2 N+L-==i|. total .hount of nerat ioh tNlr, and tis.nd Ir1t! and (b

20
2
2.O.O
TFEIIIIENT OF OATA
I. ord€r to evltu.te th€ stabltlly congtant, K, tor
a €jnol6 ayat6n .uch !6
N+L-==i|.
It 13 in thsory, pr€par€ onty 60tullon conrlinjnE a kno{n
total .hount of nerat ioh tNlr, and tis.nd Ir1t! and
Fau.rng on€ of rh6 thr* rominins unkndn conc€ntrltion6.
Ths6€....free n€u!rat ion cohc€nrrarion! th€ fr€. tj9.nd
conc€ntr.tion end rhe mt.l ]lsand conpl6x c@cen!.atjon.
This.xcaa6 of d!t. allds to do (a) !o ch.ck th6 valtdity of
the chdical nod.t chos6n ino.d€r ro contirh th. system
(b) !h. larg€ nunb€r of vatus€ of both !h..i.biliry con6iant
K .nd the erperimn!.] v..i!bl€ €€t,in.t€s of th€ or.cision
of K.nd th€ ovo..tt cohsi6r.ncy ot rh6 q!s.6€t can o€ Fd€.
such information l6 vital It w6 ar. to evatu!!€ th6
reljlbllity of strbility consr.nt, ir i€ n.c€66.ry ro rJs€ lr
to crlcul.te rh€riEdyn4ic D...mt€r3. Th. la.s€ !fiount of
data ra knon .3 econdary conc.otratlon v.ri.bt€.
2.1-o
aEc(xo Ry cot{cElTMttot vanIASLE
Ih ord.r !o evltuar€ !h€ Ft.b ily conEtrnt j! js
nece€aary to find a r€tatjonship b.tn€€n rh.m and rh€
ll
d.t mln.d v.rl.bl.. ([Nl, [U,6tc) trl. rhl.
..latlon r. ..t.bl.l.h.d vir th. d.ltnttion ol eecordlry
con.€ntrrllon vlrl.bl.. thr.. ty!.. of functlon€ can b.
6r(!€.1r€ntrl lY
2,1.1
THC
@lPlEX FOil|
ttot Frrcltd
n-
Th. coDrl!x tor||rt'lon tunctlon n- 116 flrai
int.oduc.tl by N. sJ.rru. |nd J. gJarrui [2-31. Let u.
@n6ld.r a r.trl lon L.nd. lta.nd ! ln I aolutlon of
corEt{t lo.rac .tr.neth, th. .qulrlbrl. or.€..1 aB:
N+r<Ejl.A|!
12-r,
xL+L€,Il)x!2
(2.2'
||!n-t +L15
Th. no(lrun valu.
r|ln
(
ot n, x.itt n la N, tlill
2.3)
b€
.
funclion ol both th. i5x1flxr ooordln.tion nur|ber of !h€ rb!.]
ion and th. flultld.ntl.n ol th. 1it.nd, Th. n$3 balanc.
equ.lion lor both !h. n.t.l
conc€nl;r.t lon o.n b. rrltt.n
txlr: Irl
+
0lll + [xl2l
ion .nd lotal
+, , , , , , , . . , f|.ln]
i
ig..d
(
2-4)
ttlr : tU
+tNL1 +
+........,ntrln
2tx!2
(2.5)
A tunction n_ ts d.ttn.d aa .v.rlse nunb.r of
liaanda, L, attlch..t !o !h. mt.l, x, |||y
Tot.l
tur - lll
tound Lts@d
lotal ||tll
.ubrtiluting tn.
vatue
t. frit!6n
txtr
ot
(2,6)
Eo(2.ir) .nd Ee(2.5) tn Eq(?,6
)
lll+ttlLl+2tlrlzi r ..,...,.......n(xtnl _ [Ll
txl + tr,rl! + tNt-2I +..,........txr]n
tr|ll + 2t|r2l
t|'|l + lNLl + tr!21
IN
.,n(xht
+
+
''''''"
" 'l|,tlln
(2.?)
|U|r|tion t m Eq(z,r) D.cc-.
tntxL^1
(2.4)
1r1
rh.r.
