CALORIMETRIC METHODFOR
THE
DETERMINATION OF AG, AH, AND AS
2003
Entropy Titration. A Calorimetric Method for the Determination of AG, AH,and ASfrom a Single Thermometric Titration’alb
by J. J. Christensen, R. M. Izatt, L. D. Hansen,lo and J. A. Partridge Departments of Chemical Engineering and Chemistry, Brigham Young University, Promo, Utah (Received January 14, 1966)
A calorimetric method (entropy titration) is described for the determination of AGO, AH’, and ASo from one thermometric titration. This method has been tested by determination of the pK values for proton ionization from HSO4- and HP0d2-. The results are in good agreement with literature pK data for these systems. A new and more exact method of thermometric titration data analysis is described.
Introduction Data from two different experiments are usually needed to calculate the free energy (AG), enthalpy (AH), and entropy (AS) changes associated with a chemical reaction. For example, AG may be calculated from equilibrium constant measurements, and AH either from calorimetric data or from the Gibbs-Helmholtz relationship which requires the measurement of AG as a function of temperature. A A S value can then be calculated from the AG and AH values. However, for certain classes of reactions, AG, AH, and A S values can be determined from calorimetric data alone. This determination has been attempted by several Most of the results cannot be checked because the reactions studied did not have well-known equilibrium constants. None of the determinations was done by the method of continuous thermometric titration calorimetry used to obtain the data in this paper. Thermometric titration calorimetry is particularly well suited for the determination of AG, AH, and A S from a single titration (entropy titration) since, in a single run, one obtains the equivalent of a large number of determinations by conventional calorimetry. The entropy titration method depends on calculation of the amount of reaction from the quantity of heat evolved. Its successful application to a given system depends on: (a) the equilibrium constant and the reaction conditions being such that a measurable, but not quantitative, amount of reaction takes place, and (b) the AH value for the reaction being measurably different from
zero. This means for the calorimetric equipment used in this study that log K should be between 0.5 and 3.0 and the heat due to the reaction under consideration should be greater than approximately 1 cal. The entropy titration method should be of general usefulness in studying weak complexes12 (ie., C1-, Br-, I-, and Sod2-complexes of type “A” metal ions), acids with pK values greater than 11.5 or less than 2.5, and reactions in nonaqueous or mixed solvents. I n
(1) (a) Supported by National Institutes of Health Grant RG 9430-04, and Atomic Energy Commission Contract AT-04-299. Presented at the 19th Calorimetry Conference, Washington, D. C., Oct 1964. (b) Published as a preliminary communication in Chem. Commun. (London), No. 3, 36 (1965). (c) Taken in part from the Ph.D. Dissertation of L. D. Hansen, Brigham Young University, 1965. (2) C. A. Seits, unpublished Master’s Thesis, Washington State University, 1965. (3) P. Papoff, G. Torsi, and P. G. Zambonin, Gaz. Chim. Ital., 95, 1031 (1965). (4) P. Paoletti, A. Vacca, and D. Arenare, J . Phys. Chem., 70, 193 (1966). (5) M. Bjorkman and L. G. SillBn, Trans. Roy. Inst. Technol. Stock holm, No. 199 (1963). (6) R. e n e k , Royal Institute of Technology, Stockholm, private communication. (7) M. H. Dilke and D. D. Eley, J. Chem. SOC.,2601 (1949). (8) F. J. Cioffi and S. T. Zenchelsky, J . Phys. Chem., 67, 357 (1963). (9) T. H. Benzinger and R. Hems, Proc. Natl. Acad. Sci. U.S.,42, 896 (1956). (10) F. Becker, N. G. Schmahl, and H. D. Pflug, Z . Physik. Chen. (Frankfurt), 39, 33, 52 (1963). (11) T. F. Bolles and R. S. Drago, J. Am. Chem. SOC.,87, 5015 (1965). (12) R. A. Matheson, J . Phya. Chem., 69, 1537 (1965).
