J. Phys. Chem. 19!34,98, 7906-7914
7906
Virtual Single-Particle Energy Distributions in Water versus Other Liquidst Chang-Hwei Chen Wadsworth Center for Laboratories and Research, New York State Department of Health and Department of Biomedical Sciences, The University at Albany, State University of New York, Albany, New York 12201 -0509 Received: September 23, 1993; In Final Form: May 2, 1994" Virtual single-particle energy distributions determined by the energy moments calculated from constant-volume heat capacity (C,)at a constant volume u* were presented for liquid water, methanol, ethanol, carbon disulfide, mercury, and benzene. Thedistribution curves were obtained by directly solving the coefficients in the distribution function in terms of the first three energy moments (four, counting normalization). A bimodal distribution was found for water. Conversely, methanol, ethanol, carbon disulfide, mercury, and benzene exhibit a singlepeaked distribution. The distinctive characteristic of the virtual single-particle energy distribution in water is the appearance of a lower-energy peak in the neighborhood of 1.O kcal/mol and a higher-energy peak around 3.1 kcal/mol. The energy distribution reveals molecular characteristics through examination of its dependence not only on the temperature but also on the volume. A t a fixed v*, the ratio of the higher- to the lower-energy peak increases as t increases. At a constant t, the ratio increases as u* decreases (or as pressure increases). The possibility of employing virtual single-particle energy distributions to gain insight into the structure of bulk liquids was discussed. If such energy distributions confer visibility upon the structure which is present in 1 mol of liquid, the two energy peaks in water could represent two distinguishable states in the structure of liquid water.
I. Introduction A possible advantage of dealing directly with parameters which might characterize single-particle interactions in a dense system such as water has largely been neglected, since the formation of statistical thermodynamic variables involving single-particle energies, volumes, and enthalpies is not intuitively apparent. In an attempt to explore the possible definitions of such variables and to examine their statistical distributions in relation to the correspondingthermodynamicfunctions, Stey introduced a virtual single-particle variable and its distribution through a particular set of enthalpy (H)moments.' The calculation of virtual singleparticle enthalpy distributions for liquid water revealed a bimodal character, in contrast to a monotonic character for other liquids such as benzene, NH3, and HF. Such a bimodal character suggested that there are two energetically distinguishable species in the structure of liquid water, as compared with a uniform species in the structure of liquid benzene, NH3, or HF. That work provided a foundation for investigating parameters which might characterize molecular interactions in the context of the virtual single-particle enthalpy distribution. The thermodynamic parameter, enthalpy, is derived from the heat capacity at constant pressure (C,) as a function of temperature (7'). Thermodynamic relationships show that C, is equal to the heat capacity at constant volume (CV) plus the PV (pressurevolume) term. Since the PV term differs from the energy of a species or the energy of transformation from one species to the other, its presence in C, would be expected to blur the picture that could reveal molecular characteristics. Since the energy (E) is dervived from C, as a function of T,without the term of work of expansion, therefore, the virtual single-particle energy (E) distribution has an advantage over the corresponding enthalpy (H) distribution in revealing molecular characteristics. Another advantage in the use of the energy (E) parameter is that the dependence of its distribution function on both the volume and the temperature can be delineated. Although the idea of investigating the virtual single-particle energy (E) distribution was briefly introduced,* no work in this important area has been published. This paper is dedicated in memory of Professor Henry S. Frank whose inspiration, insights, and advice made this work possible. 0 Abstract published in Advance ACS Abstracts, July 1, 1994.
In the present paper, a hypothetical virtual single-particle energy distribution function was derived on the basis of a quasipartition function for the set of energies EJu, where u is the number of molecules and El is the energy of the quantum state j of a whole mole of water or other liquids. The total energy E was obtained from CV at a constant volume v* (defined as q*)as a function of T. Virtual single-particle energy distribution curves for water and other liquids, including methanol, ethanol, CS2,Hg, and benzene, were determined by directly solving the first three energy moments (four, counting normalization) to obtain coefficients in the distribution function without worrying about higher moments or least-square fits, as employed by Stey in obtaining the virtual single-particle enthalpy distribution.' The use of to obtain energy moments offers a considerable advantage, for with a constant volume, the frequencies of the normal modes of motions in liquids ought to change much less with temperature than with a volume change. In addition, the density of the cohesive interactions is more nearly invariant under constant volume. Examination of the energy distribution at constant volume offers advantages in comparing water with other liquids and delineating the dependenceof the energy distributions not only on the temperature but also on the volume. If virtual single-particleenergy distributions of liquids confer visibilityupon the structure which is present in 1 mol of liquid, the energy distribution curves can provide insights into the structure ofwater. 11. Quasi-Single-Particle Energy Distribution Function A. Defdtion and Derivation. If Q is defined as the canonical ensemble partition function of a mole of water or other liquids,
where E, is the energy in a quantum state j, k is the Boltzmann constant, and Tis the temperature, then one can write
