J . Phys. Chem. 1990, 94, 7834-7839
7834
similar to those shown in Figure 6 can occur. Then the entropy is appreciable. Following Pauling, but with some new notation, we set W = W ,F, where Wl is the weight calculated neglecting some of the restrictions and the factor F is an estimate of the probability that a configuration is acceptable. Then for one mole, if SI = k In W , and S2 = k In F, S = k In ( W , F ) = SI + S 2 (6)
If the acid hydrogen is on O ( 4 ) with equal probability for two choices of phosphate orientation and three choices of hydrogen bond for each one, W, = 6Nand SI= R In 6 = 3.560 cal K-I mol-'. For each group of 12 water molecules there are 6 phosphate ions, each of which provides the acid proton with probability 1 / 6 . The fraction of these configurations that provide just one proton is ( 5 / 6 ) 5= 0.402. With the approximation that the probability for one group is independent of that for another, F = 0.402N,S2 = -1.81 I cal K-' mol-'. and S = 1.75 cal K-' mol-'. This value is about half thc experimental result. As mentioned above, there is evidence that each acid proton is on another oxygen atom a fraction of the time. More configurations like those in Figure 6 are possible. For molecular configurations of unequal probability it is convenient to express the molar entropy as S = - R X X i In Xi (7) I
where Xi is mole fraction of configuration i. With each acid proton distributed equally among 2 2 such structures 10% of the time, and 90% of the time on O ( 4 ) as above, SI = 4.464 cal K-I mol-I. A tabulation of the possible configurations and their probabilities for the eight neighboring acid phosphate groups of each water cluster (Figure 3 ) shows that 40.1 % of them provide exactly one acid proton to the water group. With the same approximation that this probability for one water group is independent of those for others, F = 0.401N,S2 = -1.816, and S = 2.65 cal K-' mol-'.
The agreement with the thermodynamic data is less than perfect, but not beyond possible experimental error. It is improved if the protons are more evenly distributed, but S cannot be increased above about 3 cal K-' mo1-I without serious conflict with the evidence of the bond distances. This model is not unique. The result is not very sensitive to modest variation of the various probabilities. Nor can we exclude the existence of other kinds of imperfection in small concentrations. However, we are confident that randomness among orientations of phosphate ions and among configurations like those in Figures 5 and 6 is the origin of most of the residual entropy. An item which invites further study is a report that the crystal structure of anhydrous N a 2 H P 0 , has proton disorder at room t e m p e r a t ~ r e , yet ' ~ the heat capacity for the range 10-320 K indicates no phase change or anomaly," and the evidence is convincing that there is no residual entropy in this phase at low temperature. Perhaps the crystals have a larger unit cell with additional reflections too weak to have been noticed in the diffraction experiment. For example, a doubled cell with a glide plane in place of the mirror plane (space group P 2 , / a , P2,/c, or P 2 , / n ) offers a structure that is free of disorder, but with nearly identical diffraction intensities for the reflections that were observed. Acknowledgment. We thank Dr. Lieselotte Templeton for assistance in preparing the figures. The initial part of this work was supported by the U S . Atomic Energy Commission. Registry No. Na2HP04.H20,118830-14-1; Na2HP04.2H20,1002824-7; Na2HP04.7H20,7782-85-6; Na2HP04.12H20,10039-32-4. Supplementary Material Auailable: Tables of anisotropic thermal parameters ( 3 pages) and tables of structure factors ( 3 5 pages). Ordering information is given on any current masthead page. (14)
Wiench, D. M.; Jansen, M . 2.Anorg. Chem. 1983, 501, 9 5 .
