Henry Eyring and R. P. Marchi University of Utah Salt Lake City
Significant Structure Theory of Liquids
O n e of the most eIusive properties of nature has hecn a complete and consistent theory of liquids. Since the formal development of statistical mechanics some thirty years ago, there have been two main approaches to a liquid theory. There is the formal or fundamentalist approach pioneered by Mayer and Kirkwood among others and currently being widely pursued. An alternate method of attack is the model approach. I n this method one visualizes a physical picture of a liquid, translates the picture into a mathematical equation, i.e., a partition function, and then computes the thermodynamic and mechanical properties of the liquid. If the model is reasonable and the calculated results agree with the experimentally determined values with sufficient faithfulness, then one assumes that the model is an adequate descriptionof the liquid. The formal approach provides the aesthetic pleasure of initial rigor, but a t some stage during its development, up to the present at any rate, approximations must be introduced either in the form of simplified mathematics or oversimplified intermolecular potentials. I n time such difficulties may he overcome. When this does happen, it will constitute a noteworthy advance. The model approach has several immediate advantages. First of all it gives a clear visual picture of a liquid and thus provides a physical explanation for a variety of phenomena. Second, the mathematical manipulations are greatly simplified, thus reducing the elapsed time between initial interest in the value of some particular property of a specific liquid and a knowledge of the value of this property. NaYvely, one might say that the choice between the mathematical and model approaches reduces to a choice of the time for making one's approximations. I n the model approach the approximations are made a t the beginning; in the formal approach they are made a t the end. The theory of liquids to he presented in this paper is a model approach and is analogous to the models customarily used for solids and gases. We next catalcg some of the known properties of the liquid state. As the temperature of a solid is increased, the substance suddenly melts to form a liquid. With a further rise, the density decreases until finally the liquid and vapor phases merge into one fluid-like substance a t the critical point. This would indicate that a description of a liquid should reduce to a description of a solid or gas a t the respective limits. From X-ray scattering experiments on liquids such as argon (1) we gain information of a more detailed nature. As the temperature of a solid increases, the coordination number of nearest neighbors, decreases from 12 in the solid to 10 or 11 in the liquid a t the melting point. With 562
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rising temperature the coordination numher, z, steadily decreases to approximately 4 a t about five degrees helow the critical temperature and then rises to 6 a t the critical point. While the coordination numher is decreasing to 4, the distance hetween nearest neighbors remains almost constant (3.8 A for argon). Then as z risesto 6 the distance between neighbors increases (to 4.5 A). Furthermore, the X-ray data indicates that those molecules immediately surrounding a given molecule (i.e., its nearest neighbors) are arranged in an orderly manner. The second and third nearest neighbors are arranged in a somewhat random fashion and beyond that chaos reigns. To say this in another way, a liquid possesses short range order, hut a total lack of a long range order. Excess Volume-Fluidized
Vacancies
Now may we reconcile the fact that as the temperature increases the volume increases, yet the average distance between molecules remains fairly constant? A simple way of explaining this is to imagine that some of the molecules are replaced by vacancies. For example, argon expands 12% upon melting. This corresponds to removing about every eighth molecule. This would have two desirable effects. First, it would increase the volume simply by increasing the number of sites a t the same time keeping the intermolecular distance more or less constant. And second, when two or more molecules share a vacancy, the lattice structure or orderly arrangement is destroyed so that u7e no longer have long range order. Since the introduction of vacancies into a liquid seems to he a simple and satisfactory way of accounting for expansion, constant intermolecular distance, and lack of long range order, we shall now proceed to formulate a partition function which skirts certain difficulties and yet expresses this concept. The key to the Pandora's box of liquids lies in elucidating the use a liquid makes of the excess volume, V - V,, where V is the molar volume of the liquid and V , is the molar volume of the solid. The liquid apparently utilizes this excess volume in two ways. First of all, if a molecule is rambunctious enough, that is to say if it has sufficient kinetic energy, it will take possession of some of this excess volume by pushing the neighboring molecules out of the way. By doing this it will gain entropy, or speaking in terms of a partition function, it will gain a degeneracy factor. The second use the liquid makes of the excess volume is to allow the molecule to possess translational degrees of freedom. Let us observe some particular molecule in the liquid. At some particular time it will he completely surrounded by other molecules, and so it will be vibrating about some point in space with ahout the same fundamental
