J. Phys. Chem. 1982, 86, 4948-4951
4948
Experimental Evidence for the Remarkable Behavior of the Partial Molar Heat Capacity at Infinite Dilution of Aqueous Electrolytes at the Critical Point Jeffrey A. Gates, Robert H. Wood,’ Department of Chemistry, University of Delaware, Newark, Delaware 1971 1
and Jacques R. buint Laboratoire de Thermodynamique et Cinetique Chimique, Universl de Clermont-Fenand 2, Aubsre, France (Received: April 26, 1982, I n Final Form: August 18, 1982)
Marshall and Franck have given an equation for the ionization constant of water as a function of temperature and density. The partial molar heat capacities and volumes at infiiite dilution and for the “electrolyte” H+(aq)+ OH-(aq) have been calculated from their equation. The calculated values of the partial molar heat capacity show very large and negatiue values just below the critical point of water, but very large and positive values just above the critical point. For the partial molar volume, the calculated values are very large and negative near the critical point. At higher pressures, the partial molar heat capacities and volumes show the same qualitative behavior, but fluctuations are less extreme. This is the first experimental evidence for very large and positive partial molar heat capacities above the critical point. Calculations show the Born equation predicts the same qualitative behavior found for H+(aq)+ OH-(aq). The shape of the curve and the temperatures of the maxima and the minima as a function of pressure are correctly predicted. However, quantitative prediction with a single radius is not possible. This failure is not surprising, since the Born equation is a continuum model which not only neglects the molecular nature of the solvent, but also neglects the compressibility and dielectric saturation of the solvent.
voz)
(c;z
Introduction A recent corresponding states treatment of the properties of aqueous electrolytes predicts very large negatiue values of the partial molar heat capacity of dilute solutions of electrolytes just below the critical point of water.’ This is in accord with the experimental findings. The same treatment also predicts very large but positive values of the partial molar heat capacity just above the critical point of water. Similar predictions for nonelectrolyte solutions have been made2i3but the sign of the effect depends on the solute-solvent interaction energy and the effects are much smaller. With electrolytes the effects are very much larger, presumably because of the large solute-solvent interaction energy (or alternately, because of the large shift in critical point produced by a small amount of electrolyte). This paper demonstrates ( 1 ) the experimental* heat capacity of H+(aq) + OH-(aq) has the predicted behavior, and ( 2 ) the Born equation5 predicts this behavior. The volume of ionization can also be calculated from the experimental data and confirms the behavior predicted by the Born equation. Calculations Volume and Heat Capacity of lonization of Water. The ionization constant of water has been measured by a variety of authors a t temperatures from room temperature to 1000 “C and pressures from 0.1 to 100 MPa. All these data have been correlated with high accuracy by a relatively simple equation of Marshall and F r a n ~ k .Their ~ equation is log [K,/(mol k ~ - ’ ) ~=] A + B / T + C / P + D / F + ( E + F / T + G / P ) log b / ( g ~ m - ~ (1) )l where p is the density of water and K , is the equilibrium (1) Quint, J. R.; Wood, R. H. J . Chem. Thermodyn., accepted for publication. (2) Wheeler, J. C. Ber. Bunsenges. Phys. Chem. 1972, 76, 308. (3) Rozen, A. M. RUSS.J . Phys. Chem. 1976, 50, 837. (4) Marshall, W. L.; Franck, E. U. J . Phys. Chem. R e f . Data 1981,10, 295. ( 5 ) Born, M. 2.Phys. 1920, 1 , 45. 0022-3654/82/2086-4948$01.25/0
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constant for the reaction, HzO(l) H+(aq) + OH-(aq), and the parameters of the fit (A,B, C, D, E, F, and G ) are given by Marshall and Franck. The enthalpies, heat capacities, and volumes of ionization are calculated by taking the appropriate derivatives of eq 1. The results are AH = -R[d In k , / d ( l / T ) ] , = -R[2.303(B 2 C / T 3 D / P ) ( F 2 G / T ) In p ( E P + FT + G ) ( a p / a ~ ) , / ~ i( 2 )
+
+
+ +
AC, = [aAH/dT], = -R(-2.303(2C/P + 6 D / P ) 2G In p / P ( 2 G / T - 2 ~ ~ ) ( a ~ / d T-) (, E/ ~P FT + G)[(aZp/aP),/p - ( a p / a n P 2 / p 2 i i (3)
+
+
A v = -RT(d In k,/dP)T =
-R(ET + F + G / T ) ( a p / W T / p (4) The density of water was calculated from the equation of state given by Keenan and Keyes.6 Also, since the heat capacity and volume of liquid water are known, these can be added to AC, and AV to obtain the partial molar heat capacity and the volume (Voo2) of the ions H+(aq) + OH-(aq) in their standard state. It is normally very risky to calculate heat capacities and volumes from temperature and pressure dependence of equilibrium constants. However, the wide range of temperature and pressure of the experimental data4 gives us confidence that our values are accurate (at least 110%). Since there are no data very close to the critical point and there are no data below 50 MPa at temperatures above the critical temperature, the volumes and heat capacities calculated from eq 3 and 4 are less accurate in this region. The Born Equation. The Born equation for the electrostatic contribution to the free energy of hydration of a mole of salt composed of hard-sphere ions in a continuous dielectric medium is6
(cPoz)
AGe’ = -(Le2/SmO)(R+-l+ R--l)(l - 1/D)
(5)
(6) Keenan, J. H.; Keyes, F. G.; Hill, P. G.; Moore, J. G. ‘Steam Tables”; Wiley: New York, 1978.