Volume 70, Number 6 June 1966
2004
J. J. CHRISTENSEN, R. M. IZATT, L. D. HANSEN,AND J. A. PARTRIDGE
these cases existing methods12J3are inaccurate and tedious, or the resulting data are of questionable meaning. The reactions of H + with SO4+ 14,16 and of HP042with OH- l6 were chosen to test the validity of the entropy titration method. The reasons for these choices were that (1) the reactions had pK values of the correct magnitude which appeared to be well known, (2) the reactions appeared to be simple and well understood, (3) both reactions were known to have fairly large AH values, and (4) one reaction takes place in the acid region while the other occurs in the basic region. I n the present study, AGO, AH", and ASo values calculated from entropy titration data for the reaction of H + with S042- and of OH- with HP0d2- are presented and compared with literature values. Equations are presented relating the systems studied to the various regions of the thermometric titration charts and a new, more precise method of chart analysis is described.
Experimental Section Materials. Reagent grade Na2SO4.10H20(Baker and Adamson), (n-C3H7)4NOH(Eastman), HzS04 (Baker and Adamson), HC1O4 (Baker and Adamson), HC1 (Du Pont), NaOH (Baker Analyzed), and Na2HP04 (National Bureau of Standards Sample No. 186-11-b) were used to prepare the solutions for this study. These solutions were standardized by conventional methods. The solutions used in this study were prepared, stored, and used under a pure nitrogen atmosphere. Calorimetric Equipment. The thermometric titration calorimeter used in this study together with its calibration and operation has been de~cribed.'~~'~ Procedure. Solutions of Na2S04, [ (n-C3H7)4N]2S04, and Na2HPO4 were titrated with HC1O4, HCl, and NaOH, respectively. The heat of dilution data for the titrants were taken from Swansonlg (HC1O4) and Harned and Owen2O (HC1 and NaOH). The AHo and pK values for the ionization of H2P04- and the AHo value for the ionization of HzO, which were needed in the calculations for the HP042- system, were taken to ~~~4 be O.90,z1 7.20,22and 13.34 k ~ a l / m o l e , ~respectively. The latter two values were corrected to the ionic strength, p, a t each experimental point. Data were not available for a similar correction in the case of the AHo value for HzP04- ionization. Theory of Entropy Titration. Consider the simple reaction of reactant A with titrant B to form product AB, Le., A B = AB, for which K is less than lo3and AH # 0. The thermogram obtained for this reaction can be described by eq 1-4. (For definition of the sym-
+
The Journal of Physical Chemistry
bols used, but not defined in the text, see section at end of article.) &b
K =
= AH[AB]bvb
[AB]bYABb/[AlbYAb[BlbYBb
(1)
=
[ABlb/[Alb[Blbrb (2) [ABIb
(3)
[BtotalIb = [Bib -I- [ABIb
(4)
[AtotalIb = [AIb
Combination of (l), (2), (3), and (4) results in ( 5 ) , which, if one assumes that r b is known, contains only two unknowns, AH and K.
AH
_ K
(vb [Bt~tal ]b [Atotal]brb/&b) (AH) ([BtotalIb
-
+ [AtotalIb)rbAH +
rb&b/vb
(5)
The general form of eq 5 is shown by eq 6
AH - = D(AH)z K
+ EAH + F
where D, E , and F are the appropriate constants from eq 5. Since K is a thermodynamic equilibrium constant, the quantity AH/K has the same value at any point on the titration curve provided that AH is independent of p over the p range in the titration. Therefore, eq 6 can be solved for AH by combining any two points on the curve giving eq 7
(Da- Dl)(AH)'
+ (Ez - E1)AH + (Fz - Fi)
=
0 (7)
(13) E. J. King, "Acid-Base Equilibria," The Macmillan Co., New York, N. Y., 1965, p 53. (14) H. 9. Dunsmore and G. H. Nancollas, J . Phys. Chem., 68, 1579 (1964). (15) T. F. Young and D. E. Irish, Ann. Rev. Phys. Chem., 13, 435 (1962). (16) C. E. Vanderzee and A. S. Quist, J . Phys. Chem., 6 5 , 118 (1961). (17) J. J. Christensen, R. M. Izatt, and L. D. Hansen, Rev. Sci. Instr., 36, 779 (1965). (18) L. D. Hansen, Ph.D. Dissertation, Brigham Young University, 1965. (19) J. A. Swanson, Ph.D. Dissertation, University of Nebraska, 1962; cf. Dissertation Abstr., 2 3 , 92 (1962). (20) H. 9.Harned and B. B. Owen, "Physical Chemistry of Electrolytic Solutions," 3rd ed, Reinhold Publishing Corp., New York, N. Y., 1958, pp 707-708. (21) J. J. Christensen and R. M. Izatt, J . Phys. Chem., 66, 1030 (1962). (22) R. G. Bates and S. F. Acree, J . Rea, Natl. Bur. Std., 34, 378 (1945). (23) J. D. Hale, R. M. Izatt, and J. J. Christensen, J. Phys. Chem., 67, 2605 (1963). (24) C. E. Vanderzee and J. A. Swanson, ibid., 67, 2608 (1963).
