J. Phys. Chem. 1993,97, 3058-3066
3058
Fundamental T h e ~ h r o m i Properties c of Buffered pH Indicator Solutions and the Formulation of "Atbermochrodc" Systems Dean G. Hafeman,' Kimberly L. Crawford, rad Luc J. Bortsse Molecular Devices Corporation, 4700 Bohannon Drive, Menlo Park, California 94025 Received: October 14, 1992; In Final Form. December 9, 1992
A thermodynamic basis is derived and experimentally verified for predicting the temperature-dependent spectral changes in solutions containing a pH indicator and any number of pH-buffering components. Absorbance by the acidic and basic species of the pH indicator is measured at two different wavelengths. A plot of the logarithm of the ratio of absorbances at the two wavelengths vs reciprocal temperature is a straight line with slope that varies with solution composition. The slope is used as a thermochromicity parameter to determine proton dissociation enthalpy of pH indicators and pH buffers and to calibrate solutions in "optical thermometers". The parameter is independent of optical path length, concentration of pH indicator, temperature range, and, within limits, the two wavelengths selected. We also have used the parameter to formulate solutions with temperatureindependent spectral properties. Such 'athermochromic" solutions remove temperature error from optical acidimetric measurements.
1. Introduction
species of the pH indicator, such that
Many moltcules have pH-dependent absorption spectra and are used as pH indicators.IJ These include the sulfonephthaleins such as phenol red, bromocresol purple, and bromothymol blue. Molecules such as the fluoresceins, umbelliferones, pyranines, and benmxanthenes additionally have pHdependent changes in their excitation and emission fluorescence spectra. The pHdependent absorbance or fluorescenceproperties of pH indicators are especially useful for noninvasive monitoring of pH inside ~ and ionic biological cells or their ~ r g a n e l l e s . ~Temperature strength affect the pK, of pH buffers and pH indicators and, therefore, are sourccs of error in acidometry and photometric pH measurement. In the present work, we derive and verify experimentally a thermodynamic basis for the relationship between temperature and a defined thermochromicity parameter. The results are used to formulate Yathermochromicnsystems that have temperature-independent spectral properties, thereby reducing error in acidimetric measurements. Also, the general relationships derived are used to determine proton dissociation enthalpy of pH indicators and pH buffers and to optimize and calibrate solutions for use in 'optical thermometers" which previously have been calibrated empirically.610
A,, = ctlLIH-Ind"] (4) where etl is the extinction coefficient of the acidic species of the pH indicator at wavelength XI, and Lis the light absorption pnth length. Wealsochoose a second wavelength, A*, whereabsorbance is proportional to the concentration of the basic species [Ind"'], and where absorbance is not significantly affected by the acidic species, such that
2. Tbcory 2.1. Definitionof "ochromicity. Consider the reversible dissociation of a hydrogen ion from a pH indicator:
+
H-Ind" c)Ind"' H+ (1) Theequilibrium constant, K a , for thedissociationof the indicator dye is given as Kind= [Ind"'][H+]/[H-Ind"] (2) where the brackets indicate concentration. Using the usual definitions, pH = -log[H+] and pK = -log of the equilibrium constant, eq 2 may be put in the form of the HeridersonHasselbalch equation: pH = p&,d log([Ind"']/[H-Ind"]) (3) Forany light-absorbing pH indicator,wechoosea first wavelength, AI, where absorbance is proportional to the concentration of the acidicspecies, [H-Ind],and not significantlyaffected by the basic 0022-3654/93/2097-3058$04.00/0
A,, = c!2L[Ind"'] where :e is the extinction coefficient of the basic species of the pH indicator at wavelength X2. Dividing eq 5 by eq 4, where L is identical in both equations, and rearranging give
A,, etl --[Ind"'] --[H-Ind"]
AM e*!