* fr*.nr.
ls th€ m)(trun coo.dtnlrion nrnb.r lor th. n trl tf L
i. monod€ntale iiarnd, ft. €q(2.s) .hoN. tnrt tr rill nor
h.lp I er..r d.al tn.valurt,tna n-, as (rhl and
INJ t.i nor
N
in g.n.r.t ..rdlly cbtalnrtla, Th.rarora xn and En
rr.
23
tNht
(2.s)
lxLn-tltu
IN!n
(2. i o)
txl(un
= Kt K2
Oa
(3 .... . . '..
Kn
(2.rr)
.ubatitxllns Eq(2.9) in Ea(2.?) eiv..
xttxltLt + ?xrx2trtt!I2 r .....nxrxz...xntxttrtn
txl + r(1tMl_lrt + (1K2tM]tLl2 + ...,,Kr(a...(n0iltrln \2.12'
tx1{KtlLl + 2KtrgtLt2 + ..,..nkttg...xrt!ln}
tltl{r + Kilrl + Krx2lLl2 +,,...Kira...XntUn}
l(rIL] + 2(1lgtLt2 + .....nxre...t$tr-tn
i.+ (rtrJ * Krx2tll2 + .....rrrt...(nt!tn
(2. r3)
(2.1/t)
fh. Eq(z.rt) elv.s
6itr-l + 2aztll2 +....nonlLln
i + !ttrl
Th.
Eq(2.
t5) mY
+
!2tlts +,...onlun
(2.r5)
b.,rltt r ln !u-.tlon t rB a
tnB"tLln
X*.,.r"
(2. ra)
r *{enrr.rn
X"*""
21
Th€ Eq(2.14) 6ho{s
that lhe n- i6
d€p€ndenr on
ir€€
llsand
conc€ntr.tion and ia inds0end6nt of t{lT, [L]T and f.e6 reral
ron conc€nt.ation. lhe contt.x fornltjon funcijon n- rls
fira! jniroduc€d by N. Bj€rrun [2] .nd lar€. d€votop€d by
J. 8j€r.um I3l. !l j6 rh€ .tarting point, for h.ny of th€
ibthoda u6€d in !h3 catcut.tion of stabillly con3t..r6.
T}IE OEGREE OF FOiI4ATION
Fo. a.y individu.r cdoonen! of sysr€n ! variable
dc
can be d€tin€d a6
TMLC]
Ixl
12.1r )
r
qhsr€ d,c is Dlrri!t rclE fract,jon
deqr6e
of fo.mtion of
6y6t€m
G a
o- conpone.t
|hot€
mr b.
lLc.
Th€
coneidered
.nd !noih.. varl.bto can b€ d.fin6d
4r = 2,<e
rz. ra)
/T is th. fr.crion ot tot.l rrlt bound to tigand tn
iho fo.n of a cohpt€x. 6injtar p.oc€our€
^
Eq(2.16) wa€ .dopt€d to. cltcut.tjng th. Eq(2.it). !i nay b€
Nhsr€
tc6- t Llc
' *'ie.rrr
t6-(L1c
(2. r9)
)
Bc I
r-tc
2a
The
int..6tln9 Bolurron of Eq(t.17) i3 xh€. c = o
0rl
{2.20)
[ ]r
rhe <o siv.6 rh6 Ep€ci.E dt.t.iburion for th€ fr66 @t!l
ion in !h€ 60lution. rlhsn no conpt€x formtion !ak66 pta.€
rhan do 1. unity, beca$€ [Nlr = Il|1. rh€..<c funcrion is
ua6d rn. r€c€nt atlas of fttlt ligand €quilibri! in aqu€ous
solution to 6how ar a elance th€ retarjve propo.tjons of
of th€ aD6.1€6 D.€5ent jn sotutjoh {al.