wheref(E) denotes the number of such states for an energy level Ei and Sl is the number of possible quantum levels. Accordingly,
0022-3654/94/2098-7906~04.S0/00 1994 American Chemical Society
Virtual Single-Particle Energy Distributions
The Journal of Physical Chemistry, Vol. 98, No. 32, 1994 7907
the probability density function, P(E), of observing a certain energy E is proportional to Q(E) e-ElkTand may be represented by the following empirical equation: P ( E ) = E"(a
+ b E + cE2) e-E1kT
(3)
where p(E) d E is dimensionless, which is resolved in the normalization (its integration from 0 to ~0 is 1). Also, coefficients a-c have appropriate dimensions (in the dimensions of E*, E-a-1, and E-'-2, respectively). In eq 3, P(E) 0 as E 0 or E m. This form of the approximation equation is a good choice, even though it is arbitrary. It gives a very good fit to the data in the energy distribution function as presented in section 111. The presence of additional terms such as E3 and E4 in the parentheses of the right-hand side of eq 3 fits the data nicely too (see section V). Higher terms only add precision to the positioning and the shape of the distribution curves. Other good choices of the function approximation would presumably achieve similar results. Let us define
de is dimensionless, analogous to eqs 3 and 5 . Equation 10 would then represent a quasi-single-particle energy distribution function, where the coefficients (Y and a-c could be determined by applying eq 9. The solutions of these coefficients are described below. B. Solutionsfor Coefficientsin a Quasi-Single-ParticleEnergy Distribution Function. Rearrangement of eq 5 gives
- - Substituting eq 10 into eq 1l b gives g(E) = Ea(a'+ b'E
+
= Ea(a' b'E
(4) as the quasipartition function for the set of energies Ej/v, but not for l/v of a mole. Each one equals 1/v of the energy of the quantum state j of a whole mole of water (or other liquids). In analogy to eq 3, one can define g(E/v) as the probability density function of observing a certain energy ( E l v ) . Then, g(E/v) could be represented by the following empirical equation: g(E/v) = (E/v)"[a
+ b(E/v) + c(E/v)']
+ c'E2) e-E1NkT
+ c'E2)eWEIRT
(12)
where Nk = R (gas constant), and g(e) de is dimensionless, analogous to eqs 3,5, and 10. Combination of eqs 9 and 12 gives
E" =
som
E" g(c) d E
= JOm E"Ea(a'+ b'E
+ c'E2) e-E/RTdE
(13a)
e-(E/")/kT ( 5 )
Analogous to eq 3, g(E/v) d(E/v) is also dimensionless. The nth moment of the average energy, E / v , can be written as
Jo"(Elv)" g(E/v) d(E/v) (Elv)" = Jo-g(E/v) d(E/v)
-
(6)
-
+ b(E/v) + c(E/v)'] e-(E1u)lkT d(E/v) + b(E/v) + c(E/u)'] e-(E/Y)lkTd(E/v)
(Elv)" (E/v)*[a
som
(E/v)"[a
The integration of eq 13b requires the following mathematics relation:
where r is the gamma function and m > 0 and s > -1. Integration of eq 13b leads to
E" 3 a'
r(n + CY + 1) iB"+a+l
(7)
E" =
JOm
E"(E/v)"[a
J"(E/v)*[a
b'
B" +
+ 3)
(Y
+
t1
5)
where j3 = l/RT. The first three energy moments (n = 1-3) can be represented as
Simplifying eq 7 leads to
-
r(n + + 2) + c' r(n + pn+a+2 (Y
+
+ b(E/v) + c ( E / v ) ~ ]e-(E1")/ kT d E + b(E/v) + c ( E / v ) ~ ]e-(E1")'kTd E (8)
Substituting eq 5 into eq 8 gives
As v
-
where r ( n becomes
som
N (Avogadro's number), eq 5 becomes
g(t) = (E/N)"[a
+ b(E/N) + c(E/N)'] e-(E/N)lkT
+ 1) = nr(n).
If normalization is imposed, eq 12
g(e) d E = JOm E"(a'+ b'E
(10)
where g(e) is the probability density function, e = E/N, and g(t)
+ c'E2)
dE
=1 (19) Combining eq 19 and eq 15 as n = 0, one has the normalization
7908 The Journal of Physical Chemistry, Vol. 98, No. 32, 1994
Chen
condition of
The average energy E can be determined by the integration q’with respect to T:
E= If we use the first three energy moments (eqs 16-18) to solve -8,E2, E’, and a, we obtain the following solutions: a‘-c’ in terms of
sc*d T
(29)
where q*is CV(constant-volume heat capacity) at a constant volume v‘. Cvisrarelydetermineddirectly. It must becalculated from the measured heat capacity at constant pressure (C,)using the relation
c, = c,-- azKvzbi=
,mya + 4) r(a+ 2
-
+
+
-
) [ - 2 ~ ~(4a ~ ~ 1 4 ) -~ ~ ~ 2(a 2)(a 4)E]/6
+
+
+ 3)j3E2 + (a+ 3)(a + 2)E’1/6 where 6 = 2 r ( a + 4) r ( a + 3) I’(a + 2)/B3. Substituting eqs 21-23 into the normalization - (eq 20) -gives a expressed - condition
where a and K are coefficientsof thermal expansion and isothermal compressibility, respectively. It should be noted that V, which is a function of T, appears in the right-hand side of eq 30. In order to obtain information at a constant volume, the introduction of is therefore needed, which is defined as CVat a constant volume v‘.4 The relationship between c;l* and CV can be expressed as
c*
2(a
in terms of E, E’, and E’, where E, E’, and E’ are derived from experimental heat capacity data (see below). The value of a can then be obtained by a successive-approximation method. Substituting the determined a into eqs 21-23 allows coefficients a’c’to be calculated, respectively. To test the self-consistency of the calculations, the obtained a anda‘-c‘should - be inserted into eqs 20 and 16-18 to compute E, E2, and E’, which are then compared with the input known energy moments. C. Energy Moments As Derived from Constant-VolumeHeat Capacity Data. Energy moments can be obtained from Gibbs’ recursion relation for energy fluctuation. In analogy to Gibbs’ recursion relation for a mole of water or other liquid^,^ the recursion relation for a quasipartition function Qvfor the set of energies - ( E ~ / Y(eq) 4) can be derivedas follows. The mean energy E / v can be represented as
-
(Ej/v) e-(Ejlu)fkT
E/v =
c
As Y = 1, eq 24 becomes
= (El2
+ Yk? (aE/6T)
(26)
+ vk?(aEA/aq
(27)
Further elaboration leads to
- -
E”+ 1 = E” E
As
Y
= N, we have
- E “ + ’ = E” E
=
= E’
By differentiating eq 25 with respect to T and then dividing by
+ R?(aE”/an
(28) where R = Nk. Equation 28 would represent the recursion relation for energy fluctuation for a quasi-single particle.
dV
c*
Equation 32 shows that the determination of requires PVT data in addition to CV. The first energy moment E can be calculated directly from - eq 29. From eq 28, the second and third energy moments (E2 and E’) can be represented as
e-(Ej/u)/kT
Y
-
e*= C, + s ” T (”)a?