CALORIMETRIC METHOD FOR
THE
DETERMINATION OF AG, AH, AND A S
A value of K (hence A G O ) can now be calculated from (6) and AS" obtained from AS" = (AH"- AG")/T. If AH is dependent on p, a series of runs must be made and a plot of AH vs. p extrapolated to p = 0 to obtain AH". However, in this study AH was taken to be independent of p and equal to AH" because of the low p values ( p < 0.05) used. I n those cases where p changes during a run, rb and the heat of dilution values depend on p and must be calculated for each point taken during the run. Calculation of p requires a knowledge of the equilibrium constant which is obtained together with AH by a successive approximations method. For the first calculation a value of p is estimated and r b and heat of dilution values are obtained from it. Values for Q, AH, and K are then calculated and K is used to obtain a better value of p . This iteration process is continued until repeated calculations of AH and K values agree to 0.1%. The reaction of HP042-with OH- requires modification of the preceding general treatment because of the existence of significant concentrations of HzPOd-, OH-, and Po43-a t the initial point in addition to HP04'-. The thermogram obtained from the titration of a Na2HPO4solution with a NaOH solution is described by eq 8-1 1. Qb
= A H I ( [ P O ~ ~ - - ]-~ V [P043-]iVi) ~
AHz( [HzPOd-IiVi - [H~pO4-]bVb) (8) rbK = [Po43-]b/ [HP042-]b[OH-]b [PO4total]b
+ [HP0i2-]b + [H2P04-]b [OH-Ib + [P(h3-]b -
(9)
[po43-]b
[OHadded]b=
[H2P04-]b
-
(10)
[H+]b (11)
+
where AH1 and K refer to the reaction OHHP042= Po43HzO, AHz is for the reaction OHH2P04- = HP0d2HzO, and r b = ( Y H P O , Z - ~ O H - /
+
+
+
+iP043-)be
Equation 15 can be obtained by solving simultaneously eq 8-11 and making appropriate simplifications using eq 12-14. I b
Lh
=
= [POltotalIb - [HzP04-]b
+ [ H Z P O ~ -+] ~ [H+Ib
Jb
= [OH-addedIb
Qb
- AHz([HzP04-]iVi -
(12)
(13)
[HzP04-]bVb) =
AH1( [Po.j3-]bVb - [P04*-]iVi) (14) V b -K- - -(Lb
AH1
+ AH1[P043-]iVi)/(AHi21bJbVb2-
r b (Ib
+ Jb)(LbVbml + [P043-]iVbViAH12)+ (Lb
[P043-]iViAH1)2f (15)
2005
Defining Mb and Nb in eq. 16 and 17 kfb =
JbIbVb2
-
[Po43-]iViVb(Jb
+ + Ib)
[P043-]i2Vi2 (16) Nb = 2Lb[Po43-]iVi - LbVb(Jb f
Ib)
(17)
one obtains eq 18, which is a simplified, general form of eq 15. K AH1
-=
Vb -((Lb r b
+ AH1[P043-]iVi)/ (MiAHi2
+ NbAHi + Lb2)
(18)
Now, taking two points on the curve, bl and bz, one obtains eq 19 by equating the resulting expressions for K/AHi (Mb,AHi*
+ NblAHi + Lbx2)(Lbn+
AH1 [POd3-]iVi)( V b r / r b z ) (MbzAHi2f NbzAHi
-
+ Lb,')(LbL +
AH1[P043-]iVi)(Vbi/rbl)= 0 (19) Equation 19 can now be simplified to give (Dl - D2)AHi'
+ (E1 - E2)AHiZ+ ( R - Fz)AHi
+ (GI - Gz) = 0
(20)
where D1, Dz, El, Ez,K , F2, G1, and G2 are the appropriate constants from eq 19. Equation 20 may be solved for the proper value of AH1 by Newton's method of successive approximations. I n practice, approximate values of K and p are assumed, values of [H2P04-]b, [HZPO4-]i, [H+]b, [P043--]i,and r b calculated, and AH1 and a new value of K are calculated. Better values of 1.1, [H2P04-]b, [H2P04-]i,[H+]b, [P043-]i, and rb are then calculated from K and the whole process repeated until successive