vibration frequency that it would have in the solid. Whenit is oscillating this way, we say that it is hehaviug in a solid-like manner. It then acquires some energy (by collisions with one or more of its neighbors) and transfers this energy into one of its vibrational modes of motion. The molecule then vibrates so hard in scme direction that i t pushes the neighboring molecules aside and takes possession of an additional position. I n this process of moving to a new position external vibrational degrees of freedom are converted into translational degree of freedom. When a liquid molecule possesses translational degrees of freedom, we say it is acting in a gas-like manner. One naturally asks, how are the total number of degrees of freedom to be paxtitioned between solid- and gas-like degrees of freedom? Before answering this question we should make a few comments about the excess volume. There are no stationary solid-like vacancies in a liquid. Light scattering experiments rule out such inhomogeneities in density. Furthermore, it is well known that liquids superheat when they are heated in vessels so clean that bubbles do not form on the walls. Thus it is clear that pure liquids have no "holes" large enough to act as nuclei for bubble formation. Also the structure of a liquid is very unstable with molecules rapidly shifting their position. Due to this rapid and irregular motion of the molecules, care must be exercised in speaking of a lattice or empty lattice site. Nevertheless, there is an excess volume, V - V,, in the liquid, and this excess volume can he acquired by a molecule by pushing out its competitors. Only when a molecule has enough energy to push the neighboring molecules aside, however, does it become appropriate to speak of a hole or empty lattice site coming into existence. We call these holes fluidized vacancies or sometimes simply holes or vacancies. We wish to make clear, however, that these holes are very different from the locked-in almost static vacancies found in solids. The analogy between holes in a liquid and positions in an electron sea is very good. It is immaterial whether we think of molecules or electrons moving in one direction or holes or positrons moving in the other. Xow certainly a hole by itself has no properties; but when a hole is surrounded by molecules, it becomes convenient to think of the hole as possessing properties. By moving into the hole, the surrounding molecules confer gas-like properties on the hole. It is simply a question of convenience whether we speak of the molecules possessing translational degrees of freedom or of the holes behaving as gas-like molecules; both descriptions refer to the same phenomenon. Likewise the term solidlike molecules does not imply that there are microcrystalline regions in the liquid. We simply mean that for short periods of time the motion of a molecule will be oscillatory. During other periods of time the motion of a molecule is translational. It reduces to a question of semantics whether we refer to these motions as solid- and gas-like or vibrational and translational. Returning to the division of the total number of degrees of freedom, we assume that the excess in volume between the solid and liquid, to a sufficient degree of approximation, can be measured in terms of holes of molecular size. I n one mole of liquid there are (V V , ) / V , moles of vacancies per mole of molecules. The fraction of occupied positions adjacent to a
vacancy is, assuming complete randomness, V,/V. Row as we said before, a hole surrounded by vacancies has no properties; it is the molecules around a hole which confer properties on the vacancy. Thus if we multiply the number of holes, (V - V,)/V,, by the probability of molecules occupying positions around the hole, V J V , we obtain (V - V J / V for the fraction of vacancies endowed with gas-like properties. The remaining fraction of a mole, V,/V, may be thought of as associated with solid-like molecules. Accordingly, the heat capacity at constant volume, C,, of a mole of argon should be given closely by the sum of the contributions from V,/V moles of solid and (V - V J / V moles of gas. Thus: C, = 6 .
v. + 3. v - v, V
(1)
This .expression was first suggested by Walter and Eyring (2). Its validity is tested in Figure 1. The results are sufficientlv encouraeine to cause us to develop the fluidized vacancy the&:
Figure 1 . Heat capocity a t con$tont volume for liquid argon. The solid curve represents equation ( 1 I, and the circles represent the experimental data.
