ISOSBESTIC POINTS GENERATED BY VARIATION IN TEMPERATURE
Nov., 1962
2169
FUSED SALT SPECTROPHOTOMETRY. 111. ISOSBESTIC POINTS GENERATED BY VARIATION I N TE&IPERATURE1 BY J. R. MORRES Hanford Laboratories Operation, General Electric Company, Richland, Washington Received April 8.9, 1068
A study of the temperature dependence of fused-salt spectra has revealed that several systems produce isosbestic points. As the temperature of a given sample is varied, the corresponding spectra often exhibit several wave lengths a t which the absorbance is temperature independent. A phenomenological argument is presented to show that when this happens, one and only one species contributes to the absorbance in the region of an isosbestic point and that this species does not change concentration as a function of temperature because of equilibria involving other non-absorbing species. It also is shown that spectral changes must arise from a linear temperature dependence of the molar absorptivity a t a given wave length. These conclusions are in contradistinction t o those made by consideration of isosbestic points occurring a t constant temperature. In this latter case, their existence proves that two and only two absorbing species are variables.
Introduction Isosbestic2 points often are observed in absorption spectra of solutions having a fixed concentration of metal ion but varying concentrations of a given ligand. I n such systems, the isothermally measured isosbestic points are ascribed to, and considered proof of, the presence of two, and only two, spectrophotometrically distinguishable species. Isosbestic points also have been observed in spectra of pure molten salts (Fig. 1) and mixtures of molten salts (Fig. 2) measured a t different temperatures. Temperature variation is not conducive to the formation of conventional isosbestic points because the concentration of the absorbing species changes as the fluid expands and contracts. It is the purpose of this paper to establish the conditioiis which must exist in order for temperature-generated isosbestic points to arise. As a prelude to this undertaking, the isothermal system will be considered, since, to the author’s knowledge, its treatment has not been formalized elsewhere. Discussion Constant Temperature.-If temperature is not varied, the total absorbance a t any wave length due to the concentrations of various species (Cl, Cz. . . Cn) is given by
system remains invariant, the concentration of the group of dependent species C, must remain constant and each species Dj remains constant n
co = i = l c i
(3)
and
Dj
=
D,a
Ci
for C1 in (2) produces
n
Substituting Co 2-2
n
A(X,Ci,D,)
=
EI(X)CO
+ c [€,(A)
-
2=2
+ 5 e,@)DJ
EI(X)IC~
J=l
At the isosbestic points, A,, t>hetotal absorbance remains invariant as the relative concentrations of the interdependent species change; thus, with variation of C1
dA (Xc,C,,DJ dCi
n
0
=
C e=2
[Ei(Xo)
-
dCi dC1
E I ( x ~ ) I-
+
n
A(x,C~) L
C Ei(X)Ci ?=I
(1)
where ei is the molar absorptivity of the ith species and L is the path length, hereafter assumed to be unity. However, it is important to distinguish the species which are inter-convertible, i.e., related through the law of mass action, from those which are not. To make this distinction, eq. 1 can be rewritten n
A(X,Ci,Dj) =
C ei(X)Ci a=1
+C
Ry definition, dDj/dCI = 0 and dCi,/dCI # 0 ; thus for the equality (4) to hold, the coefficients €,(Ac) - q(Xc) must all be zero; otherwise, the probability of the sum being zero is negligible. The same argument holds for each of the other species. If, in eq. 4,n = 2, then el = €2. However, if n exceeds 2, the probability of (n - 1) equations
na
j=1
Ej(X)Dj
(2)
where the various species C i are mutually dependent but independent of the various species D,. If the total concentra,tion of absorbing materials in the (1) General Electiic Co., Richland, Washington. This work was performed under contract no. AT(45-1)-1350 for the U. S. Atomic Energy Commission. (2) The words “isobestio” and “isosbestic” appear in the literature with about the sam8 frequency. “Isosbcstic” is used in this paper because i t derives from the Greek ( r . p c r s ~ r r o s )meaning “serving to extinguish.”
C i ( M
=
e~(Xc)
is remote. This becomes evident through consideration of three spectra of components CI = A, Cz = B, and C3 = C in Fig. 3. If B and C cross A somewhere between X I and h2, the probability of each crossing A in the interval (A2 - XI) can be defined as unity. The probability that C will cross (3) The cancentrations Dj could represent a series of interdependent concentrations but have been chosen t o be independent since such a choice does not result in a loss of generality.
J. R. MORREY
2170
Vol. 66
WAVELENGTH , mp.
