- Email: [email protected]
Calculation of equilibrium concentrations of complex ions from stability constant data
SHORT
Calculation
COMMUNICATIONS
577
of equilibrium concentrations of complex stability constant data
ions from
Stability constants of many complex ions have been determined. It is often necessary to be able to calculate from the stability constants the concentrations of the various ionic species at equilibrium. as for example, in the calculation of the beat changes associated with the various steps occurring in complex salt formation. The graphical solution of this problem is rather tedious and the following procedure has been developed for use with a desk machine, or better, with a digital computer. As an illustration, the mctbod is applied to the silver-pyridinc system, u and 6 being the initial concentrations of silver and pyridine respectively. AR+ (u -
x)
r.kPYl+
(x - Y)
-t-
e
PY (6 -
-t-
(6 -
x -
y)
0
+
1’): x -
(X
r-AKPYj+ -
Id1 =
F
x)(/J -
P
-
y)(b
Y) -x
-
y)
(I)
Y)
Y
k3 -
rh!l'Yal+
y)
-
-
Y
-
.r -
y)
(2)
x and y may bc found graphically but the method is tedious and of limited accuracy. A solution of this problem may be found by solving the equations kl(U - X)(L - x -.
y)
-
(x -
h&r
y)
-
y
-
y)(b
-
0 0
<
<
x
-
x <
x -t- y <
y)
=
0
-
0
(3)
a
(4)
II
Disregarding (4) for the moment, solution of set (3) yields a cubic equation and for practical values of the known parameters LZ,b, IQ, kz, this cubic may have three real roots; the computations involved in deriving the necessary coefficients for cubic solution are rather more than the few complete iterations in the method proposed here. It is easy to show that the physically meaningful root of (3) and (4) lies on the curve a -
Y ‘-
x(6
-
I
-
s)
(5)
Consequently, if no other approximation is available, x = o will do as a starting value. The iterative scheme becomes “If x is an approximation to the root, then x + It is a better one where (a -
Ir = (2%
-I- y
zy)(S -
P)(a
3_ xy --P - 2y)
fix -
lt-(0
yy
-+ cab) (6)
-
2x)(.?: And.
y)
Cltitn. A&L,
28 (rgG3)
577-579
SHORT
578
COMMUNICATIONS
where
and y given by (5)“. For example, in the silver-pyriclinc system it may be necessary to determine the concentrations of all the ionic species in solution. Commencing with an initial concentration n of silvcrion and 6 of pyridinc, the concentrations of all the ionic species can IX found by the evaluation of x and y. For this system the stability constants kl and k2 arc x00 and r4o respectively. In a particular experiment the initial concentrations CLand b were o.o+@ and 0.04.82 mol/l respectively. It is required to find x and y. As a first attempt to solve (5) put x0 = 0 and calculate yo. In this case, yo = o. Putting NO = o, YO = o in equation (6), 110is found to bc o.orG1g8. NCJWlet x1 = xg -t_ 110,that is, o.orG198, place this value for x in (5) and calculate yl. In this case yl = o.o11c)45. Putting XI = o.or6rq8, yl = 0.011945 in equation (fi), /II is found to be o.oo78ro. After five such iterations /!O is found to be -2 - IO-” and so the solution of (5) for x and y is correct to six places. The calculations are tabulated and for the initial a and 6 as indicated above, the results for x and y arc o.o26c)9r~ and 0.013768 rcspcctivcly, correct to six decimal places. The concentrations of all the ionic spccics can now bc found: A1:
=
(11 -
s)
1’1
=
(b -
s -
[&py]+
=
(X
y)
[Agpyal”
-
3
’
-
EXAMI’1.E 100. fra =
fhllu: Jr, = Sttcrfi~r~ ldtw ,X0 ant1 y0 in (4) I;irsl iterctliwr St ant1 yl in (4) Sccorld iteruliou we und yr in (4) Third ilerulimi yn ant1 y:~ in (4) i~ouvtlr ilrvcilio~r 221ant1 y.1 in (4) l:ij(li itcrution x4 and y4 in (4)
give give give give
0.0448
-
o.oz70
y) =
0.048z
-
o.oz70
3
0.0270
-
0.0138
=
0.0138
OF CAI.CUIATION
140, u =
Ill,
Xn -I- /Jo
-
Xl -I- hl
-
s:! -j- Its
-
.\‘:I -b IIn
=
.r4 + /LI
-
0.0138
(6 DECIhIAL
=
0.0x 7H mol/l
=
0.007.)