€ach
THE OE@EE OF @IIPIEX FOfU'ATIq{
The 3.d
vlriabte 0 i6 dofin€d
o
o ja
lxlT
tl
as
1+ >6nILln
of cdpt€r fo.miion.
\2.21)
alm.rizjng thea6
a€cond.ry conc€nt.ltion v.riabt6s th. n- i6 th6 @st
oe.d 1n ihe catcltation of stabil
2-2-O
d€g.€e
On
CAICULATION OF STABILIIY @NSTANI BAAED
IHE COIiPLEX FORXATION FUNCIIOII N-
o6t atability consrant calcut.rions !r€ ba6€d
on Bj€rrqn'6 conpl€x formalion funciion n-, the conpl.x
fo.nrilon luhctlon jnvotv. . pto! ol n_ lsrinrr fr.€ ris..d
26
Such
Dlor c.n bo !rep!r.d by thre6
marn
{.) Il ih€ fr.o lisand conc..i..tion ie m.3ur€d di.ectty,
for 6r.rple by an 6t6ctrod6 rev€rsibl6 to that liglnd,
theh kno{lng the totat concenrraijon of mtal and tigand
present n- c6n b€ catcutatsd, This mgthod
cah not b€
u€ed for f€at coheter6s or xhsd tbt.l n€rar ion
conc€ntrrljon js v€.y lor, sinc€ jn €irh€r c.s€
tLlr -[11 rjr] be cto6. to 2610 and h.nco n will al6o
b6 cto6€ ro 2610.
(b)
R€a.ranrenenr
or Eq(2.0) siv€6
ILlT =
n-tMlT +
tr]
\2.22)
A se.1€e ot sotutions coutd be pr6p!r6d 1n Bhtch n_
.nd [L] {.r€ consr.nt althoush unknorn, . ptot, of tLlT
against [{ly rou]d b€ a st..ight li.e of etop€ n.nd
r.r€rcspr IL]. such aol uii ona r.. catt€d !s
corr€spondin9 6otutjonsr th16 can be us€d for 6ubssqq€.t
6varuat ion of 6tabi I iry consiant's. Th6 corr6spondi
ns
mthod c.. b€ qsed ,hen ItlT - [t] i6 s...ter than
2€.o
and reas than I r]T,
(c) By follorin! soE prop€rty th.t j6 h. proporrion.t
to
lc, 6t'ch a6 rhe 6hf of rut6t ion r.v€r.1bt. €t€cr.od.
or th€ dtstribution ot h€!!t lon bslr€.n rro irniscill€
solvshts! n- can be €v.tult€d fron equllion
iros,n
_
a
lo9l l-l
Thie approach can b€ u6.d {h€n
2[
then Eq(2.7) r€duc.s
tLl + 2(t1(2t[]2
1 + KI ILl + KtK2tLl2
121
Kt
Ir'rl+lxLl+lttt21
12.23)
tLlT - [Ll i6 clos€ to
zero. Con€idor th6 c.sc wh.n il :2
IML] +
as
(2.2.)
it th€ .v€ras6 .{mb€r of tigand per n6!al
0.5 !hd, n- = O,s .nd .r any fr€€ tisahd
only tuo fr€tat containins €pecigs ...
the Eq(2.2a) reduc€s a.
I
(1[L]
r,tL]
IM] + IML]
2
1 + KtlLl
\2,26 )
tLl
l!.9.,
furth6. tE sh*n th.t KnlKn+l
ia
(n c.n
De obtaln€d from th€ valu6 0f lLl cor.espondihs to
n-:n-1l2
tha! i6
l1r
\
l-I;|.t;
ThjB r€lation6hip
at th. pointE n_ : n
-
js
€asi
=
ly
O.E th€n
12.26)
2g
:
[|aLn-11
I
ratnt
in.p.ctl6 of €q(2.9r, th. ..tlo ol th. [email protected]
ot tio coh..ct'lv. corpl.ru l! gie$ by
ay
.-
Ixlnl
11
-
n
if nrn_)n-1
txh+11
On
sutotltlt'lon ol
EC(2.211
ln Eq(2.t) rr
n-+t-n
I
Kn : (-l{-}
tor !y.t n
Kr
'
12.21)
hrye
(2.28t
N = 2 Eq(2.26) b.oones
n_I
............'_t_)
to.i)n-rO
= (l-n-)
M
lz.Za,
(2 = +{_}-r) r
for 2)n-)t
(2.30)
(2 - n-) tll
Frd Eq(2.2t) ud Eq(2.40) in ce !. .v.lu.!.d ar .I
polntr b.tr..n n_: n .nd n_: n - i. Ar.lt rutiv.