Fig. 1.-Pure
I .5
(1) T = 110'; (2) T = 145'; (3) T = 175'.
molten (CgH6)4-UUOZ(SO3)aat various temperatures:
L
1.0
1.1
I
I .o
0.5
Fig. 2.-UCl4
A within the increment 6x3 from the crossing of A and B then is given by
For S such crossings, where S represents the number of isosbestic points on a spectrum, it follows that the probability would be
p
=
6Ai
____--
A, -
An-1
and would become negligible if Ski + 0. Therefore it follows that the existence of isosbestic points is
1
2.5
I .5 2.0 WAVELENGTH, MICRONS. in KAlC14 solvent a t various temperatures.
strong evidence of only two changzng absorbing, species, especially if several such points exist. Furthermore, it is easily shown that all intersections of two spectra are isosbestic points if these spectra arise from two solutions, a and b, containing only two absorbing species in different ratios constrained by the condition expressed in eq. 3
+ +
-
+ +
Aa(A,Ci) = C~Q(A) [EZ(A) E~(A)]C~' and Ab(A,Ci) = Coel(A) [EZ(A) - c ~ ( X ) I C , ~ K where m
K
=
ej(X)Dj
j=1
ISOSBEXTIC POINTS GENERATED BY VARIATION IN TEMPERATURE
Kov., 1962
2171
At the points of intersection, As = Ab, thus
- E~(X)I
= [Q(x>
[E~(x)
-~x)Ic,~
but since CZ"f Czb, e2(X) must equal zero, the condition which uniquely guarantees isosbestic points. Variable Temperature.-If temperature is also variable, eq. 2 becomes n
A(X,Ci,Dj,tj =
i=l
ei(X,t)Ci(t)
+ m
C ej(XADj(t) 5 x 1
(5)
There may be two independent forms of temperature dependence for Ci, one due to volume expansion, f ( t ) , and the other due to the temperature dependence of the equilibrium of Ci with its environment, gi(t) Ci(tj
=
Ci'.ff(t)si(t)
(6)
The concentrations a t a reference temperature are represented by Ci0 and Djo. 8ince f(1) = 1/(1 at) where 01 is the coefficient of expansion, eq. 5 can be rewritten
+
Wavelength.
Figure 3.
equilibrium with other species, the probability of the existence of n X 8 equations 9 and m X S equations 10 is negligible if n, m, or S is greater than 1 and ei, ej, and gi are functions of 1. The only way that two or more species can exist is for eq. 8 to hold for all but one species and eq. 9 or 10 to hold for the remaining species. Furthermore, if this species is an ith species, it is improbable that S relations Ei(Xc,t)gi(t) = elo(Xc)gio(l
At an isosbestic point resulting from a temperature change, the derivative of the absorbance with respect to the temperature is zero
+ at)
will exist for one species unless either ei or gl is temperature independent. If ei is temperature independent, then
+
gi(t) = Q i o ( l at) This, however, would result in an unchanged spectrum, Le., completely independent of temperature, since 01 is not a function of A. Moreover, if the temperature dependence of gi were 1/(1 j3t) where a # j3, then independent spectra would result with no points of intersection. It therefore follows that temperature-induced isosbestic points arise only if gi = Sio; Le., there can be no change in the concentration of the species absorbing a t the isosbestic point due to being in chemical equilibrium with a non-absorbing species. I n addition, the temperature dependence of ~i must be of the form
+
Again the coefficients of C? and Djo must be zero in order for eq 7 to hold with any degree of probability. 'They therefore give rise to two differential equations, the solutions of which are either trivial, when
or are
ci(Ac,t)gi(t)
=
and ej(Xc,t)
+
=
+ at)
eio(he)gio(l
Ejo(Xc)(I
+ at)
(9)
(10)
Thus (n m) equations of the form (8), (9), or (10) must hold at every isosbestic point and S(n m) such equations must hold for a given spectrum. Since there is no implicit relationship hetween the abbsorptivity of a species and its
+
+
~ i ( L , t )= ei"(1 h(X)t) (11) Since the ith species must not be in equilibrium with any other species, by definition it becomes a jth species. It therefore is concluded that only a jth species can contribute to the absorbance a t an isosbestic point. Isosbestic points thus will occur wherever h(X) = a. Attempts to Correct Spectrafor Liquid Expansion. -If one attempts to correct the concentration term of the absorbance for expansion of the fluid, isosbestic points generated by temperature change do not vanish but, as shown by the following argument, they merely shift. A t an isosbestic point made by two spectra obtained a t temperatures tl and tz, the relationship €1
(Xc,h>Ci (tl)
=
€1
(Xojt,)
Ci ( 1 2 )
J. R. MORREY
2172 2.5
r--m
7 --~
Properties of the Temperature Broadening Function.-The function h(X) might be termed the temperature broadening function and generally will be a minimum (a negative value) a t or near the absorption peak, depending on whether there is a shift in the peak with temperature. If the transition probability is temperature independent, the area under the absorption peak should remain constant. To ensure this, the function h(X) must become positive on both sides of the wave length of maximum absorbance. Isosbestic points thus will occur whenever h(X) equals a and often will occur in pairs on each side of the peak. This is observed in Fig. 1 and 2. Figure 4 shows h(X) averaged from three independently obtained values of h(X) according to the equation ---.A
0.6
A
c
02
0 -0.2
z
05
2Lz 04
2 03 m
a 02 01 I
0 570
Fig. 4.-An
together as h(Xc’) decreases, finally coinciding when --I h(X,’) is equal to the minimum value of h(X).