values of both K and AH1 agree to 0.1%. Thermometric Titration. Data Analysis. A thermogram consists of three major regions as denoted by lettered brackets in Figure 1. In region a the titrant is off and the temperature change is due to stirring, heating by the thermistor, and heat transfer. I n region r the titrant is on and the temperature change is due to heats of reaction, heats of dilution of titrant and solution, addition of titrant at a different temperature from that of the solution, and the effects in a. I n region d the titrant is off and the temperature change is due to the effects in a. Equations 21 through 26 result if the effects of stirring and heating by the thermistor are constant and if the rate of heat transfer from the calorimeter is proportional to the temperature difference between the calorimeter and its surroundings (Newton's law of cooling).26 Volume 70, Number 6 June 1966
J. J. CHRISTENSEN, R. M. IZATT, L. D. HANSEN,AND J. A. PARTRIDGE
2006
( a / d t ) d = Cd(dB/dt) d
Table I: Heat of Reaction, Q, for the Indicated Reaction as a Function of Titrant Added" Titrant, mmoles
(0.01001F)'
(0.02007F)'
- 0.3963 -0.6219 0.8943 1.1331 1.3542 - 1.5540 - 1.7374 1 ,8994 -2.0499 -2.1856 -2.3070
-
-
Q (0.02047F ) o
-0.4587 -0.8935
- 1.3096 -1.6954 -2.0653 -2.4133 -2.7322 3.0404 -3.3270 -3.5942 -3.8415
(0.02509F)'
(0.002493F ) f
(0.01252F ) &
-0.5170 -0.8974 1.1872 1.4192 -1,6106 - 1.7969 1.9266 -2.058P -2.1810 -2.2921
-1.2619 -2.3843 -3.3887 4.2777 - 5.0532 -5.8871 - 6.3540 - 6.8883 -7.3731 -7.8044
- 0.3220
-0.9209 1.6801 -2.2916 -2.8038 -3.2326 -3.5909 -3.9033 -4.1733 -4.4220 -4.6401
-
-0.5381 -0.7051 - 0.8453 - 0.9658 1.0764 - 1.1783 -I ,2735 - 1 ,3647 - 1.4498
-
+ HP04'-
(0.005644Fjl
- 0.4555 -0.8408 -1.1786 - 1.4880 -1.7691 -2.0202 2.2400 2.4422 -2.6247 -2.7840 -2.9375
-
=
-
PO*'-
(26)
+
titrant at Br solution containing reactant at ei --+ solution containing products at ei (27) to occur by the following three steps which lead to eq 28-30. a. Titrant at e T + titrant at ei cW,/dt =
- &)*(dU/dt)
(Oi
(28)
+
b. Solution containing reactant at O i titrant at ei solution containing reactant and titrant at Bi (heat effect due to heat of mixing-no reaction takes place,
+
+ HzO (0.002140F)"
- 0.6675 -1.2506 1,7889 -2.2782 2.8702 -3.1343 - 3.5073 -3.8448 -4.1503 4.4286 4.6776
-0.2202 -0.3813 0.5281 -0.6574 -0.7793 -0.8917 -0.9882 - 1.0783 -1.1528 1.2314 1 ,3094
-
- 0) 1
-
(0.009464F)"
-
+ k(&
Equation 24 was derived by considering the process represented by eq 27
-
(0.004990F ) h
OH-
0. 1742d 0.3483 0.5225 0.6967 0.8708 1.0450 1.2192 1.3933 1.5675 1.7416 1.9158
-k Sod2-
HSO4- = H f -0.4521 -0.8811 - 1,2938 - 1,6736 -2.0359 - 2.3786 -2.6958 -2.9906 - 3.2734 - 3.5367 - 3.7838
0 .2015b 0,4031 0.6046 0.8062 1.0077 1,2093 1.4108 1.6123 1.8139 2.0154 2.2170
0.5081" 1.0162 1 ,5243 2.0323 2.5404 3.0485 3.5566 4.0647 4.5728 5.0809
Q
Q
x 10s
( w / d t ) d = [w
(25)
-
-
Initial volume: 100.0 ml. Total initial sulfate or phosphate concentration given in parentheses in each case. * Titrant: Titrant: 0.3931 F HC104. Titrant: 0.9910 F HC104. 0.3397 F NaOH. e-'L K and AH" values for these runs in Table I1 are designated by corresponding superscript letters.