a9Dro.ch l. to find !h. .toe. of tli. ?orutlon functton
cu.v. thd l. ih. elot ot n- .astFt [Ll G
(n-
6:
-+n-
-0,4941
arn[l]
By dltt.r.fltldinr ol
-:
an-
aPl
Eq(2.211
- r.lth
(2.3t)
r..p.ct ro ltg.nd
6
^t
n'
2xx_tll + 4X-2[U2 -|2*1_31113
12.32'
(r+2xx-M+rzlL]2rz
= rrlz th€n
ralll
E
r
.nd €lop€
i.
(2.33)
(l+x)
Eq(2.33) la obtrln d d.l..ci,ry ?rd.. Eq(2,32)
aubstltuttne r-tLl=t, ri onty c@€ider |rhen x ts l.rsD,
il x i. anall th€n v.lE of |(- t. .oproxjn.te, rf
Eq(2.16) j! r€srjtlen ..
+ (n--t)ttlKt + (h--2)atlzKlK2 + ...(n--N)t!lNKrq...tq : o
(2.3ll)
r|q thla valu€ of xn ,. ir.t
t Mr KtX2(3
'tI t - + i "
)trll
i(
jlr i
'Kn-r '......Xn-1
in+t Ke2 ....,.Xn+t
tu
:::(2.34)
ln!€i.r.nd h.ve vrl6 b.t..n o dd
lnt n -=n-1/2inEq(2.34)
I +t
N-n,
l+?i
fr trl rrt n=n-tl?KtX?X3 ...Krt...Xn-t
,:rt I + 2r)(tLti atn:n-!/2 xfttKni2,,,.l(n +i
::==:;===::::=(2.3!
)
30
Tht. is !h. cdv6rs.dc. lo.frl.! i. ce u{ th. h.tl
lnt erar v.tr6 ol n- !o o6tlin Xn, th. Eq(2,35) t3
(2.30)
r. D€tio. Qproxin tto?| ol Kn
trcto. t€ndlne tor.rd3 . constant v.rue 15,61,
] . vari.nt. h.v€ us.d conc€.n.d
int.s..l vrlu.. of the torDarton
th. n_ is .lnd.Ccnd.ni ot rhe
2.3.O
cR
P{tcat rGDItt
BAqED
(lr
n_
s..Phlc D€lhod i6 avail.bl. rhen
.nd
.llnirrtion ol
Eq(2.ra)
th.t
.t!
(r - n-)tuKt + (2 - n-)tl!2!tt
- n-)It-l + (2 I
rt
n_
)tll
+
n_)
tLl akz)
(2.37 )
(2
- n-)trlzi?
(2.14)
(1 - n-)tLl
n-)ILl2 {
(a - n-)(lt
!
x,
(2
(l -
-
n-)tLi?
-t_
(2.36 )
(t - n-)
(2 - n-) t Ll
+ (2)
t2
(2.39)
(2.40 )
3l
(r - n_)
(2 - n-)tll
th. r.tt
h.nd
tz
-
n_)
2
tL!
.i.L or Eq(z./()
l---
I
'Kq
ts
th. Y i.trfc.pt gtv.€ -x2 ihil.3toD.