=
0.01 32 rnol/l
=
0.0 I38 1110l/1.
PLACES
0.0448, 6 = 0.0482. . . . . . so = 0.000000 ant1 x0 Iti (3) #vcs
give fiivc
=
-
=
Xl = Irl = A.1 = ha = x:1 = IL:, = x.1 = Il.! = x,1 = /J.I-
lllOl/l
ONLY) y0 = o.oooooo
o.orG1gH
0.01G1g8 0.007810 0.024008 0.0027og 0.02G717 o.ooorHo o.orGgg7
imtl XI in (3) gives
ye -
0.01 rc).t5
and .rz in (3) gives y9 -
0.0 I .I073
wltl xn in (3) gives
o.ol3Hr4
y:~ -
;~ticl .r~ in (3) Kivcs y.1 = o.or37G~)
0.000002
0.02Gggg and ~‘4 in (3) gives y4 = 0.0137G8 --. 2. IO--~~. so that the itcrrrtion has convcrgccl
to G plWX%
In machine operation on this problem it was found that an average of 5 to 6 iterations yielded x and y correct to five significant figures. The method seems ideally suited to computer work where a large number of quartets a, b, /ZI, ka are available and the corresponding values of x and y are needed for interpretation of the experiments. For the particular machine used (I.B.M. 1620. rising FORTRAN language) the F~WZ~. Chia,l. .,1&a, 28 (rgG3) 577-579
SYORT
579
COMMUNICATIONS
very simple programmc for the above yielded therelev,ant five-figure solutions (x, y) for a number of different quartets (a, 6, /cl, kc) at muck ‘l&s &an half a.minute per -. quartet. The problem for finding equilibrium constants for such reactions as Ni’+ -+- cn + [Ni cn]3+ [Ni cn]r+ -+- cn P [Ni cnz]z+ [Ni cnz]a+ -+ cn F+ [Ni cn~]z+
has been handled similarly with a similar simple program of about the same len@h, using the same iterative technique on the three mass law equations. It is felt that attention should be paid to other similar chemical problems where an iterative method of solution can replace the usual graphical methods. Departme?it of Mathematics, lle~artme~rt Newcastle University College, Tighes Hill, h'.S.bV. ,‘Australiu) (Received Appendix: I 2
4 5 8
9
IO
15 16
July
of Chemistry,
moth, x962)
FORTRAN
Programme
FORMAT (F8.4. F8.4, l?ro.o, Fxo.0) =, FORMAT (4H DATA, 2HA =, F6.4, 2HR =, F6.4,2HC FORMAT (3HX =, F9.7. 6H Y ==, F9.7) FORMAT (GH CHECK, 16H CAI:CULATED C =, l32.2, ACCEPT TAPE I, A, R, C, D X = 0.0 Y = 0.0 PRINT 2, A, 13, C, D E = 13 + x./D 1: = A -t_ l3 + x./C G A--x./C H&M = (Y + Y - E)* (X* X + X” Y - F” X - G* HDEN = (X -i- x + Y - I;)* (E - Y - Y) $- (R - x H = HNUM/HDEN X =X+H Y = E/2. - SQRT ((E/z.)* (E/2.) - X* (R - X)) IF ((H* H) - x.oE - 12) 15, IO, IO PRINT 4, X, Y IF (SENSE SWITCH 2) 16,8 AC = (X - Y)/((B - X - Y)* (A - X)) AD = Y/((R - x -- Y)* (X - Y)) PRINT 5, AC, AD GGYTIZJS END
Notes:
C. CUKTHOYS \V. RI~ISLI’:S
F8.0, 2HD =, F8.0) 3H D ==,
F12.2)
Y + A* B) x)* (X - G)
The corrcsponclcncc bctwccn programmc and the illustration is : ABCDXY AC and AD (1 0 kl ka x y cheek-calculntcd kl and he (2) If wnsc switch z is “ON”, the computer will check the vah~cs of x and y calculating ht ancl k3 for comparison with input data. (3) The programmc is for an 1.13.M. 16ro with tape input.