e1
8,
et
Temperature. 8
Figure 1. Typical thermometric titration curve; significance of regions and points explained in text.
assuming that the dilution and effects on the solute of solution 1 are negligible) a b / d t = (4b
- h)(dV/dt)m
(29)
c. Solution containing reactant and titrant at O i + solution containing products at O i (heat effect due to heat of reaction) dH,/dt =
AH, (dn,/dt)
(30)
P
and from an enthalpy balance on the calorimeter ( a / d t ) r = -(dHa/dt)r - (mb/dt)r (ae/dt)r [w k(0,
+ +
- 011
(31)
~
(25) F. D. Rossini, "Experimental Thermochemistry," Vol. I, Interscience Publishers, Inc., New York, N. Y., 1956, pp 28-35.
The Journal of Physical Chemistry
CALORIMETRIC METHODFOR
THE
DETERMINATION OF AG, AH, AND A S
For a finite time interval, from v = 0 and t = 0 a t = Vb and t = tb at ob, eq 23 and 24 give eq 32 and 33, respectively.
ei to v
AH, = (Ca
AH, = Jrotb[w
(CAHpAnp)b P
f(t)b)(eb
-
(32)
ei)
(eT
+ ( 4 -~
- &)Wb
+
4b)mVb
(33)
-
P
eq 32 and 33 if the integral can be evaluated. The values of 4 ~ $t,, , cT, and (C. f ( t ) b ) are obtained from separate experiments or from the literature. e T is taken to be equal to 0, and measured directly for each run. Oi, ob, and tb are obtained directly from the chart and AH, is obtained from eq 32. The value of Vb is given by Vb = t b R (34) and f ( t ) b is given by eq. 35.
+
f(t)b
=
(35)
vbh'
Ls0 + t=tb
The value of the integral
[w
k(&
- e)ldt can
be calculated by approximating the titration curve using a series of straight lines. For each segment the integral can then be calculated and the total value of the integral found by summing all of these from t = 0 to t = tb. The value of [w k(e, - e) J a t any point (ob, tb) on the curve in region r is found by using the data from regions a and d in eq 36.
+
+ k(0, - e) = (dB/dt),C, + [(de/dt)dCd
If the curve in the straight line, then
s
t=tb+1
t=lb
Jrotb[(W
Sbdt
- (de/dt).C,J
eb
con-
+ k(e, - e)]dt = ban
The function [w k(0, e) Jdt must be left in differential form because 8 is a complicated function of t. The value of ( x A H p A n p ) b can now be obtained from
w
and, one can obtain (38) for the interval Bi to taining n segments
+ k(e. - e)]dt -
+
2007
where H L b is a term giving the heat effects of st'irring, heat leaks, etc. In practice the charts were analyzed for the quantities (deldt),, (dO/dt)d, Of, &, (eT - ei), and e b at 1-min time intervals. Then, AHr was calculated from eq 32, a set of values for s b was calculated from eq 36, and H L b was calculated from eq 38. The value of ( C A H p A n p ) b was then obtained from eq 33. P
For entropy titrations, ( C A H P A n p ) b =
Qb,
and the
P
procedures outlined in this paper under Theory of Entropy Titrations can be used to determine K and AH. In those cases where K values are known, AH values can be calculated for each reaction by solving simultaneously a specific set of the following general equations
These values of AH can then be used as a starting point in a least-squares program26J' to calculate the "best" AH values using all the data points.
Results In Table I entropy titration data for representative runs are presented giving the heat liberated or absorbed in the calorimeter (corrected for stirring and heat losses) as a function of moles of titrant added. The complete listing of the data is given in the form of input data and IBM FORTRAN IV computer programs in ref IC.