Polarized and Depolarized Raman Spectra of Liquid Carbon Disulfide at 0-10 k b a d 3. Interaction-Induced v2 and v3 Scattering and the Fluctuation of the Local Field S. Ikawa* Department of Chemistry, Faculty of Science, Hokkaido University, Sapporo 060, Japan
and Edward Whalley* Division of Chemistry, National Research Council of Canada, Ottawa, Ontario K1 A OR9, Canada (Received: January 11, 1990; In Final Form: April 12, 1990)
The effect of pressure on the interaction-induced Raman scattering of the forbidden u2 and u3 vibrations of liquid carbon disulfide has been measured up to 10 kbar at 295 K, and a simple model for the fluctuation of the local field has been used to describe the origin of the spectra. The scattering intensities increase with increasing pressure approximately in proportion to the density of the liquid. This is ascribed to the dominant contribution of the orientational fluctuations of the molecules to the fluctuation of the local field. The width of the u2 band decreases with increasing pressure, and its high-pressure limit is attributed to nondiffusional broadening. The diffusional line width, which was obtained by subtracting the nondiffusional part from the total line width, provided the diffusional time constant, which is linear in the viscosity of the liquid. A large part of the diffusion broadening is ascribed to the relaxation of the local field caused by rotational diffusion.
I. Introduction Pressure is an important variable for studies of the liquid phase because its effects help to elucidate the nature ofthe intermolecular interactions and to test theoretical models ofthe liquid. One direct manifestation of these interactions is the interaction-induced Raman scattering by molecular vibrations that are forbidden in
the isolated molecules and in the crystal. For example, the doubly degenerate bending u2 and the antisymmetric stretching u3 vibrations of carbon disulfide, which are Raman inactive in the isolated molecules, become weakly allowed in the liquid owing to molecular interactions,' but they are forbidden in the crystals because O f the inversion enter.^-^ Therefore, the interaction( 1 ) Evans, J . C.; Bernstein, H. J. Can. J . Chem. 1956, 34, 1127.
' N R C No. 31853.
0022-36S4/90/2094-7834$02.50/0
0 1990 American Chemical Society
Raman Spectra of Liquid CS2 at 0-10 kbar
The Journal of Physical Chemistry, Vol. 94, No. 20, 1990 7835
induced Raman scattering should tell us how the molecular environment is distorted or fluctuates in the liquid. Cox and Maddens-’ have measured the intensities and line shapes of the interaction-induced spectra of liquid carbon disulfide at various temperatures and in solution. By using the electric multipole model, they showed that the v 2 and v3 Raman bands were caused by the action of the reradiated field gradients from neighbors on the vibrationally modulated quadrupole polarizability. The same mechanism is dominant for carbon dioxide also, according to the a b initio calculation of Amos et aL8 Recently, we have measured the polarized and depolarized Raman bands of liquid carbon disulfide at pressures up to 10 kbar,9 and the present paper describes the effect of pressure on the v2 and v3 bands. The intensities are related to the density of the liquid by a simple model for the fluctuation of the local field that is produced by the electric multipoles. The line broadening under pressure is explained by the relaxation of the local field, which is caused by the rotational and translational diffusion of the molecules. In addition to liquid carbon disulfide, the interaction-induced or collision-induced Raman spectrum has been measured for the compressed gases and liquids of highly symmetric molecules such as carbon dioxide,’O.llcarbon tetrafluoride,I2 and sulfur hexaf l ~ 0 r i d e . l ~The scattering intensities of the gases were satisfactorily described by a model having two-body interactions. However, in the denser liquid phase, two-body interactions become inadequate because the intensities of the liquids are considerably lower than the values extrapolated from the gas-phase data.11*12 This is explained by the more or less symmetric configuration of the neighbors in the liquids, in which the pairwise interactions cancel to a considerable extent. One possible way to