CECILM. CRISSAND J. W. COBBLE
5390
cyo
expected since much of the data on ionic solutes pass through a maximum near this temperature. By this emperical extrapolation, entropy constants have been estimated up to 200' (Table 111). The validity of such extrapolations can be tested in a limited way by comparison of the predicted and experimental entropies for the few solutes known a t 200' (Table IV). The agreement between the predicted and observed values of a t 200' is quite satisfactory and suggests that the general validity of the correspondence principle holds up to a t least 200'. I t is to be noted that once the extrapolation method has been decided upon, values for the constants a ( t )and b, above 150' are fixed and are not arbitrarily defined. Consequently, the test given in Table IV is rigorous for the species listed, although necessarily limited.
TABLEIV COMPARISON OF PREDICTED A N D OBSERVED ENTROPIES AT 200" Ions
THE
Szo (obsd)~' Gal. mole-! deg. -1
5yo(oaiod),bcai. mole-' deg. - 1
-54 f 6 -1 f 2
-61 -1.3 4.7 -16.7 21.7 23.1 -31.7
H+, COE-' H + , C1H + , BrH + , OH-
szO
[CONTRIBUTION FROM
Vol. 86
5 -16 20 24 -32
2Ag+, S01-'
H', HCOaCa+',
*2 f4 f4 i4
a From Table I ; some data are available in the literature on the solubility of AgCl up to 200". However, at the present time it is almost impossible to account quantitatively for the hydrolysis of A g + at these high temperatures under the conditions of the experiments. * Calculated from eq. 7 using the constants given in Table 111.
DEPARTMENT OF CHEMISTRY, PURDUE UNIVERSITY, LAFAYETTE, INDIANA]
The Thermodynamic Properties of High Temperature Aqueous Solutions. V. The Calculation of Ionic Heat Capacities up to 2CO'. Entropies and Heat Capacities above 200' BY CECILM. C R I S SAND ~ J. W. COBBLE RECEIVED MAY15, 1964 Equations are given for calculating heat capacities at 25" and at elevated temperatures from the correspondence principle for entropies previously developed.
c,'
Introduction Probably the most useful function for predicting the thermodynamic properties of electrolyte solutions a t higher temperatures is the partial molal heat capacity, C,,', as a function of temperature. Consider the situation when it is desirable to know the free energy of a reaction a t some elevated temperature, t 2 , when it is known a t tl
lytes. Furthermore, although a greater number of values are known a t just one temperature, 25', such data cannot be reliably used a t other temperatures since in general varies considerably with temperature. In a previous communication, a correspondence principle for entropies for higher temperature solutions was developed. It immediately follows that whenever the entropy of an ion is known or can be accurately predicted a t two temperatures, the average value of the heat
c2'
12
capacity
between those temperatures can be
K O ] I1
calculated. The purpose of this communication is to extend the entropy correspondence principle to the calor combining terms
culation and prediction of I1
A ( A F o ) = -A.So(4,)AT
+ G o ]X
where
- Tz In
Go]'' is the average value of
and
cyo 1::for elec-
trolytes over extended temperature ranges.
I!
[AT
G,'
"1 TI
The Correspondence Principle and Heat Capacity Equations
(2)
AC,' between tl
11
and t2. The knowledge of AF' and AS' at one temperature and a value of the average heat capacity change between the two temperatures are sufficient to calculate A F ' ( f 2 ) . 3 However, there are so few data a t present on as a function of temperature that the method cannot be used for any significant number of electro-
c2'
( 1 ) Supported by a grant from the National Science Foundation. (2) Partly from the P h . D . Thesis of Cecil M. Criss, Purdue University, 1961 (3) Strictly speaking, Ep" in the linear temperature term will be somewhat different from the value of the function in the In T term. I n practice, this difference is not significant between 25 and ZOO', and in 50" steps at temperatures above 200°.
The linear correspondence of entropy for ions between 25' and t 2 can be written as4
SOU,)
= a(t,)
+ b(t,)So~dab~.)
