H. D. B. Jenkins, A. Winsor, and T. C. Waddington
578
Alkali Metal Chromates. Enthalpy of Formation, AHf' (Cr04*-)(g). Charge Distribution of Gaseous Chromate Ion and Total Lattice Potential Energies of Sodium, Potassium, Rubidium, and Cesium Chromates H. D. B. Jenkins,* Anne Winsor, Department of Molecular Sciences, University of Warwick, Coventry CV4 7AL,England
and T. C. Waddington Department of Chemistry, University of Durham, Durham City, England (Received April 15; 1974; Revised Manuscript Received September 24, 1974)
This theoretical study reports the assignment of a charge of qo = -0.61 proton units to each of the oxygen atoms in the ion. The following values for the heat of formation of the chromate ion and the total lattice potential energies of the chromates are assigned: AHfo(Cr042-) = -705 kJ mol-l; Upo~(Na2Cr04) = 1836 kJ mol-'; U p o ~ ( K ~ C r 0 4=) 1714 kJ mol-l; U p o ~ ( R b ~ C r 0 4=) 1653 kJ mol-l; UPOT(CS2Cr04) = 1596 kJ mol-l. A value for the absolute oxide ion affinity of the oxyacid CrO3, corresponding to the process CrOs(g) 02-(g) Cr042-(g), is estimated to be approximately -1224 kJ mol-l.
+
-
AE(M2Cr04) + 3RT
Introduction Considerable work has been done on the calculation of lattice energies of salts containing singly charged complex anions and cations. The complex ions, NH4+,l C032-,293 HF2-,4 N3-,5 and N03-,6 have recently been examined. Neckel and Vinek7 have also considered lattice energies of the bifluorides as have Van Gool, Bruinink, and Bottelberghs.s The CN- ion has been discussed by Laddg although there is some dispute regarding the reference energy levels employedlOJ1 in this latter work. In many cases values of the lattice energies are susceptible to direct experimental checks and, in general, agreement between values computed based on an ionic model and experiment is good. Little work has been done on the calculation of lattice energies of salts containing doubly charged negative ions and the work which does exist12-14 generally involves ions of the transition metals; the aim of this work is to extend calculations to salts containing the Cr042- ion. Though checks with experiment are much more difficult, where they can be made, agreement is satisfactory. It seems sensible to extend these calculations to doubly charged oxyanions and to relate the calculations to the estimation of the absolute oxide 02-ion affinity of strong anhydrooxyacid systems such as SO3 and CrO3. This paper represents an attempt to calculate such data for the alkali metal chromates.
Lattice Energy Calculations We now examine the Cr042- ion, where a distributed charge qcr is assigned to the chromium atom and qo to each of the four oxygen atoms. These charges are related by the equation qcr = -2[1 + 2qOl
( 1)
Total lattice potential energies, Upo~(M2Cr04), are computed as single-parameter functions of q o for the four alkali metal salts corresponding to M = Na, K, Rb, and Cs. Using the thermochemical cycle The Journal of Physical Chemistry, Vol. 79, No. 6, 1975
M2Cr04
2M+(g) + Cr042-(g)
-I'1
-AHfe(Cr042').(e)
.2AHfo(M*)(B)
2M(g)
-AH:
( M , c ~ o ~ )(c)
+
Cr(C)
f
24k)
We find that Upo~(M2Cr04)for each salt is related to LWfo(Cr042-)(g)by the equation AHfo(Cr0,23(g) = AHf0(M2CrO4)(c) -t AE(M,CrO4) - 2AH,"(M')(g)
f
3RT (2)