(24a)
where the mean value of the energy E refers to 1 mol of water or other liquids. Rearrangement of eq 24a gives
QY, one has
where P is the pressure. Integration of eq 31 results in
+ R?
e*
E + R?[
2(E
(33)
+ R T ) c *+ R f ( $ ) ]
(34)
From the value of CF.as a function - of T from experimental data, the magnitudes of E, E’, and E’ can be determined, based on eqs 29, 33, and 34, respectively. Substituting the values of these energy moments into eqs 16-18 and applying the normalization condition (eq 20) gives the coefficients in the quasi-single-particle energy distribution function (eq 10).
m.
Calculations A. Water. Values of the heat capacity at constant volume (CV) for liquid water were determined according to eq 30, where the data of C,, ranging from 0 to 100 O C were obtained from Ginnings and Furukawa5 and Osborne et aL6 More recent data by Williams et al.’ in the range from 7 to 77 OC agreed with these data within0.2%. Thaserangingfrom-20to-5 OC (supercooling) were obtained from the empirical equation of Osborne.6 Coefficients of thermal expansion and compressibility were obtained from Ke11.* According to eq 35, can be obtained from CVand PVTdata. Pressuredata for water from 0 to 140 OC at a constant volume (v‘ = 1.0018, 0.9800, 0.9600, or 0.9300 mL/g) were taken from Kell and Whalleyg and Sharp.Io They were fitted to a polynomial expression, P in terms of T. The magnitude of (a2P/aT2)vat each constant volume could be calculated by differentiating the polynomial expression with respect to T. From this, the integration in the right-hand side
c*
Virtual Single-Particle Energy Distributions
The Journal of Physical Chemistry, Vol. 98, No. 32, 1994 7909
TABLE 1: Determined Least-Square Coefficients in Eq 38 for V* T range (K) mo ( lo2) ml
q*Expressed in Terms of P
1.0018 1.0018 0.9800 0.9600 0.9300 0.9000 0
-46.350 0.40362 -1.0952 -1.8352 -3.1629 -27,544
253-293 283-373 273-393 273-393 273-393 273-333
66.389 -0.43488 1.3513 2.2460 3.9513 36.299
Units of coefficients: mo in cal/mol/K, ml in cal/mol/(K)2,
TABLE 2
m2
mz (
m3 (10-9
-355.31 2.9194 -5.0947 -9.1455 -17.239 -177.63
845.57 -8.0456 7.8990 15.959 32.702 384.99
variance (10-5)
m4
-755.17 7.6908 -4.1417 -10.061 -22.790 -312.12
2.981 2.371 5.553 16.15 54.51 34.91
in ~al/mol/(K)~, m3 in ~al/mol/(K)~, and m4 in cal/mol/(K)S.
Energy Moments and Coefficients in g(c) (Eq 12) for Liquid Water’
TOC -10 10 50 100 10 50 100 10 50 100 10 50 100 10 50 100
-
6
-
V*
E2
1.0018 1.0018 1.0018 1.0018 0.9800 0.9800 0.9800 0.9600 0.9600 0.9600 0.9300 0.9300 0.9300 0.9000 0.9000 0.9000
2.614 2.978 3.684 4.499 2.972 3.652 4.449 2.964 3.624 4.404 2.949 3.583 4.338 2.930 3.545 4.275
9.344 11.75 17.1 1 24.57 11.60 16.77 24.07 11.47 16.47 23.62 11.29 16.04 22.98 11.16 15.63 22.38
E’ 40.36 55.05 92.32 154.6 53.76 89.61 150.6 52.69 87.16 147.3 51.14 83.54 142.7 49.75 80.10 138.5
-
’,
-
a
2.744 2.735 2.807 3.085 2.774 2.798 3.190 2.774 2.768 3.396 2.622 2.678 3.871 2.289 2.533 4.311
u’(l0-1) 51.58 35.27 15.88 4.568 30.79 14.58 3.723 28.38 13.72 2.780 28.71 13.06 1.891 35.37 13.16 1.608
-b’ (1 0-2) 428.8 294.2 126.6 29.88 245.4 114.3 21.15 221.5 107.4 11.87 242.3 105.4 4.337 371.2 115.3 3.204
c’ ( 10-3) 1197.8 826.8 351.1 83.75 711.1 329.8 62.06 664.6 323.8 36.68 787.5 343.4 11.72 1305.4 406.7 4.975
Units: V * in cm3/g. 6,E and E in kcal/mol, (kcal/mol)2,and (kcal/mol)s, respectively. a is dimensionless. d-c’in the dimensionsof E”’, E*-2, and E-+’, respectively. 0
of eq 32 could be determined from the area between V and V‘ in the plot of T(d2P/dP)v versus V. Values of were then obtained by combining the integrated value with CV. Data of for water in the temperature range from -10 to 100 O C and at v* = 1.0018, 0.9800, 0.9600, or 0.9300 mL/g were fitted to a polynomial expressed in terms of T:
e*
e*
c’= m, + m,T + m , p + m , P + m 4 p
(35)
Equation 35 is a satisfactorychoice since it is used only for fitting. The determined least-square coefficients are listed in Table 1. The least-square fitting is very good: its variance has a value of (2-5)X lO-5for v* = 1.0018and0.9800mL/gand(1-3) X 10-4 for V‘ = 0.9600 and 0.9300 mL/g. Substituting eq 35 into eq 29 and performing the integration with respect to T allows E to be expressed as a function of temperature. In this integration, the integration constant was determined by knowing a value of E at a finite T. This value of E was obtained from E = H - PV, where H was determined from either the integration of C, (as a function of T ) with respect to T or directly from Landolt-Bornstein.lI The value of P V was calculated from PVTdata.g For instance, for v* = 1.0018 mL/ g, E = 2978.11, 3158.46, and 3248.25 cal/mol a t T = 283.16, 293.16, and 298.16 K, respectively. To calculate E at a different value of P, the following equation was employed:
where E l denotes the value of E for the volume VIand E2 denotes the value of E for a different volume V2 a t a same temperature T. The integration in eq 36 could be carried out using known PVT data.g The determined values of E at t = -10, 10, 50, and 100 O C and V‘ = 1.0018,0.9800,0.9600, and 0.9300 mL/g are listed in Table 2. determinations of the second and third energy moments -In the( E zand E 3 ) ,eqs 33 and 34 contain terms associated with and ( d c * / d T ) . A good fit of as a function of T should provide
e’
c’
reliable estimates of these energy moments. In general, the fitting gives very good numbers for ml and good numbers for mz. m3 is rough, and m4 is an approximation. Since ml is the first coefficient to compute ( d C f / d T ) and is very - reliable, - the determined E3 is dependable. The determined E2and E3are also listed in Table 2.