(3)
So2,(abs.)refers to the ionic entropies on the "absolute" scale; So2,(abs.) = S025(conventional)- 5.02, where 2 is the ionic ~ h a r g e . ~ The average value of the partial molal heat capacity between 25' and t z is (4) C .M. Criss and J. W. Cobble, J. A m . Chem. Soc., 8 8 , 5385 (1964). (5) Hereafter, entropies will be considered as being on an "absolute" scale unless specifically noted otherwise. (4)
Dec. 20, 1964
ENTROPIES AND HEATCAPACITIES ABOVE 200'
5391
TABLE I PARAMETERS ( EQ.6)" HEATCAPACITY -OH-
-Cations---L, o c .
8( 1 )
a(,)
60 100 150 200
35 46 46 (50)
-0.41 -0.55 -0.59 ( -0.63)
and anions-
-----Oxy
8(0
a(1)
-46 - 58
-0.28 0.000 -0.03 (-0.04)
-61 (-65)
a(,)
- 127 - 138 - 133 ( - 145)
anion-
---Acid
8( I)
1.96 2.24 2.27 (2.53)
H +b
oxy anions-
Go]';;
B( 1 )
ao)
- 122 - 135
3.44 3.97 (3,95) (4.24)
( - 143) ( - 152)
23 31 33 (35)
a Parenthetical values are derived from extrapolated values of the corresponding entropy terms from lower temperatures and are thus Entries in this column refer to the average value of the heat capacity for H + ( a q ) as derived from fixing the subject to greater error. entropy of H + ( a q ) at each temperature. Units are cal. mole-' deg.-'
TABLE I1
From eq. 3 and 4 it follows that tr
a(,,) - S026[1.000- b(,,)] In T2/298.2
(5)
BEST VALUES OF IONIC PARTIAL MOLALHEATCAPACITIES' FOR SOME COMMON IONS (CAL.MOLE-1 DEG. -1)
which can be rewritten as5t6
Ion
CPOlb
11.
where qr,)= q4)/(1n T2/298.2) and = - [1.000 bctl)J/(ln T2/298.2). Values of q t )and b(,) have been evaluated previously4 up to 200°, and the corresponding heat capacity constants, act) and /3(,), are given in Table I. From eq. 6 and the parameters given in Table I, combined with the heat capacities for pure substances,' the average value of the heat capacity change for a reaction over the temperature range 25' to t z can be calculated up to 200°, and AF' evaluated from eq. 1 over the same range. The only additionally required thermochemical information are AS'25 and A F'z~. In many cases, reliable estimates can even be made in lieu of experimental values of the entropy a t 25'. The reader is referred elsewhere for these previously published methods of estimating & o . s - 1 2 The great usefulness of this extension of the entropy correspondence principle is that it does not require any direct information on G,' for electrolytes a t any temperature. l f n
For convenience, values of C,,
up to 200' have 25
been estimated and tabulated for some common ions in Table 11. Ionic Heat Capacities at 25' The correspondence relationship can also be used in estimating point values of the ionic heat capacity, Cpto, a t a given temperature. Equation 6 suggests that as the average heat capacity is taken over smaller
1::
and smaller temperature intervals, Cplo -
a t that mean temperature, and
K , O
will approach
H+ Li + Na K+ Rb+ +
cs
+
T1+ Ag+ Be +a Mg f z Ca +z
Sr +2 Ba +2 Ra c u +2 Mn Cd +2 P b +1
(6) For reasons already discussed in ref. 4 , eq. 6 contains only an apparent error in explicitly omitting considerations of the mass and symmetry of the ions. (7) K . K . Kelley, U.S . Bureau of Mines Bulletin 476, U.S. Government Printing Office, Washington, D. C., 1949. (8) R. E . Powell and W. M. Latimer, J . Chem. P h y s . , 19, 1139 (1951). R o y . Soc. (London), A441, 80 (1957); (9) K. J. Laidler and C. Pegis, PYOC. K . J. Laidler, Can. J . C h e m . , 94, 1107 (1956). (10) R. E. Connick and R. E. Powell, J . Chem. P h y s . , 31, 2206 (1953). (11) J. W. Cobble, ibid., 41, 1443, 1446, 1451 (1953). (12) A. M . Couture and K . J. Laidler, Can. J. C h e m . , SO, 202 (1957).