where, at 298 K AE(M2CrO4) = up,, (M2Cr04) (3) where AE(M2Cr04) is the total internal energy of the crystalline chromate salt, and AHfo(M2Cr04)(c) and LWfO(M+)(g)are the standard enthalpies of formation of the crystalline chromate and of the corresponding alkali metal ion, respectively. The combined equation AHfo(Cr0,2-)(g) = AH,"(M,CrO,)(c) + Up0,(M,CrO4) - 2AHf0(M+)(g) + 3RT
(4)
can be expressed in the parametric f 0 r m ~ 9 ~ 2
AHfo(CrO,Z-)(g) =
A,qoi i= 0
Solution of equations of the above general form enables a value for LWfo(Cr042-)(g), which is alkali metal ion independent, and for qo to be assigned for the chromate ion. The total lattice potential energy is given by UpOT(M2Cro*) = U E L E C+ UD - u,
(6)
where UELECis the electrostatic lattice energy, corresponding to the process M,Cr04(c)
-
2M+(g) + Cr04"(g)
UDand URare the dispersion and repulsion energies of the lattice, respectively. The Bertaut15 method is employed to calculate the Madelung constant and energy associated with the hypothetical process M,Cr04(c)
-
2M+(g) + Cr'Cr(g) + 4090k)
579
Enthalpy of Formation of Alkali Metal Chromates TABLE I: Parameterization of Equations for the Chromatesa ~
Na2Cr04
K2Cr04
Rb, C r04
~~
C s C r04 ~
-~
2.119129 2.096907 2.239607 8.057652 8.0476 81 7.970842 12.273957 12.231810 12.148100 "2 1.6099 L 1.5895 1.5780 1,809.5 1,957.5 1,865.7 BO 6,953.4 7,034.0 7,017.7 Bl 10,695.5 10,591.8 B, 10,691 .O -6,879.2 -6,943.8 -6,946.2 Cl -10,598.9 -10,701.4 C, -10,698.5 D1 90.1 71.5 74.2 -5.9 -7.0 D2 -7.5 49.0 a for I-d 33.6 26.8 UD 32.4 e for SO4,-@ 23 .O 17.9 VD 40.7 average value taken 28.3 22.4 U, av 205.9 208.3 194.8 UFl -1,389.5 -1,380.7 -1,382 .8b AHfo(M2Cr04) (4 -1,3 88.7' 494.9b 460 .6b 514.2 609.8 W,"(M') (g) -658.9 -664 .O -749.4 (ID -632 .2b A0 (1) -638.1' A1 -76.2 90.1 71.5 74.2 A, -3.5 -7.5 -5.9 -7 .O a L is shortest Cr-0 distance to which Madelung parameters are referred.Energies quoted are in kJ mol-1. * Reference 35. Reference 34. Reference 23. e Reference 24. "0
"1
2.287772 7.903739 12.307207 1.5837 2,006.9 6,933.5 10,796.5 -7,009.8 -10,800 .O -76.2 -3.5 13.2 8 .O 10.6 226.2 -1,328.8b
TABLE 11: Intersection Points of Figure 1 Intersection
Point
Na-K(IJ) Na-K(1) Na-Rb Na-Cs Average Aver age Average
A B C D (all points) (A,C,D) (B,C,D)
AH,"(Cr042-) q o , proton (g), kJ mol'' units -689.5 -701.0 -707.1 -705.6
-0.69 -0.66 -0.57 -0.59 -0.63 -0.62 -0.61
-703 .O
-703.7 -704.6
generate dn equation for UELEC. The equations involved take the usual forms2
(10)
The dispersion energy, U,, is calculated employing the formulae u, = z[c++ss++ 1 + c-_s,--]+ c+-s,+- (11)
where. and employing the truncation procedure and convergence check suggested by Jenkins16 the values given in Table I are obtained. The Cr042- ion "self energy," U S E ,is added to U M to
c+_=
C++ = 0 . 7 5 ~ + ( Y + ~
(12)
C _ _ = 0.756-"_2
(13)
(1.5€+€-0+@-)/(€++
€3
(14)
The Journal of Physical Chemistry, Vol. 79. No. 6, 1975
H. D. B. Jenkins, A. Winsor, and T. C. Waddington
580
TABLE 111: Calculation of Absolute Oxide Ion Affinity of Cr03 and Estimate for SO3
Average -1224
-i:
46
-1244 -1202 -1227
j :
i
63 29
925 883 908'
* i
63" 2gb
-385.8d -385.8d -3 85. 8d
-705 -705 -705
-3 9 5.72d -395.72d -395.72d -395.72d -395.72d -395.72d -395.72d -395 .72d -395.72d
-741e -707f -7508 -741e
As,-+02-, k J mol-'
-1242
a
*
46
-1270 -1236 -1279 -1228 -1194 -1237 -1253 -1219 -1262
* i
+
+ izt
63 63 63 29 29 29
925 925 925 883 883 883 908' 90 8' 908'
i ~r
i iii-
63' 63a 63' 2gb 2gb 2gb
-7071
-750g -741e -707f -750'
Reference 41. Reference 42. Reference 43. d Reference 40. e Reference 46. f Reference 50. g Reference 51.