The concept of fluidized vacancies has another advantage: it explains quite nicely the law of rectilinear diameters. This law states that the average density of a liquid and its vapor is nearly independent of temperature decreasing slowly in linear fashion as the temperature increases from the melting to the critical point. If a molecule is transferred from the liquid to the vapor without displacing the neighbors, i.e., leaving a vacancy, all bonds to neighbors are broken. I n ordinary vaporization only half this energy is required since again a11 bonds to its neighbors are broken, but each bond joins two molecules and therefore only half the energy of a bond should be charged against each molecule. Thus the energy to make a vacancy just equals the heat of vaporization. Further, a vacancy in the liquid moves about as freely as does a molecule in the vapor. Thus the entropies of a hole and of a gas molecule should he about equal. Now since the vacancies and the vapor molecules have the same energy and entropy, it is to be expected that there will be as many vacancies per unit volume of liquid as there are molecules per unit volume of vapor. Thus the sum of the densities of the liquid and vapor will be constant except for the slight lattice expansion of the liquid causing the density of the liquid to deVolume 40, Number 1 1 , November 1963
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563
crease slightly with temperature. Consequently, the mean density of liquid and vapor should decrease slightly with temperature (9). A cardinal principle in significant structure theory is that molecular structures in the vapor are mirrored in the liquid as analogous structures made out of vacancies. Thus individual molecules translating in the vapor are mirrored as vacancies translating in the liquid. h t a t i n g molecules are mirrored as rotating vacancies. Association of vapor molecules is matched in the liquid by association of the corresponding v e cancies which translate and rotate like their molecular counterparts. This principle has been applied and verified for ordinary liquids, metals, and molten salts.
effectively becomes unity and we are left with the partition function of a gas. Accordingly, this partition function has the correct asymptotic behavior. Within the solid-like portion of the partition function the factor eEsjRT/(l - e-e/T)3 occurs. This is the Emstein partition function for the external vibrations of the atom. E, is the sublimation energy, and so the reference energy in this theory is for the atoms in their lattice framework rather than the separated atoms. The quantity (1 nhe-'lRT) is the geometrical degeneracy factor, and it is a direct result of a molecule appropriating excess volume. Accordingly, we expect nh to he proportional to the excess volume,
Formulation of the Partition Function
while r should be inversely proportional to the excess volume and directly proportional to the energy of sublimation of the solid. Thus
We return to the formulation of the partition function. If a molecule is to take possession of one of the fluid vacancies, it must push the neighboring molecules aside. When it does have the required energy, the additional sites become available to a molecule and there is a degeneracy factor equal to the number of sites available. The number of additional sites will be equal to the number of neighboring position, nh, which a rambunctious molecule can occupy multiplied by the probability that the molecule has enough energy, s, to move into one of these sites. Thus the number of additional sites is: Therefore, the total number of positions available to a given molecule is 1
+ nne-dkT
(2)
If we now assume that the vibrational degrees of freedom, i.e., the solid-like molecules, are adequately represented by an Einstein oscillator (which for temperatures corresponding to the liquid region is a good approximation), we can write for the partition function, j, for a monatomic liquid such as argon the expression:
The first set of brackets represents the solid-like portion of the partition function. The remaining portion is the gaelike part. I n the partition function for a nonlocalized independent system the number of complexions is overcounted due to the indistinguishability of the particles. Thus we must divide by the number of particles factorial. This explains the factorial term, [N(V - VJ/V]!, a t the end of the equation (3). Using Sterling's approximation, x! = (x/e)", equation (3) becomes
where 0 is the Einstein characteristic temperature. Note that when V = V, the exponent of the second set of brackets goes to zero leaving the partition function of a solid since nne-'jRT also goes to zero. When V is much larger than V,, the first set of brackets 564
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Journol of Chemical Education
+
nn
=
n(V - V,)/V,
(5)
We next require values for the proportionality factors n and a. These may be evaluated by calculating n* and r near the melting point where the liquid still approximates a solid lattice. Thus near the melting point the fraction of the neighboring positions, Z, which are empty and therefore available for occupancy is
Hence
is to be compared with the value 10.8 required to fit experimental data. These added positions should be available without extra energy to a neighboring molecule. However, in melting, the other neighbors to the position have gained the kinetic energy of melting by spreading into the vacancies. Accordmg to the virial theorem the kinetic energy of melting is half the total energy of melting, Em. Thus at, or very near the melting point
Hence