describe such a situation in the liquid is to consider the local structure as fluctuating around the average structure, which is assumed to preserve, e.g., the molecular inversion center for carbon disulfide, as in the crystalline phase. The interaction-induced intensity is, therefore, determined by the fluctuation of the local field, and the line broadening is related to the relaxation of the local field. This fluctuating-local-field model is an extension of the idea taken by Hyodo and FujiyamaI4 in order to explain the nonzero depolarization ratio of the Raman v I scattering of liquid CCI4. The isotopic substitution due to the natural abundance of 32S (95%) and 34S(4410) reduces the molecular symmetry to C,, and so makes the v j mode of C32S34Sallowed by mixing with the symmetric stretch. However, the v 3 scattering intensity due to this effect is estimated to be 3 orders of magnitude smaller than the observed value. If the molecule bends in the liquid phase, the symmetric stretching and bending modes are mixed and the v 2 scattering becomes allowed. If the observed intensity of v2 is wholly attributed to the effect of the bend, the angle of the bend is estimated at about 13O, even at 1 bar, which seems much too big. Perhaps the pressure affects the structure of the carbon disulfide molecule and so enhances the intensity of the v 2 band. 11. Theory of Interaction-Induced Raman Scattering The theory of electric m u l t i p ~ l e s is ’ ~applied to an assembly of molecules. The dipole and quadrupole moments of the ith
(2) Baenzinger. N . C.; Duax, W . L. J . Chem. Phys. 1968, 48, 2974. (3) Weir, C. E.; Piermarini, G. J.; Block, S.J . Chem. Phys. 1969,50, 2089. (4) Anderson, A.; Grout, P. J.; Leech, J . W.; Sun, T.Chem. Phys. Lett. 1973, 21. 9. ( 5 ) Cox, T.1.; Madden, P. A. Mol. Phys. 1980, 39, 1487. (6) Madden, P. A.; Cox, T.1. Mol. Phys. 1981, 43, 287. (7) Cox, T.1.; Madden, P. A . Mol. Phys. 1981, 43, 307. (8) Amos, R. D.; Buckingham, A . D.; Williams, J. H. Mol. Phys. 1980, 39, 1519. (9) lkawa, S.;Whalley, E. J . Chem. Phys. 1986, 85, 2538. (IO) Welsh, H . L.; Pashler, P. E.; Stoicheff, B. P. Can. J . Phys. 1952, 30, 99. ( I I ) Holzer, W.; Ouillon, R. Mol. Phys. 1978, 36, 817. (12) LeDuff, Y . ;Gharbi, A. Phys. Rev. A 1978, 17, 1729. ( 1 3) Holzer, W.; Ouillon, R. Chem. Phys. Lett. 1974, 24, 589. (14) Hyodo, S.;Fujiyama, T.Bull. Chem. SOC.Jpn. 1980, 53, 2456. ( 1 5 ) Buckingham. A . D. Ado. G e m . Phys. 1967, 12, 107.
molecule, in tensor notation, are p(i)
= po(i)
+ a ( i ) . E ( i )+ Y2j3(i):E(i)E(i) + y3A(i):F(i) + ... (1) O(i) = Oo(i)
+ A(i).E(i) + ...
(2) where po(i) and Oo(i) are the permanent dipole and quadrupole moments, respectively, a(i) is the polarizability, and B(i) and A(i) are the first hyperpolarizability and the dipole-quadrupole polarizability, respectively. The electric field and its gradient, respectively, at the position of the ith molecule are given by E(i) = E
+
T(2)(ij)-p(j)+ J(#I)
F(i) = -
y3 i(4
T(3)(ij):O(j)+
...
(3)
T(4)(ij):O(j)+ ... (4)
T(3)(ij)-p(j)- y3 j(#i)
J(#I)
where E is the electric field of the incident light and T(”)(ij)= Vn(1 /Ir(ij)l)
(5)
is the interaction tensor. An interaction-induced Raman tensor r(i)is defined by the equation
where qn(i) is the normal coordinate of the v 2 or the v3 mode. Using (l)-(5) and taking account of po = j3 = A = da/aq, = 0 for a Dmhtriatomic molecule, (6) becomessJ
(7) where only the leading terms are retained and the prime means the derivative with respect to qn. In the following discussion, we consider only the first term involving the A’ tensor, since it has been shown to be dominant, as mentioned previously. The intensity and the line shape of a polarization component are given by
where we and wo are the angular frequencies of the incident light and the center of the Raman shift, respectively, (O1qnll) is the matrix element of qn between the vibrational ground and excited states of the molecule, and the angular brackets denote an ensemble average. It has been assumed in (9) that the vibrational relaxation is independent of the rotational and translational diffusion of the molecule, which causes the relaxation of uag(t), so that the total correlation function is a simple product of both correlation functions. A . Scattering Intensity. We take the “fluctuating-local-field model” mentioned previously. The average configuration of the local structure of liquid carbon disulfide has an inversion center at the site of the molecule, as in the high-pressure crystalline phase adjacent to the liquid phase at room t e m p e r a t ~ r e .In ~ the average structure, the Raman tensor vanishes u,, = 0