Substitution of the right side of this expression into eq 3 and rearranging yield (7)
This expression reveals that when the pK and extinction coefficients of the pH-indicator are constant, a plot of log An/ AM vs pH gives a straight line with unity slope. As shall be shown subsequently, a useful parameter to characterize the thermochromic properties of a system is the slope of log Au/Axl plotted vs reciprocal absolute temperature. Taking thederivative of eq 7 with respect to reciprocal temperatures gives d(lo8 AA*/AAI) d(pH) d(pKind) I d(T1)
d(T')
d(T')
B
+
A
d(lW ~ u / ~ x I ) d(T9
(8) The final term in this expression will be neglected, at least temporarily, becausevisible absorption spectra of smallmolecules, dissolved in polar solvents, at or above room temperature are sufficiently broadened that the extinction coefficients are not strongly dependent on temperature." This is especially true for strongly allowed, purely electronic, transitions of energy much greater than kT. We later show that the extinction coefficient Ca 1993 American Chemical Society
Buffered pH-Indicator Solutions
The Journal of Physical Chemistry, Vol. 97, No. 12, I993 3059
terms are relatively small for the pH indicators used in this work. The complete expressions, including this term, are given in Appendix D. Also, we see from eq A2 in Appendix A that the standard proton dissociation enthalpy for the pH indicator,M o j n d may be substituted for d(pKind)/d(T1)giving
pH-buffering components:
2.3. Annlytical E x p ” for Thermochromicity. Substitution of eq 16 for d(pH)/d( TI)into eq 9gives an analytical expression for the thermochromicity parameter, d(log Axz/Axl)/d( ‘P): Therefore, the thermochromicity parameter, d(1og A A ~ / A ~ ~ ) / ~ (TI), is related to w i n d , and the term d(pH)/d(TI), which is evaluated below.Iz 2.2. Dep”ce of pH on Temperature. The aim of this section Because ZEIB(i)/& = 1, eq 17 may be rearranged to is to calculate d(pH)/d(T’) of a system having Ndifferent pHbuffering components. The general result is applicableto systems containing multiple pH-buffering componentsas well as to systems d ( b AAz/AAi) w(i) - wind where thesolvent or the pH indicator itself contributeappreciable & 2.303R d( TI) i= 1 buffering capacity. The pH of such systems is temperature dependent because the pK, values of the individual pH-buffering which, in a simple system having a pH indicator and one principal components are temperature dependent. Hence we can write pH buffer, becomes
-
For Ndifferent pH-bufferingcomponentsthere will beNdifferent d(pK,(o)/d( TI)terms. Each term, asshown in eq A2 ofAppendix A, is equal to M0(i)/2.303R, where AIP(i) is the standard enthalpy of ionization of an individual pH-buffering component ( i ) . To evaluate the remaining terms in eq 10,Le., the d(pH)/ a(pK,(o) terms for each pH-buffering component present in the system, we consider the N buffering components separately, each consisting of the reversible dissociation of its acidic species, with charge n, to its conjugatebase, with charge ( n - l), plus a hydrogen ion: acid;,,
-
base;;’
+ H+
(11)
Each buffering component contributes its own unique buffering capacity &) to the total buffering capacity PT,which is the sum of all &) terms in the system. When multiple buffering components are considered, the derivative of the HendersonHasselbalch equation, as written in eq B2 of Appendix B, no longer is sufficienttodescribe the pH of theentire system. Instead, the pH change caused by a change in the pKa for each pHbuffering component is found as
The quantity -dX(i)/a(pK,(i)) is the concentration of acid equivalentsreleased from buffering component ( i )per unit change in its pK,. From eq BS of Appendix B we have ax(i)/a(PKo(i)) = -B(i)
(13)
The quantity a(pH)/-dX(o is the change in pH of the entire pH-buffered system upon addition of X concentration of acid equivalents; in this case the source of the acid is component ( i ) following a change in its PK,(~).According to eq B1 of Appendix B a(PH)/ax(i) = -1 /& Substitution of eqs 13 and 14 into eq 12 gives
(14)
a(PH)/a(PKa(i)) = @(i)/& (15) Substituting theright-sideofthisexpressionintoeq 10 for a(pH)/ a(pKO(,))and substituting W(i)/2.303R for d(pK,(i,)/d( T i ) gives a final expression for d(pH)/d( TI)in a system with multiple
k$(
According to eq 18, the thermochromicity parameter for a pH indicator in any mixture of pH-buffering components is equal to the weighted average of thermochromicity parameters of the individual pH-buffering componentswith the same pH indicator. The thermochromicity parameter of each individual pH-buffering component, in turn, is equal to the difference (divided by the constant, 2.303R) in the standard enthalpies for proton dissociation from the pH-buffering component and from the pH indicator, as seen in eq 19. The weighting factor to be used with each buffering component is the fraction of the total buffering capacity, &,)//3~, that each individual buffering-component contributes.