(2.ar )
Fliiiir"
gjv.. (1(2. Th. olh€r m.lhod of
€q(2.36)qd divtd. by n- qr.n
(i -'i)tLl(i
(l -
n_
+ (2
)trlxt
- n-)t!12(tq
(2
(2.121
- n-)t!12(itt
(
(
(2
lub.tjtutlns th.i. varuo! in
2..3)
r-n-,IL]
- n_lttl2
Eq( 2.,rs )
r : ltx + f2Y
(2.11'
rtr... tt .nd t2 .r. runcrion. ot f .rE (Lt. rnts t. lolvo.t
by plottine t/11 .eaiEt 1112, trt potn! ot lnt.rrecljon
yi.ld. K1 .nd (iK2. Thi3 n lhod is t(norn ts .iininalid
32
It r€ con.ld.r th. .y.t5forx|3
f
u-:-_:IU
th.n
Eq( 2.
tn- -'rontr-:n = o
6r-2 n--x+2 "'9"1n
ll1.1n-*.t
qltll x-1-n: ;. N-t-n qr
GN-t
=
(2
th.
.45 )
|u llm.r, I glot ot l.ft hard e.ide
2)/(N-t-n)ILl13
dr.isht lim, th€
Cq(2.45)
BN-t
2.1,O
t5)
I'MTERIC
/qt
I
and
llni!ins slop. l! 6[_2/ql !s
ILI
xE?|toltA AlaEo ot{ B'rEtUtt ,t FUlcTtOt
€qu.rion (2.15) ihioh i6 non ltnllr Fy b..olv6d
ytr.n tha d.t. 9olnia .r. grxt . iho o. aqurl to i,h. nq.!€r
ol unknorn6, to. N = 3 ih. Eq(2.33!) c|n Dl. ..irt!r.n o
n- : (1-n_)tLl6t + {2-n_)tr12q + (3-n-)M36! {2.46)
,hich ia slv.bls It s h.v. thrs. s.t, o? drt. !h.t i.
I016263l
-)rtlt
fr'-_)tLt1
- ttlr
ll:.:
)
(r-n_)[L]3
l-n- ) t!12
(2-n-)tl"l z
(2-n-)ILl,
)tLlz
(3-n- ) tLts
(
(
3-n-
(2.11t
33
In nat.ix th13 i. re9r.3.rt d
by
BA = c
(2.49)
B = AC-1
\2.19)
rh!s B crn b. rosnd by inv.rrins th. A nltrtx .nd
p.€mltipling by colurh v.otor c. Ther. is . .t.pt. Fthod
due t! Btoct( lnd rGrntyrc t7l to. &rvtnr gq(2,40) rhich ir
b*ed on C..rr.rr3 rure [6]. !? r. rerrit. Eq(2.r8]
;
=
9,,-"-',, 'r".
Jn =
and
(h
- n_)trln
subalitut tli. Eq(2,6r) tn
n-
(2.81)
Eq(2.50)
= J.r,s,
Fo. thla mlhod thr€3 cFl!3fia must b. |b!:
(r) th. .!9.rlrint.l <hr. nu.! b. g@d, .. .y.n
12,62)
{r!tl €..or
yt.ld.r.66ou. or m.||t,iv. arEi.r.,
(b) th. n- v!lu.3 ctb€€n gholtd lt. id.ltly tn ur. lolrortne
can
r.2<n2-(1.6
2,2<n3-<2.6
34
2-5.O
CALCULATIq{ OF SIABILTIY cotsT^fta 6asEo
THE OEGRE€ OF THE COfIEX
2.5. I
lEDEil'S
Lend€n
I,IETHOD
[s] d6finod ! function
txjr - Ixl
Irl
TI,II
T
(2.53)
L1
ihe Eq(2.53) with herp of Eq(2.21) can b€ writr.n .6
61
12.5a)
ifN=2th6nEq(2.s4)
F(t) = st + B2trl
(2,b5)
F(L) iB ptott€d .o!in€! tL1, a str.ieh! tjne sraph
re3ul!6, int€rc€pr is Bt whit€ 6tope j6 62. For sy€t6m whe.€
N 2 3 th. relation6hip is .ot tin6a. .nd Bt has
to be found
by €x!..Dol.tjo. to zero fr.. tigand concaolr.lron.