-
-According to eqs 20-23, the energy moments (E, E’, and E’) and the normalization factor were used to determine the coefficients ( a , a’+’) in the virtual single-particle energy distribution function, g(t) = Ea(u’+ b’E + c’Ez) (eq 12). The determined coefficients in liquid water are listed in Table 2. The obtained coefficients a and a’-c’are justified by self-consistent tests, which reveal that the energy moments calculated from the determined coefficients (based on eqs 16-1 8) agree excellently with the input energy moment data. The deviation is no greater than 10 parts in 1000. The calculated normalization factor is 1.OOOf 0.002. On the basis of thedetermined coefficients, virtual single-particle energy distribution curves for water were obtained (see Results). B. Other Liquids (CSz, Methanol, Ethanol, Hg, and Benzene). Methods of calculation for liquid water as described above were applied to other liquids, including CS2, methanol, ethanol, mercury, and benzene. These calculations are described below: CV a t a Constant Volume, To determine CV of CS2 according to eq 30, values of C, and the coefficients of thermal expansion and compressibility were taken from Brown and Manov,lZ Bridgman,13 and Freyer et al.,14 respectively. PVT data were obtained from Bridgman’3 and were used to determine (i32P/dTL)v (no recent data covering the needed PT range are available). For T = 273-353 K and v* = 0.9750-1.0950 mL/g, least-square fitting (R’ = 0.998, close to 1) revealed a straight line plot of Pversus T a t a constant V. A quadratic equation did not significantly improve the least-square fitting. This result led to (d*P/dTL)v 2: 0, which implied that CV= (eq 32). Values of C, of methanol at T C 292 K and T > 292 K were taken from Kelley15and Fiock et a1.,I6respectively. Coefficients
e*.
e*
7910 The Journal of Physical Chemistry, Vol. 98, No. 32, 1994
TABLE 3 liquid
Hg methanol ethanol benzene a
Energy Moments and Coefficients in At) (Eq 12) for Other Liquids' T(OC)
cs2
Chen
25 60 25 10 50 25
V * (mL/g) 0.7638 0.0740 1.2099 1.2528 1.2528 1.1000
c'(cal/(mol.deg)) 11.54 5.71 15.35 20.78 24.78 22.75
--
E 4.963 2.481 4.420 5.347 6.257 7.010
-
-
E2(10L) 2.666 0.741 2.224 3.191 4.430 5.315
E3(102) a 0' 1.542 6.392 4.112 X 1t2 0.261 1.604 -1.042 1.268 7.311 2.265 X 2.121 9.450 1.357 X 3.529 9.280 4.599 X 10-4 4.353 12.041 5.981 X lod
-b' -2.608 X -2.223 4.472 X 2.595 X 8.605 X 8.236 X
C'
le2 -1.909 X 10-3 lt3 10-4 lt5 lt7
-1.820 X 2.772X 1.509 X 5.162 X 3.299 X
10-I
10-4
1od 10-8
',
See footnotes in Table 2 for units of E, E E ', a,and coefficients a%'.