3. Materials and Methods Obtained from Sigma Chemical Co. were HEPES (N-[2hydroxyethyl]piperazine-N’- [2-ethanesulfonicacid]), sodium salt; TRIS (tris[hydroxymethyl]aminomethane), TRIZMA; citrate, free acid monohydrate; MES (2- [N-morpholinolethanesulfonic acid); MOPS (3- [N-morpholino]propanesulfonic acid); PIPES (piperazine-N,”-bis[2-ethanesulfonic acid]), sodium salt; HEPPSO (N[2-hydroxyethyl]piperazine4’-2- hydroxypropanesulfonic acid), sodium salt; imidazole; L-histidine, monohydrochloride monohydrate; phenol red (phenolsulfonephthalein),sodium salt; bromocresol purple (S,S’-dibromo-o-cresoIsulfonephthalein),sodium salt; bromothymol blue (3’”’’-dibromothymolsulfonephthalein, sodium salt; and PBS (phosphate buffered saline), pH 7.4. Obtained from Fisher Scientific were 0.050 M potassium phosphate, monobasic and sodium hydroxide buffer, pH 6.00; 0.050 M potassium phosphate, monobasic and sodium hydroxide buffer, pH 7.00; EDTA ([ethylenedinitrilo]tetraacetic acid), disodium salt; and sodium chloride. The buffer TAPS (3-[Ntris(hydroxymethyl)methylamino]propanesulfonic acid) was obtained from Calbiochem-Behring. The buffer ethylenediamine, 99+%, was obtained from Aldrich Chemical Co. Absorbance of phenol red was monitored at 420 and 560 nm; bromocresol purple was monitored at 420 and either 590 or 600 nm; bromothymol blue was monitored at 420 and 600or 450 and 610nm. Unless otherwise noted, all solutions contained 50 mM pH-buffering components and 50 MM pH indicators. The solutions were prepared, adjusted to the desired pH, and degassed at about 40 OC with a water aspirator for 10 min to remove dissolved C02. For the data shown in Figures 1,2, and 4, optical density was measured with a Hewlett-Packard Model 8451A diode array spectrophotometer equipped with a water-jacketed cuvette holder and a 4.0-mL cuvette with a magnetic stir bar.
Hafeman et al.
3060 The Journal of Physical Chemistry, Vol. 97, No. 12. 1993 Temperature was measured with an Omega Model 5831-1 temperature meter equipped with a thermistor probe calibrated to an NBS-traceable standard to better than 0.1 OC. The thermistor probe was placed in the stirred cuvette, out of the light path, the cuvette cover placed on the cuvette and sealed with labeling tape. In other experiments, shown in Figure 3 and Table I, optical density was measured with a Thermomax microplate absorbance reader equipped with 10-nm bandpass interference filters with center wavelengths at 420,560,590,600,650, or 750 nm. Two-hundred microliters of each solution was placed in wells (from 6 to 10 replicate wells per solution) of a covered 96-well microplate (Nunc, Nunclon flat-bottom). The covered microplate was equilibrated in the temperature-controlled chamber at each temperature for at least 25 min. The cover then was removed and the optical density quickly determined at two or more wavelengths. Altematively, themicroplate cover was coated with an antifogging agent (Molecular Devices Corp.) and was left in place during the entire experiment. To correct for small amounts of light-scattering, mainly by the microplate cover, the optical density at 650 nm (for phenol red) and at 750 nm (for bromocresol purple and bromothymol blue) was subtracted from all optical density measurements to obtain the corrected absorbance values. The temperature of the solutions was taken to be the air temperature of the microplate reader as indicated by the instrument. In some cases, small temperature corrections were made by using an ethylenediamine-bromocresol purple solution and the thermochromicity curue shown in Figure 3 to determine the actual temperature of liquids in reference wells in the microplate.
Temperature (‘C)
35
30
20
25
0.15 1
15
r
0.12 .0.09
--
0.08
--
0.03 -:
3.20
3.25
3.30
3.35
iflemperature (x
3.40 lo3
0 3.50
3.45
K)
Figure 1. Plot of log A590/A420 vs T I for bromocresol purple in 50 mM sodium acetate buffer, pH 5.80 at 25 “ C (lower curve). The upper curve shows the results after addition of 0.100 M NaCl. In each case, ( 0 )
indicates the data taken while increasing temperature and (H) indicates the data taken while decreasing temperature. Temperature (‘C)
35 -0.95
’
30
25
I
20
15
,
I I
I
-0.95
b
f-l
4. ExperimentalResulta
