentration of ammonia,
b =
+ [SH,'] + 22
["I]
= (b -
~z)/s
(39)
a,
+
IVhen n > 1, the last term of Equation 47 may he neglected, and the result is
(38)
so that ["I]
(49)
in which s is defined as 1 K a / [ O H - ] . [OH-] E so that with b = 0.2 (the recommended concentration for the Liebig-DBnigbs titration, I ) , s E 1.009. Combining Equation 39 with the definition of K ' , we obtain
x S ~ ( -n 1)/2q
~ ( -n l)K'/2b'
(50)
similar to Equation 18. For a practical approximation of the inflection point near n = 1, H may be taken as a constant, according to Equation 45, since the term 2xq/b is very small compared to 1 ; accordingly, r, or 1 H / A , will also be taken as a constant ( T ) 1.013). Then with [ C Y ] e m x , Equation 47 becomes
+
whence From this we see that as long as x if b = 0.2, z