(10)
because the electric fields from the surrounding molecules completely cancel. The thermal motion of the molecules induces fluctuations of the local structure and disturbs the inversion symmetry. Then the cancellation becomes imperfect and the Raman tensor nonvanishing. The orientational fluctuation of the molecules induces a fluctuation in the components of A’(i) and a(j)projected on the laboratory-fixed coordinates, and the fluctuation of the relative positions of the molecules perturbs the components of T3(ij). To a first approximation, the Raman tensor is given by the sum of the individual contributions r = O(A) + O(T) + O(a) (1 1) where
7836 The Journal of Physical Chemistry, Vol. 94, No. 20, 1990 O(A) = -y3AA’(i):
T3(ij)-aav(j)
( 1 2a)
AT3(ij)-aav(j)
( 12b)
Ti),)(ij).Aa(j)
( 12C)
Ikawa and Whalley Substitution of ( 1 8) into ( 1 5 ) yields
-
I(#[)
O ( T ) = -y3A’av(i):
(10udA)12)
J(#I)
O ( a ) = -y3A’Ji): I(#[)
The amplitudes of the fluctuations are related to the orientational and translational fluctuations, 6Q and 6r, of the molecules. The lowest-order terms are
a
AA’(i) = 6Q(i) -A’(i) an(i)
-
-
-
kETp (19a)
([Omo(T)12) kETr4 kETp-4/3 (19b) where p is the density of the liquid. Therefore, as the pressure increases, the contribution to the induced intensity by the orientational fluctuations increases, whereas that from the translational fluctuations decreases. B. Line Broadening. The broadening of the interaction-induced Raman band is caused by the relaxation of the local field, which is associated with the rotational and translational diffusion of the molecules. If the rotational motions of the molecules are mutually independent and the rotation-translation coupling is neglected, the time correlation function of the Raman tensor is written as .*,&))
- c ze y6f y’6‘d j
(
If a correlation in the orientational fluctuations between neighboring molecules and a correlation between the orientational and translational fluctuations of a molecule are neglected, the mean-squared Raman tensor reduces to
-
(IOuB(a)12) k B T / r 3
[(A’,,,(O) A*,,&))
x
q?JU?o) 7 y U f j J ) )( “cs(o) a*,y(t))I
(20)
The Cartesian tensors are expressed in terms of the spherical tensors as =
CA’12/,m( 1,m
1213mly6a)
(ala)
Il,mItP)
(21 b)
and One simple way of considering the effect of pressure on the structure of the liquid is that pressure reduces the size of the average structure, keeping its form similar, and also reduces the amplitude of the fluctuations of the structure. Since the interaction concerned with ?r is short-ranged, we consider only the nearestneighbor interaction. Therefore, the pressure dependence of the intensity is described by the changes in the average nearestneighbor distance, r , and the fluctuation amplitudes of the molecules. The compressibility of r is given by the cube root of the volume compressibility, and the magnitude of the interaction tensor, T ( 3 ) ( i j )is, proportional to the negative fourth power of r . Assuming the molecular fluctuations to be isotropic, the terms in (14) can be written as (10,~(A)12) a 6 R 2 / r 8
(I5a)
(IOu8(T)I2)0: 6r2/rI0
(15b)
The positional and orientational fluctuations of a molecule are caused by its thermal motion in the potential field U. The amplitude of the fluctuations is given approximately by the width of the potential at the height of the thermal energy as 6r2
and
6R2
-
(16a)
kBT(a2U/ar2)-l
(16b)
kBT(d2U/aR2)-‘
E
U(i)=
c
where (12l,m(y6a) and (11,mltP) are the elements of the Cartesian spherical transformation matrix given by Stone.” Next, the spherical tensors in the laboratory frame are related to those in the molecular frame by means of Wigner rotation matrix elements D‘,,(R) as A’l2/,m
=
zA’M/,mGn,m(Q) m‘
aI/.m
=
CaY,mGn,m(Q) m’
and (22b)
For carbon disulfide, the nonzero components of the polarizability tensor are and a:,o, and those of the dipole-quadrupole polarizabilities are and A’fi3,l for the u2 vibration and and A’E3,0for the u3 ~ i b r a t i o n .Using ~ (21) and (22), the time correlation functions of A’ and a become ( A ;am(0) A r*y,a,u(r)) = C C [ A ’ M / , m ( A ’ M k , n , ) * ( 121,m‘y6a) x I.m,m’ k.n.n’