of thermal expansion and isothermal compressibilities were calculated from Bridgman's PVT data.13 Values of CV were determined based on eq 30. Least-square fitting of PT data at a constant Vexhibited an excellent straight line (R2= 0.9997). More recent data from G ~ o d w i n confirmed '~ the straight line fitting (R2 = 0.9999). Therefore, ( d 2 P / d P ) p 0 at V' = 1.29631.1111 mL/g, implying that CV = For ethanol, data of C, at T < 294 K and T = 3 13-353 K were obtained from Kelleyls and Fiock et a1.,16 respectively. Values of Cv were taken directly from Rabinovich and Nikolaev.lEPVT data were obtained from Bridgman.13 At a constant V, data of P were fitted to a quadratic equation as a function of T in a least-square regression (R2 = 1). The magnitude of (d2P/dP)y was then determined. The integration in the right-hand side of eq 32 was determined under the area between Vand v* in the plot of T(d2P/dP) versus V. The integtated value ranging from -0.15 to -0.18 cal/mol/deg was found for T = 283-343 K. This value was then corrected to obtain the value of q*.More recent data from Sun et al.19 were also used to compute (dzP/dP)y. Least-square fitting showed an excellent straight line of fitting pressureversus temperature (Rz = 0.999 92), implying that (@P/ dF)v= 0 and consequently CV= Since the above difference between and q obtained from Bridgeman's data was small and essentially constant over the temperature range studied, the use of Sun's data did not affect the qualitative nature of the following computation of the energy distribution function. Values of C, and CVfor mercury were taken from Douglas et aZ.;20 PVT data at T = 30-150 OC and P up to 9500 bar were taken from Lind.2' Data of P a t a constant V were fitted to a quadratic equation as a function of Tin a least-square regression ( R 2 = 1). The magnitude of (d2P/dP)y was then determined from the polynomial equation. A plot of T(d2PIdP)Yversus V was performed to determine the integration in eq 32, which had small values ranging from 0 to -0.028 cal/mol/deg for T = 3080 OC. These values were corrected to obtain q*. For benzene, values of C, and C, were from Oliver et aZ.22 PVTdata a t T > 45 OC and P u p to 3500 bar wereobtained from BridgmanI3 and those at T < 45O from Gibson and Kincaid.23 A straight line least-square fitting (Rz = 0.9993) implied that (d2P/dP)u= 0 at P = 1.140-1.080 mL/g and consequently Cv = More recent data of G ~ o d w i nwere ~ ~ also employed to compute (d2PldP)v. Least square regression confirmeda straight line fitting (Rz = 0.992) and the above results. Energy Moments. The above-determined values of G*for CS2, methanol, ethanol, mercury, and benzene were fitted to a polynomial expressed in terms of Tsuch as eq 35. E was obtained by integrating with respect to T (eq 29). The integration constant was determined using known values of E at a finite temperature. Known values of E were obtained from the relationship E = H - PV; for instance, E = 4963.18 cal/mol for CS2 at 298.15 K, 2576.12 cal/mol for Hg at 350 K, 4147.9 cal/ mol for methanol at 280 K, 5281.8 1 cal/mol for ethanol at 280 K, and 6607.08 - cal/mol for benzene a t 280 K. The second energy moment E' for these liquids could be determined by eq 33,which includes a term containing The third energy moment E3was obtained from eq 34, which contains terms in conjunction with
0.30
-
e*.
e*.
e*
e*.
e'
e*.
2 0.20 m
0.10
0
2
4
6
a
10
E (kcalhnole)
Figure 1. Virtual single-particle energy distribution for liquid water at VS = 1.0018 mL/g. (I) T = -10 O C , (11) T = 10 OC, (111) T = 50 O C , and (IV) T = 100 OC.
e'
and ( d q ' / d T ) . A good fit of q*as a function of T should give a reliable-estimate - of the value of (dG*/dT). The determined E, E2,and E3 are listed in Table 3. Energy Distribution Function. Inanalogyto those for water, the determined energy moments (8, E', and E3) for these liquids were used to determine the coefficients (a,a'-c') in the virtual single-particle energy distribution function, g(c), according to eqs 20-23. The energy moments and the determined coefficients in g(c) are listed in Table 3. A good fitting of eq 35assuresthe reliability of the determined energy moments (E, E2, and E3). The obtained coefficients, a and a ' 4 a r e also justified by selfconsistent tests as described in the case of water. The energy moments calculated from the determined coefficients agree excellently with those of the input data. On the basis of the determined coefficients,virtual single-particle energy distribution curves for these liquids were obtained.
IV. Results
A. Water. Virtual single-particle energy distributions for water in the temperature range from -10 to 100 OC and at VS from 1.0018 to 0.9300 mL/g are plotted in Figures 1 to 4. These figures show that two distinguishable peaks appear in the distribution curves at T = -10, 10, and 50 OC and V' = 1.00180.9300 mL/g. In contrast, a single peak is found a t T = 100 OC (except a t VS = 1.0018 mL/g, where a second peak may be arguable). The two distinguishable peaks are assigned as the lower- and higher-energy peaks. The lower-energy peak occurs in the neighborhood of E = 1 kcal/mol. The dependence of E on T and V' can be found in Table 4. At a constant v*, the value of E is increased by 0.15-0.30 kcal/mol as T increases from 10 to 50 OC. At a constant T (for instance, 10 "C), the value of the
The Journal of Physical Chemistry, Vol. 98, No. 32, 1994 7911
Virtual Single-Particle Energy Distributions
TABLE 4 Lower- and Higher-Energy Peaks in tbe Virtual SinaleParticle Energy Distribution for Liquid Water E (kcal/mol) lower-energy higher-energy T ( " C ) V * (mL/g) peak peak hE(kcallmo1)"
0.30 I
-
2 0.20
-10 10 50 10 50
0,
0.10
10 50 10
50 2
0
4
6
0
10
a
1.0018 1.0018 1.0018 0.9800 0.9800 0.9600 0.9600 0.9300 0.9300
0.95 1.oo 1.25 1.10 1.30 1.20 1.35 1.10 1.30
2.90 3.20 3.65 3.05 3.55 2.90 3.50 2.90 3.45
1.95 2.20 2.40 1.95 2.25 1.70 2.15 1.80 2.15
AE = E(the higher-energy peak) - E(the lower-energy peak).
E (kcalimole)
Figure 2. Virtual single-particle energy distribution for liquid water at v1 = 0.9800 mL/g. (I) T = 10 "C, (11) T = 50 "C, and (111) T = 100 OC.