TABLE IV: Literature Values for Charge Distribution on Cr042- Ion Source
+
YO
Reference 36 -0.66 Reference 22 -0.69 Reference 37" -0.69 References 38, 39 -0.67 Av lit. value -0.67 This work -0.61 a Assigning an oxidation state of Cr042- ion as +0.78 and linearly interpolating.
where RM+M+,, Rcrcr,, and R M + care ~ ~the distances of the ith M+ ion and j t h Cr atom a reference M+ ion or chromium atom (which is excluded from the summation). c+, the characteristic energy of the alkali metal ion M+, is taken as 0.75 of the second ionization potential of the ion (M+ M2+), values being taken from Moore.17 6- is the characteristic energy of the Cr042- ion, the value of which is taken from Von Halben and Litmanowitsch,l8 Viste and Gray,lg Waggoner and Chambers,20 Campbell,21 and Oleari, De Michelis, and Di SipioZ2all of whom have examined the most intense absorption band in the ultraviolet spectrum of the ion and found it to lie a t X 270 mp. a+ and aare the polarizabilities of the cation and anion, respectively, the former pdlarizabilities are taken from MayerZ3but the latter, have not, to our knowledge, been measured. Values of a- for the chromate ion were used which (a) assumed a- to be the same as that for an I- ion and (b)
-
The Journal of Physical Chemistry, Vol. 79, No. 6, 1975
values of a- for s04'- in the corresponding sulfates as calculated by Tessman, Kahn, and S h o ~ k l e y . ~ ~ URis calculated using the equation u, = UR+* uR-- + CY+,(18) where, following HugginsZ5 URtt = '/zbc,, exp(2rt/p)Cexp(-RM+~tI/p) i
u,+- = 'bc,,' uR--
exp([rt +
(19)
~ J Pf) C ~ X P ( -(20) R~~~~/
= '/zbc--' exp(2r_/p) [Cexp(-ROok/p)
-
k
exP(-Rooa / P ) - exP(-Rooa 0 / P ) exp(-Rooe ##/P)I (21) where the model taken for the Cr042- ion consists solely of the four oxygen atoms, the chromium ion at the center of the tetrahedra not specifically being considered. The Huggins radii for the alkali metal ions are well established; for the oxygen atoms, comprising the Cr042- ion, a value of the "basic radii,'' r- = 1.23 A, is taken from previous work on the NO3- ion described elsewhere by Jenkins and Waddington.6 p was taken to be 0.345 8-l. Root, Roo.., and roo^^^ are the distances between the reference Os0 ion and the other three oxygen atoms in the same Cr042- ion unit. b is is a constant and following Pauling26
ctt' = [1 + (2qt/nt)l
(22)
c--' = [1 + ( 2 q h L ) I
(23)
ct-' = [1 + (qt/nt) + (4./K)I (24) where q+ is the cationic charge (+1) and n+ the number of electrons in the valence shell of the M+ ion (=8). q- is the anionic charge of -2 and n- was assigned the value 8. Combining eq 6 and 10 up o ~ ( M , C r 0 4 ) =
Enthalpy of Formation of Alkali Metal Chromates
where Eo = (Bo
and
+
UD
5.81
- UR)
(26)
0
(27)
E,, = 0,f o r n
#
estimate a value for the absolute oxide ion affinity of the anhydro oxyacid Cr03, Acro302CrOs(g)
+
OYg)
-c
Cr042-(g)
ACro30Z-
consequently in eq 5
A , = Eo
+
AHfo(M2Cr04)(c)- 2AHf0(M+)(g)+ 3RT (28) Ai = E, for i # 0 (29)
Figure 1 shows AHfo(Cr042-)(g)as a function of
- 7.5q02
(31)
- 5.9q02 (32) U p o , ( C s 2 ~ r 0 4 )= 1644.3 + 74.2qo - 7.0qo2 (33) UpOT(RbzCr0,)
4-
02-(g) ---c s042-(g) ASOgoZ-
1,
9o.lqo
so,(g)
40.