( 12k,nly’b’a)*(D!,,,m[R(0)1 Dk;[Q(t)l
)I
(23a)
and (as&0)a*,,&)) =
~cr~M.o(ar”k.o)*c I ~ , O l c P ) (Ik,ol~’P)*(Dbm[fi(o)lD$:lQ(t)l ) I
(23b) Averaging over the random distribution of initial orientations in an isotropic liquid, the reorientational correlation function reduces to’*
U(j,i) (%,m[Q(o)l
/(+I)
I( #i)
Cal/,m( 1.m
/.m k.n
For a simple potential based on the Lennard-Jones 6-1 2 potential and the electric quadrupole interaction16
=4c
at/Y =
ofi;[Q(t)l )n(o) =
I 21+(Df“n(GQ))s/$m,n’
(24)
where 6Q represents the Euler angles of rotation from the orientation Q ( 0 )to Q ( t ) . According to the theory of rotational diffusion, the correlation function for a linear molecule becomesi8
[(&)I2-(&)]+
so that the leading terms of the fluctuation amplitudes are 6r2
-
6n2
(18a)
kETrI4
- kBTrS
( 18b) ~
(D!,,,(sR)) = 6,, exp[-[l(l + 1) - m2)R,t - m2R,t] (25) where R, and R, are the rotational diffusion constants of the molecular axis and the direction of the degenerate bending vibration, respectively. The latter process must be regarded as a kind of vibrational dephasing rather than the rotational diffusion of the molecular frame.
~~~
(16) Hirschfelder, J . 0.; Curtiss, C F.: Bird, R R Molecular Theory of Gases and Liquids: Wiley, New York. 1954
( 1 7 ) Stone, A. J . Mol. Phys. 1975, 29, 1461. ( 1 8 ) Steele, W . A . Adu. Chem. Phys. 1976, 34. I
Raman Spectra of Liquid CS2 at 0-10 kbar
The Journal of Physical Chemistry, Vol. 94, No. 20, 1990 7831
The translational diffusion induces relaxation of the interaction tensors. The Cartesian components of the third-order interaction tensor are expressed by means of the spherical components as
3( q%(ij) = -C%(Qij)( 123,mlyW rij4
m
liq. CS2 Polarized scattering
(26)
where rij is the distance between the ith and j t h molecules and R, specifies the orientation of the intermolecular axis. Using the relationship (24) for averaging over the initial random distribution of a,, the time correlation function of the interaction tensor is given as
.-a
-E L
300
(27) where 0 is the an le between the orientations specified by Rii(0) and f l i j ( f ) , and DW(0) is a Legendre polynomial of degree 3, i.e., P,(COS0). The correlation time
f
I
I
I
350
400
450
500
v/Cm-'
Figure 1. Polarized Raman scattering of the v2 vibration of liquid carbon disulfide at several pressures and 22 OC. The intensities are normalized by the peak height of the polarized v , scattering.
has been calculated by Lynden-BellI9 using the simple diffusion model. The result is rtr = & / 2 1 D
(29)
where D is the self-diffusion constant of the molecule and d the minimum value of the intermolecular distance. For simplicity, the correlation function of the interaction tensor is approximated by a single-exponential function with the decay time T~~~as
-
( q%(ij,O) q?jt,(ij,t)) e-+[r
(30)
The vibrational correlation function is also approximated by a single-exponential function
-
( q n ( 0 )q,,(t)*) e-ll'nv (31) where the correlation time T,, is associated with the dephasing due to the collision-induced modulation of the vibrational frequency. Equation 9 can be rearranged in the form
I,@(w) 0: Re ~ m e i ( w ~ ) f C D ( t ) CdtJ t )
(32)
where cD(t)describes the relaxation of the local field that is induced by the orientational and positional diffusion of molecules and CV(t)the vibrational relaxation, which includes the relaxation of the direction of the bending vibration. By substituting ( 2 0 ) , (23)-(25), ( 3 0 ) , and (31) into ( 9 ) , the following expressions for CD(t) and c,(t)are obtained:
cD(t)=
e-f/Ttr[X,e-RJ+ X2e-5Rd + xje-7RJ
+ X4e-llRxf+ x5e-17Rxf] (33)
and
c,(t)= e-[&+(l/r2v)lf for the v2 scattering and c D ( t )= e-f/rtrlyle-2Rxf+ y2e-8Rxf+ yje-12Rx' + y4e-18RJ1
(34) (35)
and
c,(t)= e-r/73v (36) for the vj scattering, where xi and y i are constants determined by the molecular tensor elements. 111. Results and Discussion The details of the Ramafi polarization measurements and the spectra of liquid carbon disulfide under pressure at room temperature were described in the first paper9 of this series. Figure (19) Lynden-Bell, R. M . Mol. Phys. 1977, 33, 907.