A ne,
function cln bo d€finod
IL]
a6
:6,
+
t B,trtl-r
(
2.56)
Tho procedu.e can be cohtinuld unrit 6n is found, The nerhod
h6e di3.dv!n!!s. bBcaugs th. tll and lxl n.v6 io b€ kho*n.
r!
is po$ibl€ ro round thi3 systdn rhers il:2.nd
d6fini.9 . nd va.iabl.
t(t))x2
by
tLlr - (Ll
(2.6t)
[rlr - Irl
rh6r€ t, l! aveaag€ nuib.. ot lle.||d p.r |.l.l ld .nd N :
rnd Kt )) K2, rlll b. allghtly s..at€r th.n unlly. It t :
[Ltr- tLl : [rlT_
txJ
(2.!6)
l'rLl + 2[ 12!
(2,5s)
Irlr - txt : [xU + [xt2]
(2. ao)
th€ bound lie.nd i6
tllr
conplex
m.lal
_
tLlr -ILl
tt'l
=
repr6s6nt!
sub6titulln! Eq(z.59)
and
Eq(2..o) in Ea(2,57)
t|rll + 2lrtzl
lMLl +
th. fi.at t ..
tHU IrJ
IHLl
lLl2l'|l
ol
(2.61)
[Xt-2 )
r.xt
by
tam
by
Eq(2,n1)
ztxl2ltttlllll
[xl] + ____:_--:-____l/t________
I LI
-t_____-
[xlIrI
.a@nd
lxrl tLI zlrt '''
thlch on sinpl itlcat,ion eiv..
t||ltLI
tN!,1(NucLl
trLttLtztNt'
t2.42)
o, + 2a2 [!l
(2.63)
6i + 6?tll
Eq(2.aS) l. u..d to d.rive Bt .nd 62 troi tro. lisand
2.4.2
F!X,|EA,'Atc:a|{ttt
Fron au
tlol
u..rt
x(L) = o:
, !.
a atralehi
lumtld
I +56.ILtr
fi
\2-64)
rh. dltl.r.niirl forr| ol Eq(e.a4), o r.arr.nltEnt siv6
tnx(t) : Fn(2-a5)
L
'tu-dlll
Fron h.r! tho ibthod Drocxd. .t.tllrty to !.nd.n'. Ethod,
aa alip1. in!.rr€lation.hip .xl.!!:
)((l): F(t)tL! + I
(2.d0)
liffv.r, ttr.. l! n6.tDl. r.t.tto. b.trxn.tth.r of rlr.2"4.0
lllE |.tSEn lt{)
s0tuTIotl
t
TUiE OF rB€ apECtEa
llt
B.for€ an .t,trhpt, ls .ad€ ro drt..r.tne rho
.qulll!.lun ol .'|y .otutlon ..!orton, . cl.r?ly ch...tcrt
37
@del N.t be *6uBd. lnordor ro do rhi€, both !h€ nu.b€.
and nltur. ol th€ va.iorrs ch6nic.t speci6s in .olution 6hould
of mlhod! lor d€t€rninlna th.a€ d.t! a.e
avajlabl€, it my not bo possjbt. ro der6rnjn€ th. nunb6r of
chemic.l apocis6 and th.ir fo.nula€ directty .nd recou..6
nust b€ nad€ to cheric.l r€.60nlbteness lnd !o st.tistical
evalo.tion of the d.ra using varjous rcd€t.sy.!.m.
b€ known, a nunb€r
2-6-1
THE NUIiBER OF APECIEA
IN
SOLUTIOI{
I^ order to fjnd rh€ t,otat nunbor of sp6cie3
pr6.nt in the solution .i oquitibiim, th. t.chnjqu€ ru6t
physlc6i D.rrno!..6, for !his..ch so.ciog
should mk6 i!6 o*n uniqu. contribution to th. observ.bte
param€lar a6 rn intgnsiv€ tactor. Fo. thi6 DurDos. th€ no3t
widely u6€d techniqu€ is !p.c!ropho!d.r.y.