4.1. Determination of AHO’ Valued3 for Dissocition of H+ from p H Indicators. First, we examined spectral overlap of the acidic and basic speciesof the pH indicators. For the basic species of phenol red, f420/6560 = 0.020. For the basic species of bromothymol blue, t420/e,500 = 0.152 and €450/€610 = 0.052. In experiments with these pH indicators, theappropriate corrections were made to the absorbancevalues of the acidic species to remove the contribution due to the basic species. No corrections were made for bromocresol purple because the basic species of the pH indicator contributed less than 1%to the total A42o. For all three pH indicators, the acidic species contributed less than 1%to the total absorbance at the wavelengths used to monitor the basic species, therefore no corrections were made. (See also Appendix D for a general method of correcting for spectral overlap.) Next weexaminedthetemperaturedependenceoftheextinction coefficients of the pH indicators. When plotted vs reciprocal temperature, log ~560/€420 for phenol red, log €590/€420 for bromocresol purple, and log t610/6450 for bromothymol blue had slopes, i.e., d(log eF2/etl)/d(TI) values, of 68, 48, and -25 K, respectively. This was determined by measuring the absorbances of the pH indicators, as a function of temperature, in 0.05 M sodium carbonate buffer at pH 10.1 or in 0.05 M sodium citrate buffer, pH 3.6. At 25 OC, these values correspond to d(1og e!2/s:l)/dT values of -0.O00 74, -0.00053, and O.OO0 27 per OC. Although these values are small, we chose to correct for them, as shown in Appendix D, in order to determine ionization enthalpy values for the pH indicators and pH buffers. As shown in eq 19, the thermochromicityparameter of a simple system having a pH indicator and one principal pH buffer is q u a l to the difference (divided by the constant 2.303R) in standard ionization enthalpies of the pH-buffering component and the pH indicator. Figures 1 and 2 are plots of experimental data that we obtained with the pH indicators bromocresol purple (43 r M ) or phenol red (60 pM), respectively, each in 50 mM sodium acetate buffer. Acetate was chosen as the principal pHbuffering componentbecause it has an apparent standard enthalpy of ionization, AH’”,under the present experimental conditions,
-1.25 3.20
3.25
3.30
3.35
3.40
3.45
-1.25 3.50
lflempershrre (x 105 K)
Figure 2. Plot of log A J ~ / A vs ~ ~T Io for phenol red in 50 mM sodium acetate buffer, pH 6.23 at 25 OC (lower curve). The upper curve shows the results after addition of 0.100 M NaCl. In each case ( 0 )indicates the data taken while increasing temperature and (H) indicates the data taken while decreasing temperature.
near zero.14-16 Thus, the slopes of the plots, after subtracting the extinction coefficient component, should be q u a l to W ’for dissociation of the pH indicator divided by 2.303R. The W ’ for dissociation of bromocresol purple, calculated from the slope of the plot in Figure 1, was 1.49 kcal/mol and with addition of 100 mM NaCl was 1.53 kcal/mol. At 25 OC, the W ’ v a l u e s determined correspond to d(pK’,)/dTvalues of -3.7 X 10-3 and -3.8 X l t 3 per OC, respectively. From the plot shown in Figure 2, the AHo’ for dissociation of phenol red was similarly determined to be 2.96 kcal/mol and with addition of 100mM NaCl was 3.07 kcal/mol. At 25 OC, the Lwo’ values correspond to d(pK’,)/dT valuesof-7.3 X 10-3and-7.5 X l e 3per OC, respectively. These values, and similarly determined values for 62 r M bromothymol blue in 50 mM sodium acetate buffer, are summarized in Table I. 4.2. Determinationof Tbermochromicity,Change in pH with Temperature, rad Apparent Ionization Eathrlpy of p H Buffers. Table I1summarizes thermwhromicity measurementsof solutions containingpH buffers with either phenol red, bromocresol purple, or bromothymol blue. The first column under each pH indicator gives the spectral-overlagcorrected thermochromicity of each system as d(log AA2/AAl)/d( TI).The data obtained in section 4.1, above, were used to make this correction. The second column under each pH indicator gives the thermochromicity of each
The Journal of Physical Chemistry, Vol. 97, No. 12, 1993 3061
Buffered pH-Indicator Solutions
TABLE I: Ionization Enthalpy and Temperature Coefiicients of DH Indicators AH-‘ temp coeff d(pK.’)/ DHindicator@ (kcal/mol) dTat 25 OC (OC-I) ~~
~
bromocresol purple +0.10 M NaCl phenol red +0.10 M NaCl bromothymol blue
~~
1.49 1.53 2.96 3.07 1.16
-0.0037 -0.0038 -0.0073 -0.0075 -0,0028
a Measurements were made in 0.050 M potassium acetate (0.05 ionic strength).