[(ob
-
In Table I1 are given the results for the entropy titrations of NazSOr with HC104 and Na2HP04 with NaOH, respectively. Those runs for which entropy titration data are given in Table I are indicated by -~
~~~
(26) D. Dyrssen, N. Ingri, and L. G. SillBn, Acta Chem. Scund., 15, 694 (1961). (27) L. G.SillBn, ibid., 16, 159 (1962).
Volume 70.Number 6 June 1966
J. J. CHRISTENSEN, R. M. IZATT, L. D. HANSEN,AND J. A. PARTRIDGE
2008
Table 11: Equilibrium Constant and Heat of Reaction Results Determined by the Entropy Titration Method (Each Value Is the Average from a Single Determination with the Uncertainty Expressed as the Standard Deviation)"
AH',
PK
1.89 f 0 . 0 3 1.87 f 0.08 1.89 f 0.06g 1.89 f 0.06 1.89 f 0.07' 1.90 f 0.04
AH',
AHo,
kcal/mole
PK
kcal/mole
PK
kcal/mole
-5.66 zk 0.26 -5.88 ztO.70 -5.63 f 0 . 4 7 -5.73 f 0 . 4 8 -5.71f0.53 -5.66 f 0 . 2 9
1.92 f 0.04 1.88 f 0.02 1.90 f 0.01h 1.90 f 0.01 1 . 9 3 f 0.04 1.92 f 0.02'
-5.54k0.16 -6.02k0.13 -5.73 f 0.08 -5.73 1 0 . 0 5 -5.33 f 0.26 -5.36f0.12
1 . 9 3 f 0.07( 1.89 k 0.01 1.88 f 0.03' 1.88 f 0.02 1.92 f 0.01 1 . 9 3 f 0.01 1.93fO.Olk
-5.41f0.46 -5.85f0.06 -5.89k0.21 -5.92 1 0 . 1 4 -5.50f0.08 -5.50f0.05 -5.45 f 0 . 0 4
log K
AH', kcal/mole
Average p K =
1 . 9 1 f 0.01
AH" = -5.64 f 0.08
HPOrZ-
+ OH-
AH',
log K
kcal/mole
1 . 6 4 f 0.05 1.59 f 0.10 1 . 6 1 f 0.06" 1 . 6 3 f 0.07
-8.77f0.55 -9.78 f 1 . 1 6 -9.75f0.74 -8.94f0.74
a
= !?Or3-
+ H20
AH',
log K
kcal/mole
1.59 k 0.03 -9.00f0.31 1 . 6 1 f 0.01' -8.97ztO.16 1.58 f 0.02 -9.14 i k 0 . 2 4 1.59 f 0.02 -9.lOk0.24 Average log K = 1.61 f 0.03 AH" = -9.14 f 0.31
1 . 6 1 f 0.07 1 . 6 1 f O.Ogm 1.58 k 0.04 1.58 f 0.03
-8.90f0.64 -8.97 f l . 1 1 -9.06 f 0 . 3 8 -9.12 1t0.29
For entries with superscript letters e-n, see the corresponding footnotes in Table I.
number. The log K values are thermodynamic values valid at p = 0; however, the AH values obtained by the entropy titration method are valid only for the range of p covered in the titration. I n this study, AH does not change significantly with changing p ; consequently, all values were averaged and the result was taken to be AH". The standard state was taken to be an ideal 1 M solution behaving as an infinitely dilute solution.