This value for a is to be compared with the value 0.00534 which is chosen to fit experimental data. The fraction (n - 1)/Z results from the fact that a molecule expanding into the vacancy acquires 1/Z of its kinetic energy of melting and there are only n - 1 molecules to compete with the molecule in question. Our model thus fixes all the parameters in liquid theory except those which are properties of the solid. An essential point in any model of the liquid is that it explains the utilization of excess volume introduced by melting. A model which omits this explanation is either a superheated solid or chilled vapor theory and not really a liquid theory at all. Obviously, equation (9) is only approximate, but it does give a good estimate for the value of a. The error in the value of a is due to two causes. The
quantity E, (V - V,)/V gives a value for the heat of fusion which is about 10% too low. On the other hand the value obtained from the virial theorem for the needed kinetic energy is undoubtedly only approximate. A molecule in the liquid state is apt to lower its potential energy as it is pushed out of the excess volume. Therefore, the factor in equation (9) should he smaller. The errors in these two factors probably about compensate each other. The theory also leads to a radial distribution function. Knowing the molar volume of the solid and the type of packing in the solid phase, we can compute the size of the molecules. If we now arrange these molecules in the volume V in the most random way possible (in ohtainmg the fraction of solid- and gas-like molecules, we have assumed complete random~less),we can compute a radial distribution function. Due to the faet that the molecules are not rigid spheres, but rather are elastic molecules, and since there is a distribution of Emetic energies, the peaks of the distribution function will he smeared-out, as is observed experimentally. Using equations (5) and (6), equation (4) can be rewritten
Inert Gases ( 4 )
For the liquid inert gases we use the partition function given in equation (11). E,, 8, and V8 are taken from the properties of the solid phase; n is determined from equation (8) and a is estimated from equation (10). Some of the results are summarized in Table 1. It will be noted that the calculated critical pressure is in all cases too high. Inspection of equation (11) produces a possible explanation for this. As the critical point is approached, clustering becomes important. That is, concentrations of the dimer and trimer in the vapor phase become significant and thus should be included in the partition function. These clusters have the effect of decreasing the pressure. Equation (11) only includes monomers in the gaslike portion of the partition function. Presumably, inclusion of a dimer term would improve the results. Table
1.
Calculated and Observed Thermodynamic Properties of the Liquid Noble Gases
-
Tm("K) V , (em' mole-')
P, ( a h ) AS, (cal mole-' deg-I)
Ta (OK) Va (cma male-')
This is the form in which the partition function for Significant Structure theory is most frequently written. Equation (11) is for liquids composed of monatomic molecules. Naturally for systems which possess internal vibrations and rotations, the appropriate expressions must be added to the solid- and gas-like portions of the partition function.
AS* ( e d mole-I deg-I) ~1:(OK)
V , (cma mole-')
(83.85) 83.85 28.90 28.03 0.679 0.674 3.263 3.34 87.29 87.29 29.33 28.69 19.04 17.85 149.7 150.66 83.68 75.26 52.93 48.00
talc obs calc obs calc obs calc obs calc obs
talc obs
19127 17.99 208.33 210.6 88.32 69:68 54.24
19:& 18.29 287.8 289.8 113.52 113.8 74.89 58.2
ealc obs calc
oh eale obs cak obs
Calculation of Thermodynamic Properties
Diatomic Liquids
With the basic partition function given iu equation (11) we are in an excellent position to calculate thermodynamic properties from the melting point to the critical point for a wide variety of substances ranging from such near-ideal liquids as argon to nonideal systems such as water. The relation between the Helmholtz free energy, A , and the partition function is
Significant Structure theory has been applied to a ilumber of diatomic liquids: nitrogen (4),chlorine (5), fluorine, bromine (6), iodine (6), and also hydrogen (7) which requires the use of quantum statistics. The theory is presently being applied to liquid oxygen. Let us examine the application of the theory to chlorine for which the results are typical. The unusually high entropy of fusion and the absence of transition peaks in the heat capacity curve for solid chlorine indicate that the chlorine molecule does not rotate freely in the solid state. Also the density data on liquid chlorine suggest that rotation is hindered in the liquid state. To avoid complicating the partition function by considering libration-type degrees of freedom, the six degrees of freedom in the solid-like portion of the partition function are assigned to a five degree Emstein oscillator and one internal vibration. Free rotation was assumed in the gas-like portion of the partition function. The partitioil function for liquid chlorine, then, is:
Since we now have A as a function of V and T, we can calculate all other thermodynamic properties from
eEs/RT
1
fa*= { (-I- - e-B/T)S. (1 - 0-WbT)
.
and similar expressions for the remaining thermodynamic properties. Volume 40, Number 1 I , November 1963
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565
Figures 2 and 3 compare some of the calculated and observed properties. The critical temperature is about 11% too high, the critical volume about 6% too high, and the critical pressure is too large by 38%. The results for chlorine are typical of the results obtained for many other liquids: from the melting point to the boiling point the agreement with experiment is very good; above the boiling point deviations occur so that T , and V , are 5 1 0 % too high and P. is 3W50% too large.