system, at 25 OC, as d(1og A~z/Axl)/dT. These values were obtained by multiplying the corresponding values in the first column by -1 / Tz for 298 K. The third column under each pH indicator gives the resulting apparent enthalpy of ionization of each buffering component, w ’ ( b u f f c r ) , calculated from eq 19 employing the m o ’ i n d values for the pH indicators shown in Table I and employing a correction (as shown in Appendix D) for the temperature dependence of the extinction coefficients, determined in section 4.1 above. Reported in the fourth column under each pH-indicator is the corresponding d(pH)/dT value for each pH buffer at 25 OC. The final column in Table I1 gives for each pH buffer an average of d(pH)/dT values, at 25 OC, determined with the different pH indicators. 4.3. Properties of Tbermochromicity Parameters. The thermochromicity parameter d(1og &/A~l)/d( TI)is expected to be independent of the wavelengths chosen for A1 and A2 provided that XI is chosen such that absorbance is proportional to the concentration of the acidic pH-indicator species, and independent of the basic species, and Xz is chosen such that absorbance is proportional to the concentrationof the basic pH-indicator species and independent of the acidic pH-indicator species. To verify this experimentally, we chose bromocresol purple as the pH indicator because this indicator had the least spectral overlap of the basic and acidic species. Figure 3 shows a plot of log (&/ Ax,) vs TI, employing either 590 or 600 nm for 1 2 , with XI held constant at 420 nm. The results are shown for two different pH-buffering systems, one having a very large thermochromicity, the other having nearly zero thermochromicity. As shown in Figure 3, thechangein Xz had a rather large effect on theintercepts of the plots of log (&/Ax]) vs T I but, as predicted by the theory, had no significant effect on thermochromicity, Le., the slopes of the plots. As may be deduced from eq 7,the intercept offset is expected to be equal to log t5m/c- for both pH-buffering systems. To the extent that extinctioncoefficientsfor either the basic species or the acidic species of the pH indicator are independent of the pH buffer used and independent of pH, the intercept offset is expected to be the same for both buffering systems. As can be seen from Figure 3, this is experimentally the case. The same change in h2, with either buffering system, caused a Y-intercept offset of about 0.120 pH units. 4.4. Systems Having More Than One Significant pH Buffer. Figure 4 shows the effect on the thermochromicity parameter, of combining two pH-buffering components, PIPES and phosphate in a solution with the pH indicator bromocresol purple. Plotted is the thermochromicity parameter, d(1og A ~ / A 4 2 0 ) / d( TI), vs the fraction of total buffer capacity contributed by PIPES. The sum of both buffer concentrations is 50 mM; bromocresol purple is 128uM, pH 6.0. Because the pKa values of phosphate and PIPES are nearly the same at 0.05 ionic strength, at all pH values where the buffering capacity of H20 can be neglected the fraction of the buffer capacity contributed by PIPES isthesameasthemolefractionofPIPES,i.e.,[PIPES]/([PIPES] [phosphate]). The figure shows that the bromocresol purple solution is athermochromic at 0.22 mole fraction (1 1mM) PIPES and 0.78 mol fraction (39 mM) phosphate. According to q 18, the thermochromicity parameter of any complex mixture of multiple pH-buffering components is equal to the weighted
+
average of thermochromicity parameters of the individual pHbufferingcomponents with thesame pH indicator. The weighting factor to be used for each individual buffering component in calculatingthe averageis B(i)/BT, the fractional buffering capacity that each individual buffering component contributes to the total buffering capacity of the system. The theoretical result is shown by the line to which the experimental data fit closely. In cases where the pKavalues of the buffering componentsare not identical, B(0l6.rwill vary with pH. Given in Appendix E is a step-by-step example demonstrating how to use the thermochromicity data of Table I1 to formulate an athermochromic system. 5. Discussion 5.1. Tbermochromicity. We have derived a simple thermodynamic basis for prediction of temperature-dependent spectral changes in any system containing a pH indicator and any number of pH-buffering components. Although only chromophoric pH indicators were used in the examples, the principle should be valid for any measured spectral property, including fluorescence, phosphorescence, and luminescence. At least two different wavelengths must be employed in the optical measurements. A first wavelength is used to monitor the acidic species and a second wavelength is used to monitor the basic species of the pH indicator. As shown in eq 18 or eq 19, the thermochromicity parameter d(logAu/Axl)/d( P )is related to the difference in the enthalpies of ionization of the pH indicator the pH-buffering components and has units of