0.30
1
0
2
4
6
8
10
E (kcalimole)
Figure 3. Virtual single-particle energy distribution for liquid water at v* = 0.9600 mL/g. (I) T = 10 "C, (11) T = 50 "C, and (111) T = 100 "C. 0.30
--
0.20
m
0.10
0
2
4 6 E (kcalimole)
8
10
Figure 4. Virtual single-particle energy distribution for liquid water at v1 = 0.9300 mL/g. (I) T = 10 "C, (11) T = 50 "C, and (111) T = 100 "C.
lower-energy peak is insignificantly increased (by only 0.10 kcal/ mol) as P decreases from 1.0018 to 0.9600 mL/g. These results indicate that the lower-energy peak is not sensitive to variation in T o r P. In contrast, the higher-energy peakoccurs in the neighborhood of E = 3 kcal/mol. The value of the higher-energy peak is more sensitive to a change in T or P than the lower-energy peak. When P is kept constant, the value of this higher-energy peak is increased by 0.45-0.60 kcal/mol as T increases from 10 to 50 OC. When T = 10 OC, the higher-energy peak shifts from 3.20 to 2.90 kcal/mol, as P decreases from 1.0018 to 0.9600 mL/g.
The ratio of the higher- to the lower-energy peak also depends on T and P.This can be found from the relative heights of the lower- and higher-energy peaks (Figures 1-4). At a constant P, the ratio increases as T increases (see Figure 1). Moreover, a t a constant T,it increases as P decreases. For example, for T = 10 OC, the height of the lower-energy peak is higher than that of the higher-energy peak at P = 1.0018 mg/mL (curve 11, Figure 1). On the other hand, the height of the lower-energy peak is lower than that of the higher-energy peak at P = 0.9300 mL/g (curve I, Figure 4). A decrease in P is equivalent to an increase in pressure. These results therefore indicate an enhancement of the higher-energy peak as the pressure increases. This implies that, if these two peaks are associated with two differently structured domains, then the structure associated with the lower-energy peak has a lower density than that associated with the higher-energy peak. The difference (AE)between the energies of the lower- and the higher-energy peaks is also listed in Table 4. The table shows that, at a constant P, AE increases as Tincreases. For example, AE = 1.95 a t -10 OC as compared with 2.40 kcal/mol of cooperative unit at 50 OC for P = 1.0018 mL/g. The value of A,!? changes from 2.20 to 1.80 kcal/mol at 10 OC and from 2.40 to2.15 kcal/molat 50°C,as Pdecreasesfrom 1.0018to0.9300 mL/g. From Table 4, interestingly, AE is estimated a t 2.30 kcal/mol a t 25 OC and 1 atm (P= 1.0018 mL/g). This value is approximately equal to A H since the P V term is relatively small. It is close to the enthalpy difference of 2.5 kcal/mol which is obtained by applying the two-state approximation to water.25 B. Other Liquids (CSs, Hg, Methanol, Ethanol, and Benzene). Virtual single-particle energy distributions for CS2 at 25 O C , Hg a t 60 OC, methanol a t 25 OC, ethanol at 10 and 50 OC, and benzene a t 25 O C are presented in Figures 5-9, respectively. In all of these figures, a single energy peak is found, in contrast to two energy peaks for water. The value of E is in the range of 2-6 kcal/mol. The energy distribution curve exhibits a single peak in spite of a change in P or T. For instance, Figure 9 shows that, as Tvaries and P is kept constant, the distribution curve remains a single peak. The magnitude of E for ethanol increases significantly as T increases. Comparison of the results for these liquids with those for water reveals a distinguishablecharacteristic in virtual single-particle energy distribution between water and these liquids.
V. Discussion In the virtual single-particle energy distribution function g(t) + c'E2)e-E/RT), as shown in eq 12, the first term Ea leads to g(e) 0 as E 0, the third term e-EIRTguides g(e) 0 as E m, and the second term (a'+ b'E + c'E2) allows g(e) possibly to yield two solutions for E (Le. two energy peaks). In Figures 1-4, two energy peaks (the lower and the higher) are found for liquid water in the range of T = -10 to +50 O C and P = 1.018-0.9300 mL/g. Conversely, only a single energy peak
- - - (= Ea(a'+ b'E
7912 The Journal of Physical Chemistry, Vol. 98, No. 32, 1994
Chen
TABLE 5: Energy Moments and Parameters in g(c) = E O(a’ + b‘E T(T)
E
100 50 10 0 -10
4.499 3.684 2.979 2.798 2.616
-
-
-
~ ~ ( 1 0 1 ) ~’(101) 2.457 1.711 1.176 1.053 0.935
---
’, ‘,
-
-
~ ~ ( 1 0 2 )~ ~ ( 1 0 3 ) 11.00 5.600 2.912 2.402 1.962
15.45 9.232 5.496 4.733 4.041
+ c’E + d’E + e’E ‘)e-WRT(Eq 37) for Liquid Water
8.775 3.755 1.738 1.375 1.072
a
0‘
-b‘
c’
2.347 2.501 3.988 4.599 5.179
1.565 1.385 5.883 6.441 6.691
2.090 0.991 5.985 6.180 6.141
0.903 0.207 2.139 2.068 1.955
-d’(IO-l)
e’(1O-2)
1.231 0.416 2.916 2.724 2.493
0.851 0.040 1.360 1.220 1.085
’
Units: E, E ’, E E and E in kcal/mol, (kcal/mol)2, (kcal/moV, (kcallm~l)~, and (kcal/m~l)~, respectively. a is dimensionless. the dimensions of E”-’, Eo’, Eo-’,W ,and E O s , respectively. a
a’+’ in
I
0.30 0.30
-
0.20
w
(51
0.10
E (kcalhole) 0
2
0
4
6 E (kcal/mole)
.-
Figure 7. Virtual single-particle energy distribution for methanol at T = 25 OC and V. = 1.2963 mL/g.