54
52 44 40 59 57 54
HSOa-
44 89 88 86 72 79 98 93 56 59 - 57 -59 -61 -62 -57 - 60 - 30 -40 - 56 -115 - 139 -226 - 151 -35b -43b
HCOaHapodHPO4-*
- 132b
~ 1 + 3
Cr +3 Fe +3 G d +I
sc Zr u uoz +I +3
+4
+4
Fe( OH) +*
Fc1BrIOH SH-
c104NO8 -
sod-¶ so3-2 POI-'
(7) I t is of interest to test this relationship a t 25' where sufficient data exist over a wide variety of ions. How-
28 41 37 31 30 26 28 35 76 62
cos -2 HSO4-
-48b
- 55b
23 36 35 27 25 24 25 30 62 51 45 43 38 34 49 47 45 37 72 71 70 59 64 78 75 47 49 -47 -51 -53 - 55 -47 - 52 -23 -33 -49 - 99 - 121 -200 - 132 - 13 - 16 - 27 -31 -118
31 47 41 35 32 31 32 39 82 67 59 57 50 44 64 63 60 49 95 95 93 78 85 104 100 63 64 -58 -58
-58 -58 - 58 - 58 - 25
-36 - 63 - 108 - 132 -216
- 146 - 10 -11 - 25 - 23 129
-
33 47 41 35 31 30 31 39 84 69 60 57 50
44 66 64 61 49 99 98 96 80 88 108 104 64 66 -61 -62 -61 -61 -61 - 62 - 25 -39 - 55 - 105 - 127 -205 - 138 - 18 -21 - 34 - 39 - 138
35 52 45 39 35 34 35 43 92 75 66 63 55 49 72 70 67 54 108 107 105 88 96 117 113 69 72 -65 - 66 - 65 - 64 -65 - 66 - 25 -39 - 57 -114 - 138 - 227 - 151 - 18 -21 - 35 -41 - 146
Calculated from eq. 6 and Table I; in some cases minor adjustments of a few cal. mole-' deg.-l in the heat capacities have been made to agree better with direct experimental determinations and t o smooth out the variation of heat capacity with temperature. These 25" values are simply recorded from temperature coefficient measurements of cell potentials alone and are subject t o large errors.
CECILM. CRISSAND J. W. COBBLE
5392 -0
6020
C p a (onions)
0
-20 -40 -60
-80 -100
-120 -140
-
Vol. 86
"absolute" scale a t 25' are plotted against the "absolute" ionic entropies. The predicted linearity probably is as good as the errors in the heat capacity data, considering that there are only a few salts for which the values of were obtained calorimetrically a t dilutions below 0.1 M.i4,15The parameters of eq. 7 are tabulated in Table I11 for the ion types known a t 25' and are useful in the estimation of G'z,.
G,'
TABLE I11 HEAT CAPACITY PARAMETERS AT 25" (EQ.7) C p t o ( ~ + , E 28 cal. mole-' deg.-l I
Ion
CO;-'
9 -10O t
\;.
1
\'
3-20
e uoi'
-30-40
-
-"4 -60
I
20
30
40
50
t&,
60
70
80
90
100
I
(catlonr)
diagram of Cpol,for ions with the absolute
-
A
41.6 -56.5 - 145 - 136
Cations OH-, anions Oxy anions Acid oxy anions
Fig. 1.-Correlation = 28 cal. mole-' deg.-l; So,r(~+.sq) = entropy. CP210(~+.aq) -5.0 cal. mole-' deg. Half-filled circles represent values based solely upon equilibrium constant data and are to be considered as preliminary estimates.