temperature (e.g., kelvin). This parameter is the same at any temperature and also is independent of both the light-absorptionpath length and theconcentrationof pH indicator. The parameter also is independent of the wavelengths chosen for XI and X2, provided that there is no significant spectral overlap, Le.. the measured quantity at XI is not affected by the basic species of the pH indicator and the measured quantity at X2 is not affected by the acidic species of the pH indicator. This relationshipremains true in the presence of spectral overlap, provided that a spectral overlap correctionis first applied to the data as shown in Appendix D. Thealternative thennochromicityparameter, d(log A ~ / A A ~ ) / dT, is related to the temperature-independent parameter by a factor of -1/Tz, as shown in Appendix A. The two thermochromicity parameters have similar properties except that the latter varies with temperature. The temperature-dependent parameter also has a simple physical meaning as it is equal to the difference in temperature dependencies of the pKa values of the pH-buffering component(s) and the pH indicator, as shown by eq A5 in Appendix A. As such it has units of ApKa/OC, or equivalently, Celsius-’ or kelvin-’. 5.2. Athennochromic Formulations. The expressionsderived for thermochromicity apply to any simple pH-buffered system, containing a pH indicator and one principal pH-buffering component, and for higher-order systems containing a pH indicator with any number of pH-buffering components. The higher-order systems may be formulated to have any desired thermochromic parameter within a wide range, including having zero temperature dependence. Figure 4 shows that an athermochromic mixture of phosphate and PIPES, with bromocresol purple, is 0.22 m d fraction PIPES and 0.78 mol fraction phosphate. A system will be athermochromic whenever the general expression given by eq 18sums to zero. Taking the systems shown in Table I1 as examples, any pH buffer with a negative thermochromicity, e.g. citrate, phosphate, or PIPES, with the pH indicator phenol red, when mixed with a buffer with positive thermochromicity, i.e., all the other buffers shown, will produce an atherochromic system when the buffers are combined in the correct proportions according to their individual, buffer-capacityweighted, thermochromicities. Given in Appendix E is a step by-step exampledemonstratinghow to formulateathennochromic systems. Furthermore, mixtures of three, or more, pH buffers may be used to produce thermochromic systems. At least one
3062 The Journal of Physical Chemistry, Vol. 97, No. 12, 1993 TABLE II: Tbermochromidty, Change in pH with Temperature, md Appvcar A H * V d w s of pH Buffan' (R phenol red
bromocresol purple
gfi
pH bufferh citrate
pH 6.4
+1.0 M NaCl
-752 -702 -433
PIPES EDTA
MOPS MES HEPES HEPPSO histidine imidazole TAPS
ethylenediamine TRIS +1.0 M NaCl
6.0-7.3" 6.M.2 6.0-7.3* 6.0-7.3r 7.2-7.4 7.4 6.1-6.Y 6.1-6.V 8.4 5.9-7.2r 6.9-8.4'
21
zi! 28
thermochromicity dRl w' dTat buffer 25°C (kcall (K) ("C I) mol)
d(pH)/ dTat
it4)
avgd
dpH)/ 6Tat
it4)
25OC ("C I)
d pH)/ 6Tat 25°C ("C 1)
25°C ("C 1 )
0.0085 0.0079 0.0049
-0.80
-0.46 0.66
0.0020 0.0014 -0.0016
0.0058
0.40
-0.0007
-0.0007
-645
0.0073
-0.20
0.0008
O.Ooo8
-54 72 91 416 380 550
O.ooo6 -0.0008 -0.0010 -0.0047 -0,0043 -0.0062
2.39 2.97 3.06 4.55 4.38 5.16
-0.0059 -0.0073 -0.0075 -0.0112
phosphate 5.9-7.1' phosphate 0.01 M 7.3-7.4 +0.1 I M NaCl + -513 0.003 M KCI 1 . 1 1 MNaCI+ 0.003 M KCI
log A,JA,,)
bromothymol blue
thermochromicity thcrmochromicity dRl AHO' d(pH)/ dR/ AHo' dTat buffer dTat dTat buffer 25°C (kcall 25°C 25°C (kcall (K) ("C I) mol) ("C I) (K) ("C I) mol)
iF()
-
Hafeman et al.
1295 1302 1494 1508
-0.0146 -0.0146 -0.0168 -0.0170
8.57 8.60 9.48 9.65
-78
O.ooo9
0.91
0.0020 0.0014 -0.00 19
-0.0022
246
-0.0028
2.39
-0.0059
490 639
-0.0055 -0.0072
3.51 4.20
-0,0086 -0.0103
1144 1505
-0.0129 -0.0169
6.50 8.16
-0.0160 -0.0201
1276 1728
-0.0194
7.11 9.18
1771
-0.0199
9.38
-0.0230
1968 2064
-0.0221 -0.0232
10.28 -0.0253 10.72 -0.0264
-0.0108
-0.0127 -0.0211 -0.0211 -0,0233 -0.0235
-0.0144
-0.0175 -0.0226
-0.0059 -0,0073 -0.008 I -0.0107 -0.0108 -0.0127 -0.0167 -0.0213 -0.021 1 -0.0232 -0,0248 -0.0235
The apparent standard enthalpy, AH'",values for ionization of the pH buffers were calculated from the measured thermochromicity values and the most appropriate AH"' values for the pH indicators shown in Table I. In the presence of atmospheric C 0 2 the apparent W ' values for the pH buffers vary significantly above pH 6.4 (see Discussion). All buffers are 0.050 M unless otherwise indicated. e The symbol R denotes log A A ~ / A ~ ~ , The average values for d(pH)/dTat 25 'C are the mean of the values determined with each pH indicator employed. Higher pH values were employed with phenol red as a pH indicator because of its higher pKa. /Slightly higher pH values were employed with bromothymol blue as a pH indicator because its pKa is slightly higher than that of bromocresol purple. Tenperahre (C)
2501. . .