Figure 5. Virtual single-particle energy distribution for CS2 at T = 25 “C and V. = 0.8321 mL/g. 0.40
I
0.30 I
A
-- 0.20-
I1
Y
m
0.10 -
8
E (kcallmole)
Figure 6. Virtual single-particle energy distribution for Hg at T = 60 OC and V. = 0.0740 mL/g.
is found for other liquids including CS2, Hg, methanol, ethanol, and benzene (Figures 5-9), even though a distribution function with possibly two energy peaks is imposed. Moreover, based on the energy distribution function of eq 12, our calculations on the virtual single-particle energy distribution for ice reveal a single peak whose energy lies around 0.40 kcal/mol a t -10 0C.26 These findings reveal a unique feature of the virtual single-particleenergy distribution for liquid water, resulting from the hydrogen framework in the structure of water. An important question to ask in association with such a unique feature of two energy peaks for liquid water is whether such a characteristic is restricted to the choice of the above functional form of g(t). This question can be answered by using a more complicated form of g(e) such as
+
+
+
g(t) = Ea(a‘+ b’E c’E2 d’E3 e’@) e-EIRT (37) Equation 37 provides g(t) with a potential for yielding four energy peaks in the distribution curve, and it contains six unknowns (a, a’+’). In addition to the normalization factor, E, E’ and E3, - p ,and E5are needed to give direct solutions for these unknowns. In analogy to the computation procedures as described in section
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The Journal of Physical Chemistry, Vol. 98, No. 32, 1994 7913
Virtual Single-Particle Energy Distributions 0.30
0.30
--
t AYi
0.20-
w
r n
0.lOt
/
\ 0
E (kcalimole)
Figure 9. Virtual single-particle energy distribution for benzene at T = 25 OC and VI = 1.100 mL/g.
2
4 6 E (kcalimole)
a
Figure 11. Comparison of virtual single-particleand two-particle energy distributions for liquid water at T = 10 OC and VI = 1.0018 mL/g.
-
-
As v N , eq 5 becomes eq 10. Then g(e) would represent a quasi-single-particle energy distribution function. If v N / 2 , eq 5 becomes g(2c) = ( 2 ~ ) ["a + 6(2e) c(2e)2] e-(Zf)/kT,where g ( 2 t ) would represent a quasi-two-particle energy distribution function. Figure 11 illustrates the plot of g ( E / v ) versus E as v N / 2 (curve 11) in comparison with that as v N (curve I) for liquid water at P'= 1.0018 mL/g and T = 10 OC. Curve I1 in the figure also reveals two distinguishable energy peaks in a quasi-two-particle energy distribution curve. However, the resolution for the lower- and higher-energy peaks is not as good as that in curve I for a quasi-single-particle energy distribution curve. The above finding suggests that g ( E / v ) can be regarded as if it were a proper partition function as an artifice, like an electron microscope, to confer visibility upon structure which is present in the energy distribution of the true molal partition function but is unobservable there because of the small size of k in the exponential term E / k T as shown in eq 1 . This implies that the energy distribution function in a mole of water must really have "structure" that is not present in the energy distribution function in a mole of other liquids such as CS1, Hg, methanol, ethanol, and benzene, since the process of introducing v as shown in eq 4 does not add any information. In a comparison of the presently obtained virtual single-particle energy distribution with the corresponding enthalpy distribution as previously reported by Stey,' it was found that the bimodal energy distribution function of water is comparable with the findings in the enthalpy distribution function as determined by an optimized weighted Laguerre expansion of the enthalpy moments.' Our obtained lower- and higher-energy peaks occur in the neighborhood of 1.0 and 3.1 kcallmol, in contrast to 0.4 and 2.7 kcal/mol in the enthalpy distribution,' respectively. Unlike the enthalpy distribution, the use of which variable for studying molecular characteristics is limited to the temperature, the present energy distribution can provide further information by examining its dependence on the volume. In a comparison of the results in other liquids, a single distribution was found in both the energy and enthalpy distribution curves, where the peak appears around 3.9 kcal/mol for methanol and 6.4 kcal/mol for benzene. Nevertheless, no enthalpy distribution for ethanol, carbon disulfide, or mercury is available for comparison with our obtained energy distributions for these liquids. If thevirtual single-particleenergy distribution of liquids confers visibility upon structure which is present in 1 mol of water, the low- and higher-energy peaks as shown in Figures 1-4 could represent two distinguishable states. The figures obviously cannot provide information about water clusters involved in each state. The dependence of these two peaks on T and P'seems to suggest that the lower-energy peak possess a lower density than the higher energy peak. Water clusters such as large protonated water
+
-
E (kcalimole)
Figure 10. Virtual single-particle energy distribution for liquid water at VI = 1.0018 mL/g, where a more complicated functional form of &e) is used (see eq 40). (I) T = -10 OC, (11) T = 0 "C, (111) T = 10 OC, and (IV) T = 50 O C .
Figure 10 shows that the use of eq 37 sharpens up the lowerenergy peak, as compared with Figures 1-4, which employ eq 12. Even though this more complicated form of g ( t ) in eq 37 can provide the opportunity, in principle, to display four maxima, the data do not support it. These results demonstrate that the observation of two energy peaks in thevirtual single-particle energy distribution for water is not restricted to the choice of eq 12. Another issue which should be discussed is the quasipartition function QYfortheset of energies Ej/v (eq4)and thecorresponding energy distribution function g ( t ) (eq 12) where t = E / N and v = N . If t is a physically realistic single-molecule energy, then t is not independent of 0 ( = l / k T ) in the way that E is in the global partition function. This makes it seemingly inappropriate to derive (Elv)" in the recursion relations that Gibbs got for En in a canonical ensemble, which is a necessity if experimental information is used to obtain numerical values for the energy moments. Nevertheless, the present use of and the subsequently determined energy moments may offer an excuse for justifying the use of a quasi-single-particle partition function in the recursion relations, since at a constant volume the frequencies of the normal modes of motion ought to change much less with temperature than with a volume change. The presently defined quasi-single-particle partition function could therefore be more than a phenomenological term. The third question to be addressed is the appearance of the energy distribution function in the case of v not approaching N .