A, B, and C are the principal moments of inertia and the factor 12 is the symmetry number for molecules having the configuration of a CHa molecule. Since the partition functions for the internal vibrations and the rotations occur in both portions of the partition function, we have factored them out for convenience. Some of the results for methane are given in Table 2. Table
2.
Calculated and Observed Pro~ertiesof Methane
Cale
Thermodynamic
Obs
Molten Metals (9)
TFKl
Figure 2.
V o p o r pressure versus temperature for liquid chlorine.
Figure 3.
Molar volume versus temperature for liquid chlorine.
Since the conducting (outer) electrons of a metal are spead out over the positive ions, only vacancies for the ions need appear in the liquid. In general, ions are only about a third the size of atoms, therefore the vacancies are also smaller. Thus when metals melt, they expand only 3 or 4%, that is, about a thud as much as a normal liquid. However, the entropy of melting is almost the same as that of a normal liquid, so it follows that n must have a value three times that of a normal liquid. In the vapor above a molten metal appreciable concentrations of the dimer species exist. Arguments similar to those used in explaining the law of rectilinear diameters lead one to conclude that the vacancies in the liquid are also paired. Due to the relatively high dimer concentration in the vapor a dimer term has been included in the gas-like portion of the partition function for molten metals.
Organic Liquids
The theory has been applied to a number of organic liquids ranging from such simple hydrocarbons as methane, ethane, and benzene (4) to the balocarbons, carbon tetrachloride and carbon tetrafluoride (8). The experimental heat capacity curves for both CHI and CCI, indicates that the molecules rotate in the solid state. We therefore assume that they rotate in the liquid state. The partition function for liquid CHI or CClr then is
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Journal of Chemical Education
where
and m,, mz,I, D, and v are the masses of the monatomic and diatomic metal atoms, the moment of inertia, the dissociation energy, and the ground state vibrational frequency of the dimer. As T-T,, the electrons are no longer smeared-out over the positive ions, and the atoms no longer behave as if they were ions. Thus the value of n must now be near to that of a normal liquid. The partition function in equation (16) has been used to calculate properties for liquid sodium, mercury, copper, and lead. Some typical results are
given in Figure 4, which plots molar volume versus temperature for liquid mercury over a four hundred degree temperature span. At 600°K it is approximately 3% too small.
Table 3.
Thermodynamic Properties of the Fused Alkali Halides
Melting-Point Properties of the Alkali Halides NaCl
T, ("K) V , (cc)
AS, (eu)
calc obs calc obs cale obs
1070 1074 38.14 37.74 7.75 6.3
KC1
NaBr
KBr
1023 (1023) (1008) 1049 1023 1008 49.06 43.84 56.13 48.80 44.08 56.03 5.40 6.16 4.83 5.8 5.96 6.94
Boiling-Paint Properties of the Alkali Halides
Ta ("K) V s (. c c. )
AS.(eu) Figure A.
Molar volume versus temperofwe for liquid mercury.
ealc obs ealc obs eale obs
IiaCI
KC1
NaBr
KBr
1750 1738 51.15
1684 1680 71.20
1671 1665 60.15
1661 1653 81.81
2i.55 23.5
21.63 23.1
2i.02 23.2
20.89 22.4
Critical Constants of the Alkali Halides NaCl
KC1
NsBr
KBr
Fused Salk ( 10 )
The observed percentage change in volume for argon upon rnelting is 12%. For molten salt such as NaCl the per cent expansion is approximately twice that of argon. The entropy of melting comes from the randomness introduced by the excess volume. Since positive ions can occupy only half of the fluid vacancies while negative ions can occupy the other half, it should require twice the per cent expansion to obtain the same entropy increase for each kind of ion. This is in accord with the experimental observations. The fact that only half the excess volume provides vacancies for each kind of ion means the factor n in the degeneracy should only be about half as large for salts as it is for argon. The partition function for a mole of alkali halide molecules is
Figure 5. Vopor pressure of liquid para hydrogen, hydrogen deuteride and orlho deuterium.