:
. .
.
:
.
:
.
. .
.
:
-
.
.
250
200 L
0.4
-.
-- 0.4
0.3
-.
-- 0.3
0.2.;
.-0.2
150
1
100
a
8
50
0.1
-.
-- 0.1
0
-.
-- 0
0 -50
-100
4 0
' '
'
; 0.2
'
'
'
:. . . 0.4
; .
. . : . .
0.6
0.8
'
t -100 1.0
B P m
BPIm+BmOsk.
Figure 4. Plot of the thermochromicity parameter, d(log Am/A420)/ d( T i ) vs , fraction of total buffer capacity contributed by PIPES for mixtures of PIPES and phosphate buffers. The sum of both buffer concentrations is constant at 50 mM, pH 6.0 at 25 "C. Bromocresol purple is 128 pM. Because the pK: values of PIPES and phosphate are nearly the same, the fraction of total buffer capacity contributed by PIPES is the same as the mole fraction PIPES in this case when the buffering capacity of water and the pH indicator may be ignored. The figure shows that the bromocresol purple solution is athermochromic at 0.22 mole fraction ( 1 1 mM) PIPES at 0.78 mole fraction (39 mM) phosphate.
The d(pK',)/dTvalues at 25 OC determined by this method for phenol red and bromocresol purple were 4.0073 and -0.0037 per OC,respectively. Thesevalues areclose to theolder literature values of -0.007 and -0.005 per OC,respectively.lJ7Je In the primary literature these pK6 values were determined by visual color comparison of the pH indicators, which the authors state, is more difficult for bromocresol purple than for phenol r d 1 8 Because of the experimental difficulties in the older visual methodology, the values shown in Table I may be more accurate. With the same reasoning, our determined d(pK',)/dT value at 25 OC for bromothymol blue of -0.0028 per OC may be more accurate than the older literature value of ~ e r o Our . ~ results ~ ~ ~
Buffered pH-Indicator Solutions rely heavily on the AHO' values reported in the literature for sodium acetate, which were determined with relatively modern electrochemicaltechniques in evaluating sodium acetate as a pH standard.14 The most likely source of error in our measurements comes from C02 present in the air. We did not take any precautions to exclude atmospheric COZfrom contact with the samples. The error, however, should not have been appreciable in the acetate buffers when the pH was not substantially greater than the 6.35 pKa of carbonic acid and when the samples were heated to 40 "C before the measurements in order to drive off
c02. Table I1 shows thermochromicity of various pH buffers with three different pH indicators. The data may be employed to formulate athermochromic systems. The data also allow direct calculation of d(pH)/dT under the experimental conditions employed. The d(pH)/dTvalues will be identical to d(pK',)/dT values for the pH buffers when the effect of atmospheric COZ is negligible. The most reliable literature data for comparison have come from the careful examination of acetate and phosphate, as pH s t a n d a r d ~ . ~ From ~ * ~ sthese data the d(pK',)/dTvalue, at 25 "C, for 0.050 M acetate is expected to be zero and for 0.050 M phosphate q u a l to-0.0028 per "C. We have used acetate as our standard and, as shown in Table 11, 0.050 M phosphate gave a d(pK',)/dT value of -0.0022 per "C with the pH indicator bromocresol purple and -0.0016 with the pH indicator phenol red. The greater deviation of d(pK:',)/dTvalues determined with phenol red compared to the literature values likely is due to a greater effect of atmospheric COZin the determinations done with phenol red. Because of the higher pK', of phenol red, experiments with this indicator were run at pH values higher than 7.0 where atmospheric C02 is expected to have a greater effect. Therefore,for precisedeterminationof pH buffer d(pKi)/ d T or enthalpy values above pH 7.0, C 0 2 should be rigorously excluded. The enthalpy values given in Table 11, therefore, are tobetaken as apparent valuesonly. Thed(pH)/dTvalues shown inTable 11, however, areactual, under the experimental conditions employed. Theobservedd(pH)/dTvaluesaresimilar tod(pK',)/ dTvalues found in manysources.16,2k26(Thesesources,however, commonly fail to disclose the experimental temperature, which is important because this value is proportional to 1/Tz.) The thermochromicity method, if carried out with exclusion of COZ would appear to offer a method of acceptable analytical accuracy todetermineionizationenthalpiesof pH buffers and pH indicators. 