In sp€ct.oohotd.t.y r. conc€r. ,ith tha lb&rbanc€
A, ihich ia a lih€a. lunclion ol conc.ntrlljon of.giv€n
sD.ci63 .! apecified r.v€t€nsth, ihlt i5
"tr. "1., -.i. - .......i""
(2.67
)
A!b./l i6 th€ ob3!rv.d abaorbanc. p.r unit p.!h t€nsth
lrr
al rlv€l.heth, .,,.r,.:
.......r; is mtar lb.o.ptivily ot
ap.ciaa l, 2, 3 ahd n at, rav.lengih l .nd ci, cZ, 63 ...,.cn
uher.
3a
a.€ their .€6p€ctjve conc€ntratjon. Th€ Eq(2.67) i6 rin€ar
d€pend€nc6 0f A.bs on the conc€nrration ot th€ va.io!B
at€cie5 pr€aonl. If d sDocie has z€.o ml!. abso.ptjvit,
th€n it doss not absorb jh sp6.rral r€gion.nd do€s not
ihfluence on r€ve length. In 6quiribriun 6tudie€ to rJ6€
constaht ionic backsround to adjust th6 €quitibrja rlth
solut,ion of alrons nineral .cid6 and .tt!t.i€ is u€u.l in
aolqt,ion. Th66€ 6p€cie6 do not absorb in !h. 6p.ctrat .6gion.
In the sinpl€Bt c.66 th€ Uv-Vi6ible.pectrlh is.ecord€d ?or
a nunber of solutiohs contsjnlng a conslaht anouht of .n6!.1
with incre!6ing tigand concantrations, Th€ spectr. ar€
alp€rinposgd by the pre66nc€ ot anr sp€ctr!tty invaria.t
points, Ihe6€ points ar€ knorn a6 isob€e!ic poinis. Tho6o
polnt3 indlcAt6 the pr6E€nc€ of at tea6t iro ind€p€ndeht
j,
L€t !5 consider a €e.t66 0f 60tution nuhber€d r b
6.ch ha. ite absorb.nce m.sured ar a nuhb.. of diff6r6nt,
wavelonsth. 1
to i th6. dat. s6! o.y b6 t.id jn the forh ot
"i"
t1j
a
,23
'i
taJ
'l'
39
Bh€re aij r€p.6E€nrs th€ .b6o.banc€ ot 5ol!tion J .t
wavelensth i. va.ious rerhod6 a.e avaitabl€ in th€ .literaturo
to determin€ lh€ number of sp€cjes t14-15j jn eoiurion.
2.6.2
@|IPUTATIONAI IiETHOO FOR DETERNIIINC TH!
llUtlBER OF SPECIES IN 6OLUTIO|{
It the.6 a.e nor€ speci€s j.votvina jn 6q!jtibria
the computation.l n€rhod€ are used ro d€t€rmine rh6 .!nk
natrix [16-19]. conpurer pro!rans t2o-211 catcurats !he
nlnber of sp6cjes ih sotulion.
mat.ix j6.educ6d by. s€ri€€ of e
opera!rohs to an squivat6nt, r6duced n.!.ix in
Th6
largest. et6renr. are on rh€ teading djagonal and al
belo, this are z€ro.