Discussion The average pK and AH" values summarized from results in Table I1 together with selected literature data are given in Table 111. The pK value found in this study for HS04- ioniaation is somewhat lower than, but in substantially good agreement with, previous value sreported in Table 111. There are two difficulties in obtaining a pK value for the HS04- system. These are (a) the possible existence of an MS0,- (M = Naf, K+) c ~ m p l e x , and ~ * ~(b) ~ ~the choice of activity coefficient^.^^ I n an attempt to learn whether a significant cation effect exists, a preliminary investigation has been made of the pK value in [(n-C3H,)4N]2S04--HCl solutions. The pK value obtained (Table 111) is significantly higher than any of the literature pK values, indicating a considerable cation effect in this system. Although The Journal of Physical Chemistry
the results are approximate, they indicate the desirability of a more complete investigation of HS04ionization as a function of the cation used. The numerous recalculated values appearing in Table I11 are indicative of the second difficulty; e.g., the pK value a t infinite dilution depends on the values of the activity coefficients and thus on the values of the parameters used in the Debye-Huckel extrapolation. Covington, Dobson, and Wynne-Jones30 have recently shown that for all emf methods regardless of the cell solutions used, the pK value of HS04- depends on the value chosen for the ion-size parameter in the DebyeHuckel formula. They also show that because of indicator salt effects the spectrophotometric method is not superior to emf methods in this respect. They state that with the combined uncertainties of the choice of the ion-size parameter and the extrapolation of results the pK value calculated from emf measurements is between 1.94 and 2.01, but that neither emf nor spectrophotometric methods allow the pK value to be fixed very closely. I n addition to these uncertainties, (28) E. C. Righellato and C. W. Davies, Trans. Faraday SOC.,26s 592 (1930). (29) I. L. Jenkins and C. B. Monk, J . Am. Chem. Soc., 72, 2695 (1950). (30) A. K. Covington, J. V. Dobson, and W. F. K. Wynne-Jones, Trans. Faraday Soc., 61, 2057 (1965).
CALORIMETRIC METHOD FOR THE DETERMINATION OF AG, AH,AND A S
2009
Table 111: pK and AHo Values for Proton Ionization from HS04p K at 25O
1.91 z!c 0.01 2 . 2 f0 . 1 1.94 1.92
1.99 1.99 1.99 1.99 1.99 1.94 1.96 1.96 1.99
p K as f [H +tad 1.99
1.97
AHa,
Methoda
ionization
Entropy titration - 5 . 6 f0.1 (NazSO4, HClOI) Entropy titration - 4 . 1 f0 . 5 { [(n-CaH~)4NlzSOd,HCl) Conductivity (HzS04, NaHSOa) Emf -2.2 ( Pt,H2[NaHS04,NazSO+NaC1/AgCl[Ag) -5.2 Spectrophotometry -5.2 (NazS04, HCl) Emf -5.2 ( P ~ , H ~ ~ H P S O ~ , H CAg) ~IA~C~[ Recalculation of data in ref b Recalculation of data in ref c (Corrected for NaSOl+, K = 0.19) Conductivity (HC1, K2S04) Solubility (AgzSO4, Emf -5.6 (Pt,Hzj HzSO4,HClIAgClIAg) Recalculation of data in ref g -5.5 Solubility (AgzS04, H2S04) -5.7 Glass electrode
-5.4
P
Method
Reference
0
Entropy titration
This work
0
Entropy titration
This work
b 0
pK as f( 2')
0 0
Calorimetry pK as f( T )
0
PK
= f(T)
C
d e, f 9 9
9
h
i 0
PK
= f(T)
j j
0
Calorimetry
k
1 0 Calorimetry m (Corrected for NaS04+, AH = - 1 . 1 kcal) 1 pK as f ( 2') n
Recalculation of data in ref j Emf (Pt,HzINaHSO4,NaB041Hg2S041Hg)
0
P
I n parentheses is given the electrolyte present in the determination. M. S. Sherill and A. A. Noyes, J. Am. Chem. SOC.,48, 1861 (1926). W. J. Hamer, ibid., 56, 860 (1934). K. Pitzer, ibid., 59, 2365 (1937). I. M. Klotz, Thesis, University of Chicago, 1940. C. R. Singleteny, Thesis, University of Chicago, 1940. C. Pi. Davies, H. W. Jones, and C. B. Monk, Trans. Faraday Soc., 48, 921 (1952). * M. Kerker, J. Am. Chem. SOC.,79, 3664 (1957). J. Kenttamaa, Suomen Kemistilehti, 30B, 9 (1957). V. S. K. Nair and G. H. Nancollas, J. Chem. Soc., 4144 (1958). A. J. Zielen, J. Am. Chem. SOC.,81, 5026 (1959). M. H. Lietzke, R. W, Stoughton, and T. F. Young, J . Phys. Chem., 65,2247 (1961). J. M. Austin and A. D. Mair, ibid., 66,519 (1962). A. N. Fletcher, J. Inorg. Nucl. Chem., 26, 955 (1964). H. S. Dunsmore and G. H. Nancollas, J. Phys. Chem., 68, 1579 (1964). See ref 30.