10
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7914 The Journal of Physical Chemistry, Vol. 98, No. 32, 1994
clusters H+(HzO), (n = 4-45) have been reported.27 The recent probe of weakly bound clusters using far-infrared vibrationrotation-tunneling spectroscopyhas raised interesting possibilities concerning the existence of transient local chiral structures for clusters larger than a trimer in liquid water.28 The present investigation has demonstrated that the virtual single-particle energy distribution function can provide structure information on liquid water. Recent reports on the presence of two distinguishablestates in liquid water include the isomerization between clusters such as polycyclic, cubic-shaped octamer,29 and fluctuations of water clusters between the two states, the longbond to short-bond cooperative process ("geometric relaxation") of water pentamer or tetramer.3O-32 Moreover, the addition of the geometric-relaxation feature to random H-bond connectivity was found to offer a basis for explaining two-state behavior of water in numerous observations,*8including the bimodal distribution of the enthalpy.' It is hoped that the present work will stimulte further interest in exploring the distributions of singleparticle parameters, including their theoretical basis and implications.
Acknowledgment. The author acknowledges his great indebtedness to Professor Rufus Lumry of the University of Minnesota for his many valuable criticisms and suggestions in the course of preparing this manuscript. He is grateful to Dr. Yi-der Chen of the National Institute of Health for his helpful comments on section I1 of the manuscript. The author specially thanks Tracy Godfrey and Maryellen Carl for their devotion and enthusiasm in preparing this manuscript. References and Notes (1) Stey, G. C. Ph.D. Dissertation, University of Pittsburgh, 1967.
(2) Stey, G. C.; Chcn, C-H; Frank, H. S . Am. Chem. Soc. National Meeting Abstract (Phys. Chem.) 1974, 167, 52. (3) Hill, T. L. Inrroduction to Statistical Thermodynamics; AddisonWesley Publisher: Reading, MA, 1960; p 35.
Chen (4) Chen, C-H.; Wooten. W.J.; Frank, H. S.Am. Chem. Soc. National Meering Abstract (Phys. Chem.) 1974, 167, 51. ( 5 ) Ginnings, D. C.; Furukawa, G. T. J . Am. Chem. SOC.1953,75,522. (6) Osborne, N. S.;Stimson, H. F.; Ginnings, D. C. J . Res. Natl. Bur. Stand. 1939, 23, 197. (7) Williams, I. S.;Street, R.; Gopal, E. S.R. Pramana 1978,11,519. (8) Kell, G. S . J . Chem. Eng. Data 1967, 12, 66. (9) Kell, G. S.; Whalley, E. Phil. Trans. R. Soc. London 1965, A258, 597. (10) Sharp, W. E. Report of the Lawrence Radiation Laboratory, University of California, UCRL-7118, 1962. (1 1) Landolt-Bomstein Zahlemverte and Funktionen aus Physik; Springer Verlag: Heidelberg, 1961; usw. 6. Aufl. I1 Band 4 Teil, pp 394-427. (12) Brown, 0. L. J.; Manov, G. G. J. Am. Chem. Soc. 1937,59, 500. (13) Bridnman, P. W. Proc. Am. Acad. Art Sei. XLIX 1913, 1 and 62. (14 Frey&, E. B.; Hubbard, J. C.; Andrews, D. H. J. Am. Chem. Soc. 1929, 51, 759. (15) Kelley, K. K. J. Am. Chem. SOC.1929,51, 180 and 779. (16) Fiock, E. F.;Ginnings, D. C.;Holton, W. B.J. Res. Bur. Natl. Stand. 1939, 6, 893. (17) Goodwin, R. D. J. Phys. Chem. Re/. Data 1987, 16, 799. (18) Rabinovich, I. B.; Nikolaev, P. N. Dokl, Akad. Nuuk. USSR 1962, 142, 1335. (19) Sun, T. F.; Ten Seldam. C. A.; Kortbeek, P. J.; Trappeniers, N. J.; Biswas, S.N. Phys. Chem. Liq. 1988, 18, 107. (20) Douglas, T. B.; Ball, A. F.; Ginnings, D. C. J . Res. Bur. Natl. Stand. 1951, 46, 334. (21) Grindley, T.; Lind, J. E., Jr. J. Chem. Phys. 1971, 54, 3985. (22) Oliver, G. D.; Eaton, M.;Huffman, H. M. J . Am. Chem. Soc. 1948, 70, 1502. (23) Gibson, R. E.; Kincaid. J. F. J. Am. Chem. SOC.1938, 60, 511. (24) Goodwin, R. D. J. Phys. Chem. Re$ Datu 1988, 17, 1541. (25) Walrafen, G. E.; Chu, Y. C. J . Phys. Chem. 1991, 95, 8909. (26) Chen, C.-H. Unpublished data. (27) Yang, X.;Castleman, A. W., Jr. J. Am. Chem. Soc. 1989,111,6845. (28) Pugliano, N.; Saykally, R. J. Science 1992, 257, 1937. (29) Benson, S.W.; Siebert, E. D. J. Am. Chem. SOC.1992,114,4269. (30) Lumry, R.; Battistel, E.; Jolicoeur, C. Faraday Symp. Chem. Soc. 1982, 17, 93. (31) Lumry, R.; Gregory, R. B. The Fluctuating Enzyme; John Wiley Publisher: New York. 1986. (32) Hameka, H. F.; Robinson, G. W.; Marsden, C. J. J. Phys. Chem. 1987, 91, 3150.