2.1-O
STEP
SISE FORIIATIdI OF @|IPL€XES
It is fundameni. y assumed th.t conptex form.tion
is a st6p riEe proc€ss. rl it js t,ru€ lh€n ihe rario o,
various stabitiry consranrs co!td in rheory be pr6dlct6d by
6iat,rstical considerarjons. rhe exDerin€n!.t ratio wi]] b€
dep6ndeht upon ihe et6ct,ronic and steric flctor in addition
io a sinpl€ staristicat effect, for conpt€x MLn Bh6.€ L i6
uhidehtat€ ligand, jt Is ueqatly assum6d th!t:
a0
(.) arr poa.tbl. coo.diEtid ..1t 6 (x) .r. lrLntic.l,
(b) th. tsd.ncy o?. corolex NLn to t$. . ltsand, I, t3
Frorortlonrl t,o lhe nunb.r ol li€aod occupj.d 61t.. (n),
(c) th. i6hd.rcy to. G4Dlo. rln to trk uD mther ltaard
l. groDo.tlon l to rh€ nu*ar o? v&ur ctr65 0{-n)r n
r. Ma€cuttv. .r.btltly con tant, xl,t2,,.....,xN rill
b. pr€dict d t6 tE gropo.tlon to N/t, (X-t)/2, ,...,(Nn+r )/n, (tr-n)/(.+r ),.. ....21 ltr-1 t, i/
r..p6!iv.ly, .o
th. r.tio ol aucc€s.lv. cri.r!t!nt5 du. to atltisttcrl
(rFn+l )(n+1)
(2.44)
(|Fn)n
(|.neral]y to quot6 th. toacrithiic r.tlo of 3trDi tjtt
.oBr.nt r it c.n b€ d.lln.d s
(
-n+r )(n+r)
(
(x-n)n
2. a9)
.o Ec(2.64) b.c.|r.s ..
(2.70 )
loih+t :
loetx
l. cd b. .xpaot.d, it irtliC.nt t
ratlo dltfara oonsldambly.
tieand. ar. u..d th.
REFERINCES
1.
F.R, HErt6ly, C, Aurg.6a .nd R. Atcocr, sotution
€quilibria , EtIra Hor*ood Chichester, rohn krle,
N€v York {1940) pp 54.
2.
3.
N. Bj€..um, Z. Anorg, A|gsn. Chem., tl9, t79 {t921).
J. Bje..un, " Metal Amine Fo.Btion jh Aqu6ou6
Solution ", cooenhag6n, p. H.ase and 6on, (r941) and
(1957).
4.
J. Kragteni Artas of M€ral Lisand Equitibria in
Solution , chich.sr€r, E j6 Ho.wood (1976),
5. M,T. Beck, Ch6mjstry of ConDtex Eq!itjbrj. , Vah
No6trand, rondon (1970).
0.
7.
a.
t.s. Ro66otti, - Th€ D€!€rnjnatlon
of Stability Constantg , trccrar-Hiit! London (t96j).
8.p. Blolh and G.H. ltclnlyre, J. An. Ch6n. Soc.,
75, 5667 ( t953) c,s, chaDra and R.p, c.nale, Eds. ' Nurerrc.t Methods
ror rngrneer6 , 2nd Edn., l{ccr.t_Htll, N€, york
( 1949)
F,J.C.. Rossottj and
!D
21a-221
.
9. l. Lgnden, z, Physjk, Ch€n., Atss, 160 (1941).
1O. A, Froneau, Acta, ch6n. scand.,4, 72 {1950).
r1. T,-Noxicka-JEnkorEk., J. Inorg, Nucl. Ch€n,,33,2Ot3
(r97r).
12.
,3.
chyle$ki, AnOrd,
to, 1s5 (t971)-
Ch.
Ch. grynestad .nd
72, 296 ( 1963).
Ch€m.
ht.,
c.p. snith, J. phys. cheh.
14. F€f. 2.1 pp 45-50.
r5. H.M. Irving ahd H,s, Ro.6otti, J.
(19s3).
3397
Edn.,
t
ch6n. aoc.,
a2
t6. R.tl. hll&e dd a,t (ltz, J, phya. chir.,
at, 3690 ( | 96,1).
17. l.P. v.r9a lnd F.c. v.rtcht An!t. chon.,
39, I tOl (r96?),
18. 0. Krllkie, an!1. ch.n.,37,376 (19a!).
19. Z.Z.
Husa ffd A.A.EI, Aildy, J. pht.. Ch.h., r!.
( tttl ).
20. Ral. I DD 3t5-3i7.
2r.
A.1l.l(. Khah:.da .nd
A.t. Iutti,-
Proeru. " in ot-g^atc !.neu!!€.
Unpubti.tEd
Z95ra
Caur€r