'
none of the previous workers considered the effects of a possible complex of S042- or HSOa- with Ag+. This could be an important factor since most of the pK values were either calculated from emf data obtained using Ag-AgC1 electrodes or calculated from Ag2S04 solubility data. I n summary) the results presented for the HS04system show the entropy titration method to give a pK value comparable to that obtained by other methods and a AHo value which is in good agreement with reported literature values.
I n Table IV are given average log K and AHo values HP042- + H20 Po43for the reaction OHsummarized from the results in Table I1 together with available literature data. The log K results from the present study for the phosphate system are in excellent agreement with those found in the literature. The AHovalue determined in this study agrees well with that reported by Papoff, et aL13but not with that reported by P i t ~ e r . How~~
+
+
(31) K. 9.Pitzer, J . Am. Chem. Soe., 59, 2365 (1937).
Volume 70.Number 6 June 1966
J. J. CHRISTENSEN, R. M. IZATT, L. D. HANSEN,AND J. A. PARTRIDGE
2010
Table IV: Log K and AH" Values for the Reaction OH-
+ HPOa2- = HzO + PO2AHo,
Method
kcal/mole
1 . 6 1 & 0.03 1.625 2 ~ 0 . 0 1
Entropy titration Spectrophotometry
- 9 . 1 rt 0.4
1.61
Calorimetry
Log K
0
Method
Ref
Entropy titration
This work 16 31 3
0 0 0.1
-9.9 i 0.5 -8.9
ever, in the latter study an incorrect log K value (2.07) was used and not enough data were given for a recalculation. From the results on the HS0,- and H P 0 2 - systems, the entropy titration method is estimated to be capable of an accuracy of 3=0.05pK unit. This, of course, is dependent upon the magnitude of the pK value being determined. Most of the uncertainty originates from the uncertainties in the calorimetric data, the remainder resulting from the calculation of the activity coefficients. An attempt was made to evaluate this latter source of error by using different it values in the Debye-Hiickel expression for the activity coefficients in the H P 0 2 - system. The pK values calculated using 5 and 10 A for d differed only by 0.008 pK unit. This low dependency of the pK value on the r function is the result of r appearing as a ratio I'bl/I'bp which is not appreciably affected by small errors in the I' calculation because the errors in I'bl and r b z are of the same direction and magnitude. It has been demonstrated with two different systems that the entropy titration method is capable of determining pK values with good precision and accuracy. The method is simple and rapid, the only stringent requirement being that the calorimetric data must be of high accuracy.
Definition of Symbols 01, Ob, Or, tb, point b, and regions a, r, and d are defined by Figure 1. The subscripts a, r, and d, refer to regions a, r, and d, and subscript b refers to point b.
The Journal of Physical Chemistry
P
A AB B
Calorimetry Calorimetry
Reactive component in calorimeter (molesfliter) Reaction product in calorimeter (molesfliter) Reactive component in titrant (molesfliter) C Total heat capacity (cal/deg) CT Heat capacity of titrant per unit volume (cal/ml) H Enthalpy or heat content (cal) A H p Enthalpy change per mole of the pth reaction (cal/mole) k Modulus of heat exchange between calorimeter and surroundings (cal/sec deg) KP Equilibrium constant Concentration o f reacting species in titrant (moles/ml) m Moles of product produced by the pth reaction (moles) nP Heat produced in the calorimeter by chemical reactions Q except heats of dilution (cal) Titrant delivery rates (ml/sec) R Rate of heat loss a t point b (cal/sec) s b t Time (sec) Volume of solution in calorimeter (1.) V 2, Volume of titrant added (ml) Heat input from external effects including stirring and W thermistor heating (cal/sec) Activity coefficient of indicated subscripted species Y Products of activity coefficients of reactants divided by r b products of activity coefficients o f products Temperature of calorimeter ("C) e Temperature o f surroundings ("C) es Temperature of titrant ("C) eT Apparent partial molar enthalpy of reacting species in $b titrant after dilution to the ionic strength a t point b (cal/mole) Apparent partial molar enthalpy of reacting species in 4JT titrant (cal/mole) Ionic strength P (c, f(t)b) = heat capacity at point b
+