(I)
O~JtXiinCd
by
Calculation
COMMUNICATIONS
577
of equilibrium concentrations of complex stability constant data
ions from
Stability constants of many complex ions have been determined. It is often necessary to be able to calculate from the stability constants the concentrations of the various ionic species at equilibrium. as for example, in the calculation of the beat changes associated with the various steps occurring in complex salt formation. The graphical solution of this problem is rather tedious and the following procedure has been developed for use with a desk machine, or better, with a digital computer. As an illustration, the mctbod is applied to the silver-pyridinc system, u and 6 being the initial concentrations of silver and pyridine respectively. AR+ (u -
x)
r.kPYl+
(x - Y)
-t-
e
PY (6 -
-t-
(6 -
x -
y)
0
+
1’): x -
(X
r-AKPYj+ -
Id1 =
F
x)(/J -
P
-
y)(b
Y) -x
-
y)
(I)
Y)
Y
k3 -
rh!l'Yal+
y)
-
-
Y
-
.r -
y)
(2)
x and y may bc found graphically but the method is tedious and of limited accuracy. A solution of this problem may be found by solving the equations kl(U - X)(L - x -.
y)
-
(x -
h&r
y)
-
y
-
y)(b
-
0 0
<
<
x
-
x <
x -t- y <
y)
=
0
-
0
(3)
a
(4)
II
Disregarding (4) for the moment, solution of set (3) yields a cubic equation and for practical values of the known parameters LZ,b, IQ, kz, this cubic may have three real roots; the computations involved in deriving the necessary coefficients for cubic solution are rather more than the few complete iterations in the method proposed here. It is easy to show that the physically meaningful root of (3) and (4) lies on the curve a -
Y ‘-
x(6
-
I
-
s)
(5)
Consequently, if no other approximation is available, x = o will do as a starting value. The iterative scheme becomes “If x is an approximation to the root, then x + It is a better one where (a -
Ir = (2%
-I- y
zy)(S -
P)(a
3_ xy --P - 2y)
fix -
lt-(0
yy
-+ cab) (6)
-
2x)(.?: And.
y)
Cltitn. A&L,
28 (rgG3)
577-579
SHORT
578
COMMUNICATIONS
where
and y given by (5)“. For example, in the silver-pyriclinc system it may be necessary to determine the concentrations of all the ionic species in solution. Commencing with an initial concentration n of silvcrion and 6 of pyridinc, the concentrations of all the ionic species can IX found by the evaluation of x and y. For this system the stability constants kl and k2 arc x00 and r4o respectively. In a particular experiment the initial concentrations CLand b were o.o+@ and 0.04.82 mol/l respectively. It is required to find x and y. As a first attempt to solve (5) put x0 = 0 and calculate yo. In this case, yo = o. Putting NO = o, YO = o in equation (6), 110is found to bc o.orG1g8. NCJWlet x1 = xg -t_ 110,that is, o.orG198, place this value for x in (5) and calculate yl. In this case yl = o.o11c)45. Putting XI = o.or6rq8, yl = 0.011945 in equation (fi), /II is found to be o.oo78ro. After five such iterations /!O is found to be -2 - IO-” and so the solution of (5) for x and y is correct to six places. The calculations are tabulated and for the initial a and 6 as indicated above, the results for x and y arc o.o26c)9r~ and 0.013768 rcspcctivcly, correct to six decimal places. The concentrations of all the ionic spccics can now bc found: A1:
=
(11 -
s)
1’1
=
(b -
s -
[&py]+
=
(X
y)
[Agpyal”
-
3
’
-
EXAMI’1.E 100. fra =
fhllu: Jr, = Sttcrfi~r~ ldtw ,X0 ant1 y0 in (4) I;irsl iterctliwr St ant1 yl in (4) Sccorld iteruliou we und yr in (4) Third ilerulimi yn ant1 y:~ in (4) i~ouvtlr ilrvcilio~r 221ant1 y.1 in (4) l:ij(li itcrution x4 and y4 in (4)
give give give give
0.0448
-
o.oz70
y) =
0.048z
-
o.oz70
3
0.0270
-
0.0138
=
0.0138
OF CAI.CUIATION
140, u =
Ill,
Xn -I- /Jo
-
Xl -I- hl
-
s:! -j- Its
-
.\‘:I -b IIn
=
.r4 + /LI
-
0.0138
(6 DECIhIAL
=
0.0x 7H mol/l
=
0.007.)