Results The alkali metal chromates show various structure types.27 Sodium chromate is orthorhombic having a space group, Vh17 (Cmcm) and unit cell lengths (a0 ='7.138 A, bo = 5.861 A, co = 9.259 .&).z8 Potassium, rubidium, and cesium chromates have the K2SO4 arran ement with the following cell lengths: K2Cr04 (a0 = 7.61 bo = 5.92 A, co = 10.40 A);29-31RbzCrOd (a0 = 7.983 A, bo = 6.288 A, co = 10.704 and Cs2CrO4 (a0 = 8.429 A, bo = 6.302 A, co = 11.19 Table I cites details of the coefficients and energy parameters calculated in this work. There are several points to note, (i) The value of tiD, the dispersion energy used in these calculations, corresponded to the average value of UD(*),calculated assuming the Cr0d2- ion to have the same polarizability as an I- ion ( M a ~ e rand ~ ~ a) value U D ( cal~) culated taking the polarizability for the ion to be that of the S042- ion in the corresponding alkali metal sulfate (Tessman, et ~ 1 . ~ 4 )(ii) . For K2Cr04, the value of AHfo(KzCr04)(c) as cited by Muldrow and H e ~ l e generr~~ ates curve I while the older NBS data generate curve 11. The fact that the interaction point B is closer to points C and D generated by the other chromates suggests that the values of Muldrow and Hepler are more reliable. This illustrates a feature of this type of calculation, which we have noted b e f ~ r e , lnamely, ,~ that when there are two differing values of the lattice parameters or two differing values for enthalpy data (as in this case) then from the curves of AHfo(Xn-)(g) us. distributed charge it is often possible to make a value judgment between them. (iii) The equations for the total lattice potential energies of the chromates take the form Upo~(Na2Cr04)= 1792.3 - 76.2qo - 3.5q02 (30) UpoT(K2Cr04) = 1771.6
and for the absolute oxide ion affinity of the oxyacid SO3, AS03°2-
1699.2
(iv) Table I1 gives the ordinates and abcissae of the intersection points of the graph of AHf0(CrOq2-)(g)as a function of 90. The average value of aHfo(Cr042-)(g)obtained by using the points (B, C, and D) which are generated by using the more modern data (avoiding the older NBS values) is probably the most reliable. Accordingly we assign a value for AHfa(Cr042-)(g)= -705 k J mol'' (34) which is probably accurate to f20 kJ mol-l, and a value of q O for the chromate ion q o = -0.61 proton units (3 5) (v) Since data are available for AHfo(Cr03)(g) and AHfo(S03)(g)40and for Lvrf0(024)(g)41-43it is possible to
whereupon
ACr030z-= AHfo(Cr042-)(g)- AH,"(CrOJ(g) AHf"(0'7 (g)
(36)
with an analogous equation for Aso3O2-,and as shown in Table 111 a value of -1224 k J mol-l emerges for Acro302with a probable error of about f 5 0 kJ mol-l and a value of -1242 kJ mol-l can be estimated for AS^^^^- with a rather larger uncertainty owing to the uncertainty about a value for ~W;(S04~-)(g).
Discussion AHfo(Cr042-)(g) is a thermodynamic quantity, the value of which has never previously been assigned to our knowledge. The value obtained for qo seems satisfactory although the present work places considerable reliance on the curve for Na2CrO4 since this curve is the discriminant, used when examining the mutual intersections of the curves. The crystal structure of Li2Cr04, had it been determined, would have enabled us to ascertain with even more confidence a value for AHfo(Cr042-)(g). The values of U p o ~ ( M ~ C r 0 4 ) , the total lattice potential energies, are found to be Upo~(NElzCr04)= 1836.5 kJ mol-'
(37)
UpoT(K2Cr0,) = 1713.8 k J mol-'
(38)
Upo,(Rb2Cr04) = 1653.4 k J mol-'
(39)
UpOT(CSCr04) = 1596.4 k J mol-'
(40)
and correspond to a value for 40 of -0.61 proton units. The only values of lattice energies of chromates which appear in the literature are those quoted by S a m s a n o ~ 4which ~ originate from Yatsimirskii's work45 and are, for Na2CrO4, 1979 kJ molb1 and, for K2Cr04, 1816 kJ mol-l and differ from the current values by 6-8%, which is the order of accuracy claimed for the Yatsimirskii empirical equation. Agreement seems satisfactory. Values of -705 kJ molF1 for AHfo(Cr042-)(g) and of -1224 kJ mol-l for A c ~ can ~ be~ compared ~ ~ -with an estimated value from -707 to -750 for aHfo(S042-)(g)and of -1242 kJ mol-1 for AS^^^^-. These figures show clearly that Cr03 is a Lewis oxyacid of quantitatively very similar strength to SO3 perhaps the strongest uncharged Lewis acid known. Unfortunately, at the present time, there are insufficient data to make a comparison with Moo3 and SeOn_ _Dossible. A number of calculations for 40 exist in the literature. These range from semiempirical e ~ t i m a t e s ~ ~ , ~to 6 , 3a 7full ab initio c a l c ~ l a t i o nand ~ ~ our ~ ~ ~current value is in fair agreement with these (Table IV). Acknowledgments. We wish to thank Professor Yosio Sakamoto of the University of Hiroshima for his interest The Journal of Physical Chemistry, Vol. 79,No. 6, 1b75
J. H . Stern and L. R. Beeninga
and kindness in reading the manuscript prior to publication and for his helpful suggestions.