0 1982 American Chemical Society
C, at
The Journal of Physical Chemistry, Vol. 86,No. 25, 7982 4949
Infinite Dilution of Electrolytes
TABLE I: Experimental and Predicted Values for
-
cpoz and P, for H+(aq)and OH-(aq) a t 22.1 MPa -
Cp",i( J K-l mol-')
TiK
expP
400 450 500 550 600 645 646
-171 -238 -396 -850 -2930 -6.6
X lo5 -1.9 x 106
1 . 7 X lo6 7.4 x 105 3770 1000 410 190 84 22 -18
649 650 700 750 800 850 900 950 1000 a
Experimental values for H+(aq)+ OH-(aq).
V",/(cm3 mol-')
Bornb
exp t
Born
-100 -155 -283 -689 -2930 -9.9 x 105 -3.0 X l o 6 Critical Point 4.6 X lo6 2.1 x 106 1.6 x 104 5400 2900 1900 1300 960 730
-8.20 -18.0 -37.8 -85.0 -251 -6710 -11600
-8.3 -14.2 -26.9 -60.7 -204 -8870 -16400
-31700 -23800 -5770 -4960 -4750 -4720 -4790 -4900 -5040
-92300 -71000 -15500 -10900 -8650 -7260 -6310 - 5630 -5110
Predicted by the Born equation with R = 0.228 nm.
where L is Avogadro's number, e is the protonic charge, eo is the vacuum permittivity, and R+ and R- are the radii of the cation and anion. Taking appropriate derivatives eq 1, we find AHe' = - ( K / R * ) ( l - 1 / D - T(dD/aT),/D2) Ac,e' =
(6)
cpo2 =
-(K/ R*)(2 T ( a D/ dT);/ D3 - T(a2D/aT2),/ D2] (7) AVe' = 9'2 = - ( K / R * ) ( ( d D / d P ) T / D 2 }
(8)
where K = L e 2 / 4 m oand 1/R* = (R+-l+ R--l)/2. We set AC;l = CPo2and AVel = Vo2because the electrostatic contribution to the heat capacity and the volume of the ions is zero in the gas phase. We compare values of and P2calculated by eq 7 and 8 with the experimental values for H+(aq) OH-(aq). For these calculations, we used the equation of Uematsu and Franck7 for the dielectric constant of water as a function of temperature and density, together with the equation of Keenan and Keyes.6 For these calculations we use
+
( a D / a v , = (warn + (aD/ap),(ap/an,
(awn
= (swap),(ap
/wT
(9)
(10)
where the partial derivatives are calculated from (1)the equation of Uematsu and Franck7 for the dielectric constant as a function of p and T and (2)the equation of state for pure water. T h e Born Equation Near the Critical Point. It is worthwhile pointing out here that the Born equation predicts infinities in the enthalpy, volume, and heat capacity of hydration of electrolytes a t the critical point of a fluid, provided that the dielectric constant of the fluid is a continuous function of temperature and density, with continuous derivatives with respect to temperature and density. The experimental data, as well as the equation of Uematsu and Franck,' show that the dielectric constant is a smooth and continuous function of temperature and density (although not of temperature and pressure), so that it is likely that these necessary conditions are satisfied for water and for many (if not all) solvents. The enthalpy of hydration depends on (aD/aT), and this partial derivative is calculated with eq 9. A t the critical point (dD/aT), is minus infinity because ( a p / a T ) , (7) Uematau, M.; Franck, E.U.J.Phys. Chem. Ref. Data 1980,9,1291.