the solid-like portion of the partition function is raised to the 2N VJV power because a mole of alkali halide contains 2N particles. The alkali metal ions and the halide ions are in the same environment so a common Einstein temperature is used for both. E , / N is the sublimation energy per molecule. The E, required in the partition function is the sublimation energy per ion so we must divide by two. To account for the long-range nature of the coulombic interionic potential the sublimation energy must be altered. For liquid argon (see eq. (11))the energy falls off as E, VJV, that is inversely as the volume which was what van der Waals suggested for gases long ago. This is not true for the molten salts, however, since in one of the three dimensions a Na+ and a C1- cling together, and so there is expansion only in the other two directions. Therefore, the energy should decrease about as l/V"'; thus in the partition function we multiply E, by ( V . The gas-like portion of the partition function is composed of molecules whose moments of inertia and ground state vibrational frequencies are assumed to be the same as for the vapor phase. A
summary of the results is given in Table 3. The agreement with the observed data is good. Recent experimental work has indicated that vapors of the alkali halides are strongly associated to four and six atom complexes. Inclusion of these terms in the partition function should impmve the results. Liquid Hydrogen (7)
The theory has been applied with excellent results to liquid hydrogen, deuterium, and hydrogen deuteride. It provides an interesting test for this appmach to liquids since changes in the concentrations of ortho and para hydmgen cause slight changes in the thermodynamic properties. Because these liquids exist a t extremely low temperatures a Debye partition function is used for the solid-like degrees of freedom and equation (7) is used for nh. The gas-like molecules are treated as a slightly degenerate gas which obeys Bose-Einstein statistics for the cases of hydrogen and deuterium and Fermi-Dirac statistics for the Volume 40, Number 1 1 , November 1963
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567
case of hydrogen deuteride. Henderson thus obtained the follo&g partition function eEa/RT f = (1 + n n e - = ~ ~ v d ~ ~ c vf,- v d ) .
IT
U, = [(V./N)'/a - (b/4N)'/.I" (21) .. . . ~, where b is the van der Waals constant and b/4N is the net molecular volume. We, therefore, replace the simple Einstein arti it ion function in etrudnn 1 1 1 )
where f, is the Debye partition, f, is the rotational partition function and depends upon the species, and y is related to the translational partition function. The reader is referred to the original paper for more details on the partition function, since it is inherently complex. Table 4 lists some calculated and observed properties a t the melting, boiling, and critical point. Figure 5 shows the vapor pressure curves versus temperature for several species.
When 1 e >> T , equation (22) reduces to the Einstein partition function in equation (11); and when T >> 1 e, it reduces to a translational partition function. The equation of state obtained from equation (11) modified by means of equation (22) is
Dense Liquids and Gases ( 1 I )
where
The theory as described above works very well for normal temperatures and pressures. A few simple corrections, which are natural consequences of the model, extend the useful range of the theory to very high pressures and temperatures with highly satisfactory results. In using an Einstein oscillator in the solid-like portion of the partition function it was tacitly assumed that 1 0 >> T , where 1 is the vibrational quantum number. The exact partition function of a one dimensional harmonic oscillator terminating at the 1-th level is given by
and 1 is determined by the relation I 9 d J 3 R . The usual Einstein partition function is obtained when 10 >> T. This condition, 1 9 >> T , however, does not hold at high temperatures for substances with a low E, (i.e., low molecular weight substances such as argon and neon). When this condition is no longer satisfied, the one dimensional oscillator becomes a one dimensional gas with one translational degree of freedom:
where v, is the molar free volume in the solid. Since a t high temperatures molecules act like hard spheres, v,may he represented by Table
4.
2 =
V/V,
and
,
=
E,/RT +
1 - e-la/T
i1
e-VT
.
1
+ e - r s / ~ . (2smkT)'Ixvx,2/ h
I n addition to the quantum effect mentioned above, the pressure effect on V , must also be considered. Thus
p is the solid compressibility and A P is the excess pressure above a standard pressure, say the vapor pressure a t the melting point. Since P is rather small, 10-= to lo-' atm-', equation (25) reduces to V 8 if At' is small. For argon, as an example, the pressure effect is negligible if A P is less than 500 atmospheres. From equation (23) the second virial coefficient may be determined: B(T) = V.(r - a, - Inn)
(26)
and compared with experiment. Some results for nitrogen are shown in Figure 6. Figure 7 shows the compressibility factor, P V / R T , as a function of density for argon a t 150°C. The values obtained by Kirkwood and Hirschfelder are also shown.