-
-00 P/ kbar
O P/ kbar
Figure 2. Effect of pressure on the polarized, I,,,and depolarized, IL, Raman intensities of the v2 and vj vibrations of liquid carbon disulfide relative to that of the polarized v, intensity. The dotted line a and the broken lines b and c represent the pressure dependence of pSl3, p, and pw4/',
respectively.
1 reproduces examples of the observed polarized scattering by the v2 vibration at several pressures. The intensities are normalized
by the peak heights of the polarized v, scattering in the same scan because v 1 is strong in the vapor, and its relative change of intensity with pressure is likely to be small and can be neglected for the As the pressure increases, the peak shifts to present dis~ussion.~ low frequencies, the line narrows, and the relative intensity grows. The frequency shift was explained by the combined effect of the vibrational anharmonicity of the molecule and the intermolecular intera~tion.~ The effect of pressure on the intensities and the line widths are the subjects of the present discussion. A . Intensity. Figure 2 shows the observed pressure dependence of the interaction-induced scattering intensities reproduced from the previous paper.9 The intensities of the polarized and depolarized v2 and v3 scattering were measured relative to the intensity of the polarized v I scattering. The pressure dependence of the latter was assumed to be caused only by changes in the number density of the molecules and in the strength of the local field. Both effects must cancel in the relative intensities9 The dotted lines a and the two sets of broken lines b and c in Figure 2 give the relative values of p a l 3 , p , and p - 4 / 3 under pressure, respectively, which were calculated from the relative volume of liquid carbon disulfide measured by Bridgman.*O The observed intensities under pressure are definitely below the line a that represents the pressure dependence of the strength of the pairwise interaction: This indicates that the simple pairwise interaction is inadequate (20) Bridgman, P. W. The Physics of High Pressure; Bell: London, 1958.
7838 The Journal of Physical Chemistry, Vol. 94, No. 20, 1990
Ikawa and Whalley
25
liq. CS2
$“ 4
5t Figure 3. Effect of pressure on the width of the polarized u2 Raman line of liquid carbon disulfide. The lower broken line represents the plot of 8,, above the dotted line at 6 cm-I, and the upper broken line represents the plot of 6,, = 17RJ2uc above the lower broken line.
to describe the intensities of the liquid. The increase in the deviation of the observed intensities from line a is explained by the increase of the cancellation or, in other words, the decrease of the amplitude of the local-field fluctuation by increasing pressure. The broken lines b and c represent the pressure dependence of the intensities from the orientational and translational fluctuations, respectively, given by (19a) and (19b). The lines b reproduce fairly well the observed intensities of all the four scatterings. On the other hand, the lines c show the wrong pressure dependence. Consequently, we conclude that the fluctuation of the local field caused by the fluctuation of the molecular orientations dominates the induction of the Raman u2 and uj scatterings of liquid carbon disulfide. On the contrary, the translational fluctuations around the average position of the molecule are not large enough to significantly affect the interaction-induced scattering. B. Line Broadening. The line widths of the u2 scattering were taken from the low-frequency half of the observed bands because of their a ~ y m m e t r y . While ~ the depolarized line was always slightly narrower than the polarized, both agreed with each other to within 1 cm-I. This likely corresponds to the case of the allowed doubly degenerate vibration, for which the Raman tensor has zero trace and both the polarized and depolarized scatterings have the same shape.21 Figure 3 is a plot of the width of the polarized line versus pressure. At higher pressures, the line width seems to become constant, as suggested by the dotted line. Carbon disulfide at room temperature freezes around 12 kbar, where the viscosity is about 15 times the viscosity at zero pressure.20 Therefore, it seems reasonable that the line broadening at the high-pressure limit, which is estimated at about 6 cm-l, is attributed to some process not associated with molecular diffusion. The nondiffusional process will be discussed in later paragraphs; temporarily, its effect on the line width is assumed to be independent of pressure, and the remainder, 6: = 62 - 6 cm-I, is ascribed to diffusion processes. The translational and rotational motions of molecules in the ordinary liquid are usually described as diffusion processes. It (21) Nafie, L. A.; Peticolas, W. L. J . Chem. Phys. 1972, 57, 3145.