References and Notes (1) A. L. Goodliffe, H. D. B. Jenkins, S. V. Martin, and T. C. Waddington, Mol. Phys., 21, 76 (1971). (2) H. D. B. Jenkins and T. C. Waddington, J. Chem. Phys., 58, 5323 (1972). (3) H. D. B. Jenkins and T. C. Waddington, Nature (London),Phys. Sci.. 232, 5 (1971). (4) H. P. Dixon, H. D. B. Jenkins, and T. C. Waddington, J. Chem. Phys., 57, 4388 (1972). ( 5 ) H. P. Dixon, H. D. B. Jenkins, and T. C. Waddington, Chern. Phys. Lett., I O , 600 (1971). (6) H. D. B. Jenkins and T. C. Waddington, J. lnorg. Nucl. Chem., 34, 2465 (1972). (7) A. Neckel and G. Vinek, Z. Naturforsch. A, 28, 569 (1971). (8) W. Van Gooi, J. Bruinink, and P. H. Bottelberghs, J. lnorg. Nucl. Chem., 34, 363 1 (1972). (9) M. F. C. Ladd, Trans. Faraday Soc., 85, 2712 (1969). (IO) H. D. B. Jenkins and T. C. Waddington, Nature(London),Phys. Sci., 238, 126 (1972). (11) M. F. C. Ladd, Nature(London), Phys. Sci., 238, 125 (1972). (12) M. Mori, R. Tsuchiya, E. Kyuno, and T. Nichide, Bull. Chem. SOC.Jap., 33, 1510(1960). (13) A. B. Blake and F. A. Cotton, lnorg. Chem., 2, 906 (1963). (14) J. Beck, R. H. Wood, and N. N. Greenwood, horg. Chem., 9, 86 (1970). (15) E. F. Bertaut, J. Phys. Radium, 13, 499 (1952). (16) H. D. B. Jenkins, Chem. Phys. Lett., 9,473 (1971). (17) C. E. Moore, Nat. Bur. Stand., Circ. No. 467 (1958). (18) H. Von Halben and M. Litrnanowitsch, Helv. Chim. Acta, 24, 44 (1941). (19) A. VisteandH. B.Gray, horg. Chem.,3, 1113(1964). (20) W. H. Waggoner and M. E. Chambers, Talanta,5, 121 (1960). (21) J. A. Campbell, Spectrochim.Acta, 21, 1333 (1965). (22) L. Oieari, G. De Micheiis, and L. Di Sipio, Mol. Phys., 10, 111 (1966).