P
. 1.0
--$
Vol. 66
580
590
610 600 WAVELENGTH, m p .
620
example of the function h(X) for the region 575-625 rng of Fig. 2.
+ +
+ +
- A(X,tm)(l atnr) - A(X,tn)(l atn)trn obtained by simultaneous solution of the equations A(X,tn)(l
h(X) =
A(X,trn)(l
atn)
atm)tn
is valid. An attempt to correct Ci(t2) to Ci(t1) by atz)/(l atl) results multiplying Ci(i2)by (1 in the inequality
+
+
and thus destroying the isosbestic point a t X,. However, to a good approximation, a new isosbestic point A,’ mill occur in the vicinity of A,. That the value of h(X,’) at the new isosbestic point is zero can be obtained from eq. 12 after removing the inequality by replacing A, by A,‘ and substituting eq. 6 and 11 into it, keeping in mind that gi(t) is now unity. IIowever, if the expansion coefficient is not known accurately and an assumed value y is used in the corrective term for concentration, i.e., ci(tJ*((l y22)/(1 rtd),eq. 13 results
+ + (1 + h(hc’)h) - (1 + h(Xc’)t2)(1 + (1 + 4) (1 + + -
at210
Yt2)
$1)
(13)
By solving for h(X,’) , expanding, and simplifying terms eq. 14 is obtained h(Xc’) =
+ Ytl) + (1 + O--Y
(1
1
+ K’lz
+
LINURTEMPERATURE CORRECTION
(7)
(14)
As indicated above, when an accurate correction is made then a = y and the new isosbestic point occurs where h(X,‘) is not dependent upon tz but is zero; even when a t’ y, unless ytz is very large, h(X,’) is only slightly dependent upon tz. Thus the temptation to conclude that isosbestic points in spectra corrected to a common concentration prove the existence of two species must be avoided. It is to be noted from eq. 14 and results below that as y increases, h(X,’) continually decreases. From the nature of the function h(X), it will become obvious that paired isosbestic points will shift closer
+
TABLE I
10-6
5%K
+
+
T H E SHIFT OF ISOSBESTIC P O I N T S BY
Assumed temp. ooeff.
_ _ _ _Qtl)Yt2
K
In each case, the average deviation of h(X) was within the precision of the individual measurements, indicating that h(X) is not in itself a function of 1. Using the curve for h(X) and comparing with h(h,’) from eq. 13, one can predict where the new intersections mill be if curves 2 and 3 in Fig. 4a ytz)/(l yfl) and are multiplied by factors (1 (1 yt3)/(1 yh), respectively. It is clear from Table I that, within experimental accuracy, isosbestic points are not destroyed by such operations, unless y is large enough to separate the curves completely.
Speotral designation
&I 1 1
10-4
2 1 1 2
10-8
1
10-
1 2 1 1 2
N
2 3 3 2 3 3 2 3 3 2 3 3
h(Xc’)
x
io1
0.405
-
-
-
-
.405 .405 .287 ,285 ,283 .270 .261 .250 .775 ,773 ,675
Crossing wave length of curves, A.
6095 6095 6095 6091 6091 6091 6073 6074 6075 6024 6030 6037
Summary.-If an absorbing sample produces several isosbestic points as temperature is varied,
Nov., 1962
SECONDARY REACTIONS IN CONTROLLED POTENTIAL COULOMETRY
these conclusions can be drawn with considerable confidence: 1. Only one species contributes to the absorbance at a given isosbestic point. 2. The equilibrium of this species is not appreciably affected by temperature changes. 3. The only manner in which isosbestic points can be generated is for the absorptivity of one species at a given wave length to be linearly dependent on temperature, a t least to a good approximation.* 4. WIultiplication of a family of absorption spectra generated a t temperatures 11, tz, ts, . . . tn by the corresponding factor (1 &)/(l $I), to a good approximation, only shifts and does not destroy the isosbestic points which have resulted in the original genera tion.
+
+
NOTEADDEDIN PRooF.-Since
the foregoing paper was submitted for publication, Cohen and Fischer have concluded in a DaDer entitled “Isosbestie Points” ( J . Chem. SOC.,
-
(4) It is possible t h a t ei(h) e,O(X)exp h(X)t; in the approximation of h(X)t