Calculated and Observed Thermodvnamic Prooerties of Liauid Hvdroaen -
Tm (OK) Pm (atm)
V , (omamole-') AS, (cal mole-' deg-')
p-Hn
n-HB
H-D
0-Dz
n-D2
(13.84) 13.84 0.07388 0.06942 26.213 26.176 1.932
(13.94) 13.94 0.07589 0.07085 26.093 26.108 1.936
(16.60) 16.60 0.1236 0.1221 24.491 24.487 2.048
(18.63) 18.63 0 1706 0.1678 23.262
(18.73) 18.73 0.1724 0.1692 23.155 23.162 2.210
2.198
Tb ("K)
Vs (ema mole-])
calc obs calc obs eak obs
calc obs calc obs ale obs
AS*(cal mole-' deg-')
eale
obs
T. (OK)
calc obs calc obs
P. (atm)
V. (cma mole-') 65.5
568 / Journal of Chemical Education
...
62.8
60.3
...
cale obs
Surface Tension (12)
As noted earlier, the partition function used in this theory reduces to a solid partition function when V = V, and to a gaseous partition function when V>>V,. Thus it is reasonable to expect that this partition function will also represent the surface layer, if the density gradient is taken into account. For a simple close packed liquid such as argon, it is supposed that the molecules on the surface will orient such that each molecule will tend to have six neighbors in the
same layer, three neighbors below and three above. Only E,, the bulk energy of sublimation, should be appreciably diierent for a surface and a bulk molecules. The energy E, of a molecule in the ith surface layer is given by the relationship.
Here pi is the density of the ith layer and the subscript increases with depth in the liquid. If p i t 1 > pt, then more molecules are in the ith 1 layer with respect to the ith layer. That is, the molecules 1 layer are bound more tightly, or IEst+~l in the ith >1EJ. When i is the top liquid layer and (i - 1) the first gas layer, we can write
+
+
pj-l/Pi
Figure 6.
The second viriol coefficients of nitrogen versus tempemlure.
= el - ( E s / Z R T ) (1- T / T r ) l
(28)
This corresponds to a molecule in the top liquid layer possessing six nearest neighbors in the same plane, three in the neighboring liquid plane, but none in the gas layer a t low temperatures. A molecule in the first gas layer, on the other hand, has only the three neighbors in the first liquid layer and no neighbors in the same gas layer or in the gas layer above. Thus, the extra binding energy of a molecule in the first liquid layer over one in the first gas layer corresponds to six more neighbors out of a possible twelve or an extra binding energy of E,/2. In keeping with this result the exponent of equation (28) reduces to E,/2RT a t low temperatures and vanishes a t T = T,, as it must since the densities become equal a t the critical temperature. The Gibbs free energy per mole, G, = As PVr, for each surface layer may now be calculated by means of eauations (27) ~. and (28) . . bv using an iterative process td determine the p(s. he surface tension, may be calculated from the relationship
+
y,
Figure
versus densities at 150' C. Signiflconl Structure theory; - - r e p r e s e n t s reprerents Hinchfelder's valuer
7. Compressibility factors of argon
-represents volues from Kirkwood's valuer;
-.-.
Table 5.
where AGi is the difference in free energy of one of the surface layers and the bulk liquid. The remaining factor is the volume of a layer of surface lattice one cm2 in area assuming a face centered cubic lattice. Other types of packing would give a different factor. Surface tensions have been calculated for argon, nitrogen, and methane a t several temperatures. The results are given in Table 5. The columns labeled lst, 2nd, and 3rd are the per cent contributions to the
Surface Tension of Liauidr (dvnelcm)
-
T (OK) Argon 83.85(m) 85.5 87.29(b) 90.0 Nitrogen 63.14(m) 70.0 77.34(b) 90.0 Methane 90.65(m) 100.0 111.67(b) 120.0 --
Vz (cc)
V o (CC)
1st
%
2nd
O/o
3rd %
Surface Tension eak obs
A
(%)
p~ ~
Volume 40, Number I I, November 1963
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569
surface tension from the respective layers. The Eotvos constant, K,, is defined by KB = yV'/a/(Tr - 1')
(30)
{
' I = 7- X'fa+XOfu =
S
'I, =
The Eotvos Constants
Ks (calc) average
Argon 8385(m) 85.5 87.29(b) 90.0 Nitrogen 63.14(m) 70.0 77.34(b) 90.0
Methane 9065(m) loo0 111.67(b) 12n
n
'I.+ v- -vv
(31)
Kinetic theory considerations lead us to
and are given in Table 6. Table 6.