7P/ 7, Figure 4. Plots of the time constants T ~ i2, . and iZD versus the relative viscosity of liquid carbon disulfide at 1 bar to 10 kbar.
is well-known that the diffusion coefficients related to the Raman line widths are inversely proportional to the shear viscosity of the liquid, although the meaning of the proportionality constant is still in c o n t r o v e r ~ y . ~ ~Increasing -~~ viscosity at high pressures reduces the molecular diffusion and so reduces the Raman line width. Consequently, the correlation time associated with the line width increases with increasing pressure in proportion to the viscosity. A typical example is the reorientational correlation time of liquid carbon disulfide r, = 1 / 6 R ,
(38)
which was estimated from the observed u1 line widths in the second paper of this series26and is reproduced in Figure 4. The nonzero intercept at zero viscosity was explained by the inertial effect. In contrast with rr, a time constant r2,which is estimated from the whole line width of the u2 scatterings 72
= 1/2acs2
(39)
is nonlinear in the viscosity. On the contrary, a time constant estimated from 6 y
rp =
1/2*csp
(40)
fits well on a straight line, as shown in Figure 4. This is consistent with the above assumption that 6: is the line broadening due to the diffusion of the molecules. The line broadening due to the translational diffusion 6,, = 1 /2acr,,
= 21D/2acdz
(41)
was calculated from the observed self-diffusionconstant D of liquid carbon disulfide under pressure,*’ and the average molecular which is assumed to be independent of diameter d = 4.575 pressure, is the Lennard-Jones diameter as determined in various ways and summarized in Table I of ref 28. The calculated values (22) Kivelson, D.; Kivelson, M. G.; Oppenheim, I. J . Chem. Phys. 1970, 52, 1810. (23) Fury, M.; Jonas, J. J . Chem. Phys. 1976, 65, 2206. (24) Hynes, J. T.; Kapral, R.; Weinberg, M.J . Chem. Phys. 1978, 69,
2725.
(25) (26) (27) (28)
Zerda, T. W.; Schroeder, J.; Jonas, J. J . Chem. Phys. 1981, 75, 1612. Ikawa, S.; Whalley, E. J . Chem. Phys. 1987, 86, 1836. Koeller, R. C.; Drickamer, H. G.J . Chem. Phys. 1953, 2 l . 267. Mourits, F. M.; Rummens, F. H. A . Can. J . Chem. 1977,55,3007.
