4704
J. Phys. Chem. 1082, 86,4704-4708
internal problem in the Kerr effect to the same problem in depolarized light scattering. There appear to be possibilities of extending the methods used here to several related problems. The principal uses of Kerr-effect relaxation have been to study relatively slow motions as of biopolymers or of smaller molecules in highly viscous states. Better time resolution for faster processes should be possible by various combinations of sine wave and pulse excitations. Analyses of such situations by our method will be discussed in another paper, as will possibilities of extending the analysis to higher-order field effects for both electrooptic and dielectric responses. Acknowledgment. I was introduced to the Kerr-effect problem by Professor G. Willams and thank him for sev~
~~
~
(13)Keyes, T.;Ladanyi, B. Mol. Phys. 1979,37, 1643.
eral subsequent discussions. The work was supported by NSF Grant CHE-7822209.
Appendix The manipulations to obtain eq 15 and 16 from Z both involve the rule for differentiating a product: D(uu) = UDV + UDU,where D is either the Liouville operator Lo= d/dt for u and u not explicit functions of time or d/dps. As the equilibrium ensemble average of d(uu)/dt vanishes and f do= - Pfo(dH0/dt)= 0, repeated application of the rules gives
(fouLo"u) = (fo[(-1)"~c,"ulu, from which eq 14 follows on remembering the series expansion of the exponential. Using the rule with D = d/dpo then gives eq 15 and 16 as the integrated part vanishes at the limits p e = --m and p s = m.
Self-Ionlzation of Water at High Temperature and the Thermodynamic Properties of the Ions Kenneth S. Piker Department of Chemism and Lawrence Eerkeley Laboratory, University of California, Eerkeley, California 94720 (Received: June 2, 1982)
It is shown that gas-phase data on hydrated H+and OH- ions from mass spectrometry can be used to calculate the ionization product for water at high temperature and at high enough pressure to allow relating these results with those directly measured near 1000 K and 0.5 g ~ m - The ~ . thermodynamic properties of the hydrated H+ and OH- are discussed and the heat capacity is compared with results calculated from the Born equation for an appropriate region of temperature and pressure.
The ionization product for water has been investigated over a wide range of temperature, and Marshall and Franck' (MF) have recently proposed an empirical equation after a careful review of the various direct measurements. This equation appears to be an excellent representation of these data which are for densities above 0.4 g ~ m - ~It . is interesting to consider the behavior of the ionization product at lower density than 0.4 g in the range near or above the critical temperature. The Marshall and Franck equation can be extrapolated into this region; indeed Gates, Wood, and Quint2have used such extrapolated values to discuss the behavior of the partial molal heat capacity of ions near the critical point. But long extrapolations are dangerous. In this paper the properties of gaseous, water-related ions3are used to calculate the ion product for steam. These results appear to be valid at pressure gradually increasing with temperature until the curve can be connected with the directly measured experimental values of Quist4 at 1073 K and 0.5 g ~ m - ~ At. lower temperatures one can interpolate between these ~
(1)W.L. Marshall and E. U. Franck, J. Phys. Chem. Ref. Data, 10, 295 (1981). (2) J. A. Gates, R. H. Wood,and J. R. Quint, J. Phys. Chem., in preaa. (3)P. Kebarle, Annu. Rev. Phys. Chem., 28, 445 (1977);"Modern Aspects of Electrochemistry", Vol. 9,B. E. Conway and J. OM. Bockris, Ed!., Plenum Press, New York, (1974,p 1, and references cited in these reviews. (4)A. R.Quiet, J. Phys. Chem., 74,3396 (1970). 0022-3854/82/2086-4704$01.25/0
newly calculated curves at lower density and the equation of Marshall and Franck for the high-density region. These interpolated curves, although still somewhat uncertain, should be much more reliable than extrapolations considering only the high-density data. The thermodynamic properties, including the partial molal heat capacities, of the ions can be calculated from the ion product equation. These properties are discussed in comparison with calculations from the Born equation and the earlier results of Gates et a1.2
Gaseous Ion Equilibria The thermodynamic properties of gaseous H+ and OHand their hydrated ions are known. The equilibria between the successive hydrate ions were measured, primarily by Kebarle and his a s s o ~ i a t e s , with ~ ~ ~ a' mass spectrometer having a relatively high pressure of H20 in the ion source. The first hydration of the proton is complete under all conditions of present interest; hence, we consider as the initial reaction 2H20 = H30++ OH-
K1 = P H s 0 + P O H - / P H 2 0 '
(1)
( 5 ) Y. K. Lau, S. Ikuta, and P. Kebarle, J. Am. Chem. SOC.,104,1462 (1982). (6) M. Arshadi and P. Kebarle, J. Phys. Chem., 74,1483 (1970). (7)J. D.Payzant, R. Yamdagni, and P. Kebarle, Can. J. Chem., 49, 3308 (1971).
0 1982 American Chemical Society
The Journal of Physical Chemi$try, Vol. 86, No. 24, 1982 4705
Self-Ionization of Water at High Temperature
TABLE I : Enthalpy and Entropy Changesa for Successive Hydration of H,O' and OHn
-ASop/ (cal K T*/K mol-')
-APT*/
(kcal mol-')
model
~
a
Ions 31.6 19.5 17.9 13.0-(n-4) 1 3 . 0 - 0.8(n- 4 )
3
700 500 380
Positive 24.3 21.7 28.4
4t
250
25'0
1 2 3 4+
600 500 400 330
Negative Ions 20.8 25.0 21 17., 25 15 14 - ( n- 4 ) 25 14 - 0.8(n - 4 )
1 2
A B
A
+
B
Values from ref 3 and 5-7.
for which3" Moo = 222 kcal mol-'. The partition function product for H30+and OH- should be almost exactly the same as that for NH3 and HF which can be found, as -(Go - Hoo/RT), in appropriate table^.^ Thus, one has the equilibrium constant K1 for reaction 1. But with increasing pressure of H 2 0 most of these ions are further hydrated H30(HzO)n-1++ H 2 0 = H30(H20),+,K,' OH(H20),