is minus infinity and (dD/ap)Tis positive for water (and also presumably for most, if not all, solvents). This leads to the prediction that AHe' goes to minus infinity at the critical point (eq 6). Similarly, AVe' goes to minus infinity at the critical point (eq 8) because (dD/dP), goes to plus infinity and (ap/dP), goes to plus infinity at the critical point (eq 10). The heat capacity prediction is not so obvious. The behavior of ( d 2 D / d P ) ,at the critical point must first be understood. Certainly, (dD/dT), has an isolated singularity at the critical point because (dp/dT), has an isolated singularity at the critical point. We assume that AC, is well-behaved near the critical point, where by well-behaved we mean that the limits, lim, Tc; ( d 2 D / a P ) , and lim, + T + (a2D/dT2),, ~ ~ exist. From this and the minus infinity of (dD/dT), a t the critical point, it is easily shown that (dzD/d?"?), approaches minus infinity from below and positive infinity from above the critical temperature. (Similar statements hold for any well-behaved functions near isolated singularities.) The analysis of AC, behavior begins with a rearrangement of eq 7.
-
CPo2= ( K / R * ) ( T / D 2 )X (ao/an,[(aIn i ( a D / m , i / a n , - 2(a In D/~T),I = (KT/R*D2)(aD/aT),(a In i(awaT),/o2i/aTIP (11) It has already been assumed that (dD/dT), is well-behaved and it is known that the dielectric constant is finite; thus In )(aD/dT),/D21 must be well-behaved so that [a In I(dD/dT),/D21/aTl, approaches positive infinity from above the critical point. Thus, eq 11 shows that CPo2 approaches minus infinity as the temperature approaches the critical point from below and positive infinity as the temperature approaches the critical point from above. Results The calculated and experimental results for the heat capacity and volume of H+(aq) + OH-(aq) at infinite dilution are given in (Table I) and plotted in Figures 1 and 2. The Born radius (0.2285 nm) was chosen to fit the low temperature data. The results are plotted at a pressure of 22.1 (the critical pressure of water), 50, and 100 MPa. The experimental results calculated from the equation of Marshall and Franck at the critical pressure, 22.1 MPa, exhibit the infinities predicted by the Born equation near the critical point. This verifies the predicted spectacular shift from a very large and negative heat capacity just below the critical point to a very large and positive heat
4950
Gates et ai.
The Journal of Physical Chemistry, Vol. 86, No. 25, 1982
c
0-+l
IY
\
L
0
E
U \
O N
I>
-1
!: I i.1: I Tc;: I
300
500
700
-2
900
I
0
TIK
TI,
500
700
0
TIK
I
m5 O N
I>
t
1 -4 -4
I
I
I
t
Tc
-20
li 1
I
" I
-8-
I
300
500
TIK
r
. 7
F?
IO
I
I
I I
700
500
TC
900
-2
T/K
cpoz
900
I
t 300
700 TIK
+
Figure 1. Plot of [H+(aq) OH-(aq)] vs. temperature: Marshall and Franck; (----) Born prediction; (a) pressure = 22.1 MPa; (b) pressure = 50 MPa; (c) pressure = 100 MPa. (..e.)
capacity above the critical point. Above the critical point this comparison depends upon extrapolations using Marshall and Franck's equation outside of the range of the experimental data. The experimental data extend down to K , 5 above the critical point and so the curve in
I
I
TIK Figure 2. Plot of Poz[H+(aq) OH-(aq)] vs. temperature: (Marshall and Franck (- - - -) Born prediction; (a) pressure = 22.1 MPa; (b) pressure = 50 MPa; (c) pressure = 100 MPa.
+
.