J . Phys. Chem. 1990, 94, 7839-7842 of 6, are 1.8-0.3 cm-I in the range 1 bar to 10 kbar and are plotted above the dotted line in Figure 3. Obviously, translational diffusion makes only a minor contribution to the line broadening. The large remaining part of the diffusional broadening must be ascribed to the rotational diffusion. The rotational diffusion coefficient R, is calculated from the v i line widths.26 The “unit” line broadening associated with the rotational diffusion, R X / 2 r c , changes from 0.7 to 0.1 cm-’ in the range 1 bar to 10 kbar. Since the molecular tensor elements required to calculate the coefficients xi in (33) are not available at present, a single-exponentialfunction, exp[-nR,t], is temporarily assumed in place of the summation in the brackets in (33). A reasonably good fit is obtained when n = 17, as shown in Figure 3 where the values of the broadening, 17RX/2nc,are plotted above the contribution from the translational diffusion. This suggests that the last term among those in brackets in (33) is dominant. In our theoretical scheme, the vibrational correlation function given by (34) or (36) corresponds to the nondiffusional part of the line broadening. An infrared study will be useful for obtaining information about R, and T2”. The reorientational correlation function for the v2 infrared absorption is given, by substituting 1 = m = 1 into (25), as
Since the line broadening due to the vibrational relaxation by frequency modulation is considered to be common to the infrared and Raman spectra, the infrared line width becomes (43) Then the line width associated with the vibrational correlation function, (34), is given by
-!-( + $)
6; = 2*c R,
(44) Using the observed value 6:” = 4 cm-’ 29 and RX/2.rrc= 0.7 cm-I, 6’ at 1 bar is estimated at 3.3 cm-’. So far, no measurement of 6,?R under pressure has been reported, and so we cannot conclu-
7839
sively discuss the pressure dependence of 6;. However, it seems that 6; has not a big pressure dependence for the following reason. As pressure increases, the vibrational dephasing usually becomes faster because the frequency of molecular collisions increases. A typical example is the dephasing of the v , vibration of carbon disulfide,26where the polarized line width increases from 0.3 to 1 cm-I in the range from 1 bar to 10 kbar. The diffusion of the direction of the bending vibration is one of phase relaxation of the degenerate vibration and not a molecular diffusional process. However, it has analogy with the rotational diffusion of the molecular frame. It may be suppressed by compression and R, becomes smaller. Consequently, the changes in R, and I/r2’ cancel each other to some extent and the pressure dependence of 6; is reduced. In addition to the vibrational dephasing, inhomogeneous line broadening cannot be excluded. Various distortions of the local structure are not always covered by a diffusion process within the time comparable to the average lifetime of the local structures. Then the local field has a quasi-static distribution and provides another type of nondiffusional line broadening. By compression, the distortion of the local structure will decrease, while the lifetime of the local structure will increase. The effects of these changes on the line broadening are in opposite directions. Therefore, the pressure dependence of the inhomogeneous broadening may also be small. The line width of the u3 scattering was roughly constant around 30 cm-’ in the range 1 bar to 10 kbar. The contribution of the relaxation of the local field to the width of the v j line should decrease with increasing pressure in the same way as for the u2 line. The cancellation of this decrease can be caused by intermolecular resonant vibrational coupling, which is caused by the large transition dipole moment of the u3 ~ i b r a t i o n . , ~The line broadening that is predicted by the hydrodynamic model for the resonant i.e. bd = const X p f / T (45) is proportional to the density p and the viscosity 7. Then bd increases with increasing pressure and, consequently, may cancel out the decrease in the line broadening from the relaxation of the local field. (29) Kakimoto, M.; Fujiyama, T. Bull. Chem. SOC.Jpn. 1972, 45, 2970. (30) Wang, C. H . Mol. Phys. 1977, 33, 207.
Ionic Metallic Solutions Leo Brewer Department of Chemistry, University of California, and Materials and Chemical Sciences Division, Lawrence Berkeley Laboratory, I Cyclotron Road, Berkeley, California 94720 (Received: January 11, 19901
The variation of activity coefficients at low concentrations is examined to determine if one can demonstrate charge transfer and formation of ionic solutions for mixtures of some metals.
Introduction I have had many fruitful interactions with Kenneth Pitzer over the years, and I welcome this opportunity to participate in honoring him. He has made important contributions in recent years to the thermodynamic treatment of aqueous electrolytes. His treatment has wide application, and he has been able to extend it to fused salts and gaseous plasmas. I would like to examine an extension to ionic metallic solutions. Metals are sometimes described as cations in a sea of electrons. In typical metallic solutions, the charge transfer is accepted as 0022-3654/90/2094-7839$02.50/0
being sma1l.l Thus each cation with its surrounding electrons is considered a neutral species in metallic solutions. However, there are some examples of “ionic alloys” that are described in terms of charge transfer between metallic atoms that result in charged cations and anions. These are usually solutions of alkali or alkaline-earth metals with the relatively electronegative metals of the right-hand side of the periodic table. The demonstration (1) Pauling, L. The Nature of fhe Chemical Bond, 3rd ed.; Cornell University Press: Ithaca, New York, 1960.
0 1990 American Chemical Society