=
0.01 32 rnol/l
=
0.0 I38 1110l/1.
PLACES
0.0448, 6 = 0.0482. . . . . . so = 0.000000 ant1 x0 Iti (3) #vcs
give fiivc
=
-
=
Xl = Irl = A.1 = ha = x:1 = IL:, = x.1 = Il.! = x,1 = /J.I-
lllOl/l
ONLY) y0 = o.oooooo
o.orG1gH
0.01G1g8 0.007810 0.024008 0.0027og 0.02G717 o.ooorHo o.orGgg7
imtl XI in (3) gives
ye -
0.01 rc).t5
and .rz in (3) gives y9 -
0.0 I .I073
wltl xn in (3) gives
o.ol3Hr4
y:~ -
;~ticl .r~ in (3) Kivcs y.1 = o.or37G~)
0.000002
0.02Gggg and ~‘4 in (3) gives y4 = 0.0137G8 --. 2. IO--~~. so that the itcrrrtion has convcrgccl
to G plWX%
In machine operation on this problem it was found that an average of 5 to 6 iterations yielded x and y correct to five significant figures. The method seems ideally suited to computer work where a large number of quartets a, b, /ZI, ka are available and the corresponding values of x and y are needed for interpretation of the experiments. For the particular machine used (I.B.M. 1620. rising FORTRAN language) the F~WZ~. Chia,l. .,1&a, 28 (rgG3) 577-579
SYORT
579
COMMUNICATIONS
very simple programmc for the above yielded therelev,ant five-figure solutions (x, y) for a number of different quartets (a, 6, /cl, kc) at muck ‘l&s &an half a.minute per -. quartet. The problem for finding equilibrium constants for such reactions as Ni’+ -+- cn + [Ni cn]3+ [Ni cn]r+ -+- cn P [Ni cnz]z+ [Ni cnz]a+ -+ cn F+ [Ni cn~]z+
has been handled similarly with a similar simple program of about the same len@h, using the same iterative technique on the three mass law equations. It is felt that attention should be paid to other similar chemical problems where an iterative method of solution can replace the usual graphical methods. Departme?it of Mathematics, lle~artme~rt Newcastle University College, Tighes Hill, h'.S.bV. ,‘Australiu) (Received Appendix: I 2
4 5 8
9
IO
15 16
July
of Chemistry,
moth, x962)
FORTRAN
Programme
FORMAT (F8.4. F8.4, l?ro.o, Fxo.0) =, FORMAT (4H DATA, 2HA =, F6.4, 2HR =, F6.4,2HC FORMAT (3HX =, F9.7. 6H Y ==, F9.7) FORMAT (GH CHECK, 16H CAI:CULATED C =, l32.2, ACCEPT TAPE I, A, R, C, D X = 0.0 Y = 0.0 PRINT 2, A, 13, C, D E = 13 + x./D 1: = A -t_ l3 + x./C G A--x./C H&M = (Y + Y - E)* (X* X + X” Y - F” X - G* HDEN = (X -i- x + Y - I;)* (E - Y - Y) $- (R - x H = HNUM/HDEN X =X+H Y = E/2. - SQRT ((E/z.)* (E/2.) - X* (R - X)) IF ((H* H) - x.oE - 12) 15, IO, IO PRINT 4, X, Y IF (SENSE SWITCH 2) 16,8 AC = (X - Y)/((B - X - Y)* (A - X)) AD = Y/((R - x -- Y)* (X - Y)) PRINT 5, AC, AD GGYTIZJS END
Notes:
C. CUKTHOYS \V. RI~ISLI’:S
F8.0, 2HD =, F8.0) 3H D ==,
F12.2)
Y + A* B) x)* (X - G)
The corrcsponclcncc bctwccn programmc and the illustration is : ABCDXY AC and AD (1 0 kl ka x y cheek-calculntcd kl and he (2) If wnsc switch z is “ON”, the computer will check the vah~cs of x and y calculating ht ancl k3 for comparison with input data. (3) The programmc is for an 1.13.M. 16ro with tape input.
(I)
O~JtXiinCd
by