(23) J. E. Mayer, J. Chem. Phys., 1, 270 (1938). (24) J. R. Tessman, A. H. Kahn, and W. Shockiey, Phys. Rev., 92, 890 (1953). (25) M. L. Huggins, J. Chem. Phys., 5, 143 (1937). (26) L. Pauiing, Z.Kristallogr.,8, 377 (1928). (27) R. W. G. Wyckoff, "Crystal Structures," Voi. 3, 2nd ed, interscience, New York, N.Y., 1965. (28) A. Niggli, Acta Crystallogr.,7, 776 (1954). (29) K. Herrnann, M. Hosenfeld, and N. Schonfeid, Wlss. Veroff. Siemenskonz, 5, 119 (1926). (30) M. Y. Colby, Z.Kristallogr., 78, 168 (1931). (31) W. H. Zachariasen and G. E. Zeigier, Z. Kristallogr.,80, 164 (1931). (32) H. W. Smith, Jr., and M. Y. Coiby, Z. Kristallogr., 101, 90 (1940). (33) J. J. Miller, 2.Kristallogr.,99A, 32 (1938). (34) C. N. Muidrow, Jr., and L. G. Hepier, J. Amer. Chem. SOC.,79, 4045 (1957). (35) F. D. Rossini, eta/., Net. Bur. Stand., Circ., No. 500 (1952). (36) S.S. Batsanov, Zhur. Neorg. Khim., 9, 1322 (1964). (37) G. De Michelis, L. Oieari, and L. Di Sipio, Coord. Chem. Rev., 1, 18 (1966). (38) I. H. Hilller and V. R. Saunders, J. Chem. SOC.0,1275 (1969). (39) I. H. Hiiiier and V. R. Saunders, Proc. Roy. Soc., Ser. A, 320, 161 (1970). (40) D. D. Wagrnan, W. H. Evans, V. B. Parker, i. Haiow, S.M. Bailey, and R. H. Schurnrn, Nat. Bur. Stand., Tech. Note, No. 270-4 (1969). (41) M. L. Huggins and Y. Sakamoto, J. Phys. SOC. Jap., 12, 241 (1957). (42) D. F. C. Morris, Proc. Roy SOC.,Ser. A, 242, 116 (1957). (43) T. C. Waddington, Advan. horg. Chem. Radiochem., 1, 157 (1959). (44) G. V. Sarnsonov, Vysokotemp, 110 (1965). (45) K. B. Yatsimirskii, *OX, 28, 9 (1956). (46) M. F. C. Ladd and W. H. Lee, J. lnorg. Nuci. Chem., 21, 216 (1961). (47) K. B. Yatsimirskii, J. Gen. Chem. USSR, 28, 2655 (1956). (48) K. B.Yatslrnirskii, Zhur. Obshch. Khim., 28, 2376 (1956). (49) N. M. Selivanova and M. Ch. Karapet'yants, lzv. Vyssh. Ucheb. Zaved., Khlm. Khim. Tekhnol., 8, 891 (1963). (50) M. F. C. Ladd and W. H. Lee, J. horg. Nucl. Chem., 30, 330 (1968). (51) H. D. 8.Jenkins, J. Chem. Phys.,J8, 5969 (1972).
Partial Molal Heat Capacities of Caffeine and Theophylline in Pure Water J. H. Stern* and L. R. Beeninga Department of Chemistry, California State University, Long Beach, California 90840 (Received Ju/y 3 1, 1974; Revised Manuscript Received December 9, 1974)
Calorimetric enthalpies of solution of the biochemically important methylated xanthines, caffeine and theophylline, to very low concentrations in water have been measured from 288 to 308'K. The temperature derivatives of the enthalpies yield the appropriate partial molal heat capacity differences between the aqueous solutes and the pure crystalline solids, with values of 100 and 90 cal/deg mol a t 298'K for caffeine and theophylline, respectively. These results, combined with the estimated heat capacities of the crystalline solids, give the anomalously high partial molal heat capacities of 160 and 140 cal/deg mol at 298'K foi caffeine and theophylline, respectively, and are evidence of the complex structuring induced by the two solutes. The CH2 group contribution to the partial molal heat capacities deduced from these two xanthines is approximately the same as that for various aqueous aliphatic nonelectrolytes, showing that limited group contribution predictions for partial molal heat capacities of aqueous nonelectrolytes may be possible when more data becomes available.
Introduction Aqueous methylated xanthines are important biochemical solutes1 with very interesting but poorly understood physicochemical properties.2 A previous paper3 included a study of the enthalpies of solution of crystalline caffeine in pure water a t 298'K, as a function of concentration. This contribution reports on the partial molal heat capacities of caffeine and theophylline at very low concentrations in The Journai of Physical Chemistry, Vo/. 79, No. 6, 1975
pure water via measurement of the enthalpies of solution from 288 to 308'K. The interaction of these solutes with water and their effect on aqueous solution structure should be of particular interest since both xanthines have similar pharmacodynamical properties, particularly as diuretic^.^ 'It has been observed that the anomalous partial molal heat capacity difference ACO,, between a pore and a dissolved solute at low concentrations in water, or the absolute partial molal heat capacity of the aqueous solute Gop2,