B
-0.523 0.179 2.20 3.07
Estimation of Ionic Entropies and Heat Capacities above 200" The estimation of heat capacities for ionic solutes above 200' must be based a t the present time purely on some method of extrapolation from lower temperatures. There are a number of ways this can be done, although i t should be emphasized that all are subject to greater errors than a t temperatures below 200'. The first most obvious method is to calculate ACp 115
for a given reaction a t 100, 150, and 200' and, by inspection, extrapolate this function to higher temperatures. If there are ions present on both sides of the reaction, then ACp will tend to change more slowly than f
ever, some further choice must first be made in order to separate the experimental heat capacities into ionic values. The previous assignment of the entropy of H+ a t each temperature can be used to define the ionic heat capacity a t a given temperature. This can be done by noting that
If the absolute entropy of H+(aq) is plotted against temperature, the slope of the resulting curve a t some temperature t is
In the limit, as tl
+
tl
(dS/dt) was evaluated13 from the previously4 assigned ionic entropies of H+(aq) as 0.0922; a t 25', the "absolute" ionic heat capacity of H+(aq) from eq. 10 then becomes 28 cal. mole-' deg-l. In Fig. 1, the known14ionic heat capacitydataon.this (13) T h e graph of t h e entropy of the hydrogen ion against temperature is actually sigmoid in nature; the curvature is particularly marked in t h e vicinity of 60' where most electrolyte heat capacities show maxima, and below 25O where changes rapidly with temperature. T h e curvature may, however, be a n artifact arising from errors involved in evaluation of the absolute ionic entropy a t each temperature. T h e best t h a t can be done a t present is t o obtain only a n approximate value for t h e slope a t 25'. Fortunately, the test under discussion is not very sensitive t o t h e exact value of (dS/dl). (14) Partial molal heat capacity d a t a a t 25O are from previously summarized values and from newer d a t a collected in these laboratories a s fol-
Go
the individual
values.
K O ]
25
A better method is to extrapolate the entropy par a m e t e r ~of~eq. 3 to higher temperatures. I t had been found previously that these parameters vary approximutely linearly with temperatureI6 over the 100- 200' lows: NaCI, -18.9; BaCIz, -72.6 (ref. 15); HCL, -30.5; NaReOl, 8.8 [J. C. Ahluwalia and J. W. Cobble, J. A m . Ckem. S O L . 8. 6 , 5377 (1964)l; CsI, -36.6 [R. E. Mitchell and J. W. Cobble, i b i d . 8 6 , 5401 (1964)l; GdClr, -105 [E. C. Jekel, C. M. Criss, and J. W. Cobble, i b i d . , 86, 5404 (1964)l; KCI, -27.7; N a I , -23.4; LiCI, -15.8; K O H , -25; K B r , -28.9; KNOa, -15.5 [G.N . Lewis and M. Randall, "Thermodynamics," 2nd Ed., revised by K . S. Pitzer a n d L. Brewer, McGraw-Hill Book Co., New York, N. Y.,1961, p. 6521. Further ionic values, based upon for H * e 0, are given in the Lewis and Randall text, p. 400, a s follows: Mg+2, 59; F-, -57.5; S 0 1 - 2 , 5 2 2 ; CIOa-, -46. These values were 28 before they were entered converted t o the present scale of CPom for H + on Fig, 1. T h e heat capacity of UOz+*is from t h e UOzClz(aq) d a t a of A. F Kapustinskii and I. I. Lipilina, Doki. A k a d . N a u k SSSR (Engl. Transl.), 104, 264 (1955); the revised entropy of UOz'2 given by M . H . Rand and 0. Kubaschewski, "The Thermochemical Properties of Uranium Compounds," Oliver and Boyd, L t d . , London, 1963, p. 12. I t is perhaps worth noting t h a t i t is difficult t o find a complete and consistent set of ionic heat capacities involving both newer a n d older data. Indeed, in t h e text referred t o above, there are two sets of entries for both NOa- and O H Newer values and more recent concentration extrapolations of older d a t a have been given preference in Fig. 1. However, part of the scatter may be due t o this unavoidable use of the d a t a , Consequently, errors of u p t o five heat capacity units may be involved in some cases. D a t a on COa-land the acid oxy anions are from potentiometric determinations of t h e respective dissociation constants and their temperature coefficients as follows: HIPOI- (from HaPOd, HPOd-2 (from HzPOi-), and HSOI- (from SO,-2) [ H . S. Harned and B. B. Owen, "The Physical Chemist r y of Electrolytic Solutions," 3rd E d . , Reinhold Publishing Co., New York, N. Y . ,1958, p. 6671. T h e required CJ2" for HaPOi(aq) was estimated a t 25' t o be 24 cal, mole-1 deg.-' from the d a t a of E . P. Egan, J r . , B. B. Luff, and Z. T . Wakefield, J . P h y s . Chem., 62, 1091 (1958). T h e and HCOa- values are from the Harned and Owen reference, pp. 692 and 7-58, Heat capacities from temperature coefficient studies of equilibrium constants are, of course, subject t o even much greater errors t h a n t h e calorimetric data. (15) C . M . Criss and J. W. Cobble, J. A m . Ckem. S O C .89, , 3223 (1961). (16) I t should perhaps be emphasized t h a t this functional dependence of at11 and b(1) at higher temperatures implies an ionic heat capacity propor-
ENTROPIES AND HEATCAPACITIES ABOVE 200'
Dec. 20, 1964
5393
TABLE IV ESTIMATED ENTROPY PARAMETERS ABOVE 150"" ,--Cations--f,
oc.