5.4. Effects of Ionic Strength on "hermochromicity. Thermochromicity may be remarkably independent of ionic strength if the activity coefficients of the pH-buffering component and the pH indicator respond similarly to ionic strength changes. This effect is shown in Figures 1 and 2 and Table I where an increase in ionic strength from approximately 0.05 to 0.15 in systems containing sodium acetate and either bromocresolpurple or phenol red as pH indicators caused a rather insignificantchange in the thermochromicity parameter. Calculations from the thermodynamic data of MacInnes et al.14 show that AH"' for sodium acetate is zero at 0.020 M and is 0.23 kcal/mol at 0.200 M (equivalent to 6 X 10-4 ApK,/"C at 25 "C). Thus the AH"' for sodium acetate also is relatively insensitiveto changes in ionic strength. As shown in Figures 1 and 2, log Akz/A~iincreased by 0.034.05 (i.e., the pH increased by this amount) upon addition of 0.1 M NaCl. This indicates that the pK, of both indicators decreases slightly more than that of acetate upon an increase in ionic strength. In general, the effects of ionic strength on the pK, values of the sulfonephthaleins,such as phenol red and bromocresol purple, are only slightly greater than for small singly-charged ions.l*~4J7J9Although the charge ( n ) on the acidic species of acetate is 0 and is -1 for the pH indicators, the larger ionic size of the pH indicators apparently makes them respond similarly to changes in ionic strength.lJs17J9 In summary, the effects of ionic strength on thermochromicity are small if the effects of
The Journal of Physical Chemistry, Vol. 97, NO. 12, 1993 3063 ionic strength on both the pKa of the pH buffer and the pKa of the pH indicator are small. The effect of thermochromicity is even smaller if these two effects cancel. Although we did not do so in the present work, d(pKa)/dT or AHO values at unity activity coefficients may be obtained for either pH buffers or pH indicators provided that AHO' is known as a function of ionic strength for the other member of the pair. The method consistsof measuring the thermochromicity at several low values of ionic strength, and using the known values of AHO' of one member of the pH buffer/indicator pair to obtain the AH"' values of the other as a function of ionic strength. These last values then can be extrapolated to zero ionic strength. 6. Appendix A Temperature Dependence of Equilibrium coastnnts
The temperature dependence of equilibrium constant, K,of a chemical equilibrium, such as that shown in eq 2, is determined from the fundamental thermodynamic relationships AGO = -RT In K, and AGO = AH" - TAS" giving the standard expression
AH"
AS"
-log K , =--2.303RT 2.303R where AHO and AS" are the dissociation enthalpy and entropy, respectively, under standard conditions, T is the absolute temperature, and R is the gas law constant. The usual way of obtaining AH" is by measuring K,, as a function of T. With the approximation that AHO and AS" are independentof temperature, a plot of log K, versus 1/ T yields a straight line with a slope of -AH0/2.303R (see, for example, refs 20 or 21). Substituting pKa for -log K , in eq A1 and placing the result in derivative form gives d(pK,)/d(T1) = AH0/2.303R (A21 Alternatively, this expression may be put into its temperaturedependent form, by multiplying by d(T-I)/dT = -1/T2, that is
The same transformation may be applied to the thermochromicity to convert it into the temparameter, d(1og AA2/AAl)/d(TI), peraturedependent formd(logAAz/Axl)/dT. Thuseq 19becomes d(log AA2/AAl) = - - 1 dT l-2
wbuffer
- Moind 2.303R
Substitution of the left-hand side of eq A3 into eq A4 gives
-
d(log AAZ/AAl) d(pKbuffer) --d(pKind) dT dT dT Thus, the expression d(1og AAZ/AAl)/dT is related to the differences between d(pKa)/dT values of the pH indicator and the pH-bufferingcomponent(s). As shown by eq A3, d(pK,)/dT is not independent of temperature but instead varies inversely with the square of the absolute temperature. Thus, d(1og A A ~ / Axl)/dTalsovaries with the square of the absolute temperature. Similarly eq 18, describing the thermochromicity of a system containing a pH indicator and a mixture of pH-buffering components, may be written as
The parameter d(1og AAz/AAl)/dT for a pH indicator and any complex mixture of pH-buffering components, according to eq A6, is equal to the weighted average of the d(1og AAz/Axl)/dT parameters of the individual pH-buffering components and the pH indicator (eq AS). The individual weighting factor to be
3064 The Journal of Physical Chemistry, Vol. 97, No. 12, 199'3 used with each buffering component is the fraction of the total buffering capacity, &f)/&, that each individual buffering component contributes to the total buffering capacity of the system. Characteristic of all d(pKa)/d T values, this expression varies inversely with the square of absolute temperature. The above expressions assume unity activity coefficients. Apparent pK, values, denoted pK