e)
Figure l a is an extrapolation by eq 1to values of K, = at 700 K and loTz4at 900 K. However, at 50 MPa there is no extrapolation at 700 K and a much smaller extrapolation at 900 K. Figures l b and 2b show that the same
C, O at Infinite Dilution of Electrolytes
The Journal of Physical Chemistry, Vol. 86, No. 25, 1982 4951
Discussion Comparison of Calculated and Experimental Results. To our knowledge, this is the first experimental evidence for the very large and positive values of partial molar heat capacity above the critical point. Murray and Cobble8were the first to demonstrate the very large and negative values of the partial molar heat capacity below the critical point at the saturation pressure. Smith-Magowan and Woodg showed that the Born equation predicted that these large negative values would not be observed at higher pressures. The present results are experimental confirmation of the prediction. The very large negative values of Pz for salts near the critical point was first observed by Benson, Copeland, and Pearson.lo Their results for NaCl clearly show the same kind of large minimum in Vo2near the critical point found for H+(aq) + OH-(aq). Kirkham and Helgeson'l have previously used the Born theory (with additional terms for the "chemical effects") to extrapolate volumetric data above 200 "C and clearly show the very large and negative values of Pzpredicted by the Born equation near the critical point. Kirkham and Helgeson were even able to predict P2 for H+(aq) + OH-(aq) (derived from the ionization constant) at supercritical temperatures and 100-500 MPa. Helgeson and Kirkham's predictions are very similar to those presented here because the "chemical effects" are small compared to the electrostatic effects and their radius is only 16% smaller. Thus, their predictions are also not accurate at low pressures (Figure 2). More recently, Lukashov12gave an electrostatic treatment of AG (also AS, AV, and AC,) of ionization with the water molecule described as an ion pair. This treatment
leads to difficulty in defining the distance of approach of H+and OH- in a water molecule. The present treatment uses an electrostatic term for the ions only (instead of for the ionization reaction), and subtracts the known C'p02 and Vo2for water to obtain values for the "electrolyte" H+(aq) + OH-(aq). Lukashov's values for AC,_of ionization show the same qualitative behavior as our CPo2and Vo2. Reasons for Inaccuracy of the Born Equation. The results presented here show that, although the Born equation gives the correct qualitative features of the curve, it fails in quantitative predictions. Since the Born equation is a continuum model, it neglects the molecular nature of the solvent. In addition, the Born equation is not an accurate continuum model because it also neglects compressibility13J4 and dielectric saturation of the solvent. Compressibility is infinite at the critical point and very large above the critical point. Recent calculations by Wood, Quint, and Grolier14 have shown that the compressibility effects on the heat capacity are quite large below the critical point. Their results also indicate that dielectric saturation is an important effect. It will be important to see whether a continuum model which takes into account compjessibility effects and/or dielectric saturation effects will fit the experimental results. Additional experimental work on other electrolytes above the critical point is needed. Other Ionization Equilibria. Marshall15has used eq 1 with remarkable success to represent the density and temperature dependence of the equilibrium constant for a wide variety of ionization reactions. If the heat capacity of the un-ionized salt is neglected, thsse equations predict the same qualitative behavior for CPo2and Po2 as that shown in Figures 1 and 2, and this same qualitative behavior is in accord with the predictions of both the Born equation and corresponding states theory.' These equations may be successful in part because (1) density is a more fundamental variable than pressure, and (2) log p is used in the equation. We note that thermodynamic properties are functions of the radial distribution function which is a function of the energy of interaction between molecules which, in turn, depends directly on the distance between molecules. Thus, the volume (which is a measure of the average distance between molecules) is a more fundamental variable than the pressure. Also, the use of a term proportional to log p incorporates into the equation a term having the correct behavior near the critical point.
(8)Murray, R. C., Jr.; Cobble, J. W. Discuss. Faraday SOC. 1977,64,
Acknowledgment. This work was supported by the National Science Foundation under Grant CHE 8009672.
qualitative behavior is found although, since we are now above the critical pressure, the infinities have become large but finite maxima and minima. The Born equation predicts the qualitative behavior (shape of plot and temperature of minimum and maximum) of Cpozand Vo2.The prediction is too large above the critical temperature. The Born equation predicts these same qualitative features will be present for any strong electrolyte near the critical point of water. In addition, most (if not all) other solvents should show the same smooth dependence of the dielectric constant on density and temperature, so that the same effects should be present in most (all?) solvents at their critical points.
144. (9) Smith-Magowan, D.; Wood, R. H. J. Chem. Thermodyn. 1981,14, 15-26. (10) Benson, S. W.; Copeland, C. S.;Pearson, D. J. Chem. Phys. 1953, 21, 2208. (11) Helgeson, H. C.; Kirkham, D. H. Am. J . Sci. 1976,276, 97. (12) Lukashov, Yu. M. Russ. J. Phys. Chem. 1980,54, 792.
(13) Franck, H. S. J. Chem. Phys. 1955,23, 2023. (14) Wood, R. H.; Quint, J. R.; Grolier, J.-P. J.Phys. Chem. 1981,85, 3944. (15) Marshall, W. L. Rec. Chem. Prog. 1969,30,61. Chem. Geol. 1972, 10, 59, J. Phys. Chem. 1972, 76, 720.