200 2 50 300 a
-Anions
and OH---
-Oxy
anions-
--Acid
SOH+
a(t)
b(t)
a(t)
b(r)
a(t)
b(t)
11.1 16.1 20.7
23.3 29.9 36.6
0.711 0.630 0,548
-30.2 -38.7 -49.2
0.981 0.978 0.972
-67.0 -86.5 - 106
2.020 2.320 2.618
oxy anionst)
-70.0 -90.0 ...
btr)
2.960 3.530 ...
For eq. 3, in cal. mole-' deg.-'.
range. If one is willing to risk assuming that this approximate linearity will continue above 200°, then the entropy parameters given in Table IV can be estimated. Because of the fact that the linear AT and A log T averages of Goin eq. 1 are not the same over too great a temperature interval, calculations of the free energy above 200' should be carried out in 50' intervals." Values of the entropies for the ions involved in a given reaction can be calculated a t 200, 250, and 300'. These values can then be used in turn to generate the corresponding ionic heat capacities over 50' intervals.
1::
The resulting series of ACp
above 200' can finally
be used to estimate AF(Q using eq 1.
Discussion It is of interest to inquire into the assignment of ionic heat capacities which have resulted from the treatment just presented. The general correspondence principle itself does not provide any direct indication of either the method or validity of assigning ionic entropies from solute entropies. While it is true that the entropy (and thus heat capacity) of H+(aq) is fixed by trial and error a t each temperature in order to obtain a linear correspondence in accordance with eq. 3, this is not necessarily a unique assignment. Indeed, as originally formulated, tional t o the absolute temperature. However, this approximation does not follow from t h e correspondence principle. T h e latter is perfectly general and can accommodate almost a n y functional heat capacity variation with temperature. Indeed, the values given in Table I correctly reflect the maximum in t h a t salts show near 60°,and t h e evidence clearly indicates t h a t over large temperature intervals the ionic heat capacities will be a complex function of temperature. (17) Still another source of error can arise from not considering t h e effect of solvent pressure on AF a t t h e highest temperatures. Fortunately, in most cases this will probably not be significant below 300° and will be discussed further in the communication following this paper.
G2'
the entropies could be related through some other, nonlinear type of function, in which case the entropy of H+(aq) would probably take on different values at the elevated temperatures. Nevertheless, one can still inquire into possible reasons that might give cations positive heat capacities and anions negative values. In the entropy expression which results from the Born equation there is always a dielectric term, (1/D2)(dD/dt). In the near vicinity of an ion there is question even with regard to the sign of this term. Because of the different orientation of solvent water molecules around cations and anions, it may be possible for the dielectric term to be either positive or negative. As the temperature changes, the contribution of the dielectric term may thus be of different sign, and the resulting heat capacities could also be of different sign. Another factor which is always present is a change in hydration of the ions. Because the entropy of unbound water molecules probably increases with temperature faster than for the bound species, ions of even constant hydration number will have decreasing entropies (negative heat capacities) with increasing temperature. Thus different temperature hydration effects of cations and anions may also give rise to differing signs for the anion and cation heat capacities. It may be noted that the present ionic heat capacity divisions are in qualitative agreement with those obtained from purely theoretical reasons by others.I8 The question can probably be resolved by repetition of thermocell experiments a t higher temperatures. Although there is no satisfactory means of separating the absolute ionic entropy and ionic entropy of transfer which results from such experiments, the entropy of transfer might not be very temperature sensitive. Consequently, the trend of the absolute entropy of an ion with temperature could possibly be discerned and would be of great help in making proper assignments of absolute ionic entropies a t higher temperatures. (18) M Eigen and E Wicke, 2 Elekfrochem., 66, 354 (1951,)