v. XXin,+Xgn.= . n ' 2 -(mkT)'/. 3r'd2
(32)
where d is the effective molecular diameter, for the gas-like viscosity. 7, may be calculated by means of reaction rate theory in the following manner. If K , is the frequency of a molecule jumping into the ith neighboring empty lattice site when the shear stress, f, is zero, then, when f is not zero, the work helping the molecule to move forward is the force f X d a (where Xp and X3 represent intermolecular distances in the shear plane) multiplied by the distance the molecule has jumped, X cos er. Thus the rate of jumping into the ith site is K.e(fA=ArAcos&/zkT)
(33)
The factor 2 is inserted since it is assumed that the top of the potential barrier is halfway between the two equilibrium positions. 7, may then be written
Transport Properties (9)
Significant Structure theory includes a general theory of transport properties implicitly. It is interesting to note that when the liquid and solid phases have the same free energy, as they do a t the melting point, their viscosities differ;i.e., differentshear stresses are needed to maintain unit velocity gradients. This difference in viscosity must arise from the change in structure. Although they melt a t different temperatures, normal liquids have about the same viscosity (-2 centipoises) a t their melting point. Long ago Batschinski (14) noted that the reciprocal of the viscosity, the fluidity, is proportional to the volume of the liquid minus a volume very close to that of the solid, i.e., what we call the excess volume. This observation is a natural consequence of the significant structure approach to liquids. Consider the shear surface between two layers of molecules. Shearing occurs as molecules on either side of the shear plane move into those fluidized vacancies which will relieve the accumulated stress. Obviously, this motion can only occur during the time a vacancy is available for occupancy. The probability of such a vacancy being a t a particular point will be proportional to the number of vacancies, V - V,. Thus the rate of shear, the fluidity, is proportional to this excess volume. This explains Batschinski's results. Moreover, most liquids have about the same concentration of vacancies a t their melting points. Since the rate of jumping is nearly the same for each liquid a t its T,,,, we would expect liquids to have the same visocity a t their very different melting points. Most liquids do. Since there are different types of degrees of freedom (solid-like and gas-like) in the liquid, they both must be taken into account in calculating the viscosity. The shear plane lies between two molecular layers. If a fraction, X,, of the shear plane is covered by solid-like molecules and the remaining fraction, X,, by gas-like molecules, then the viscosity, 7, which is the ratio of shear stress, f, to rate of strain, S, is given by 570
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Journol of Chemical Education
where XI, is the distance between the two molecular shearing planes. Expanding the exponential, we obtain
Since the sites are randomly distributed over the solid angle, we may average cosel (which becomes zero) and cos2si(which becomes 1/3) :
Now according to this liquid t h e o ~ ya molecule can jump into any of the neighboring n(V - V,)/V, vacancies, and so from absolute reaction rate theory we have
where F and F* are the partition functions for the molecule in the initial and activated states, and ro = (afE,/RT)(VJV - V,) is the activation energy. F and F* are given by
taking XZh2X3/X1= V s / N and substituting equations (37) and (38) into (36), we obtain ~h
'I'
=P
.1 - 1
e-a/T
.6 En V - V,
E ~ V ~ / R T ( I ~ - V(39) S)
Substitution of equations (39) and (32) into (31) then yields an expression for the viscosity. The self-diffusion coefficient, D, may be calculated from the relationship
where is the effective number of neighbors of a molecule lying in the same plane. For a close packed structure this is six. Figures 8, 9, and 10 show- the viscosity of liquid argon, of liquid sodium, and the selfdiffusion coefficient of molten sodium.
near future. I n the article we give the reader a brief indication as to how we are proceeding on these different liquids. Binary Mixtures
A most exciting problem currently being considered is binary mixtures. If we can successfully apply the theory to mixtures, then a whole new panorama of problems is brought into view. Consider the reaction aA+bB$eC+dD
One can show that the net rate, r, of this reaction is
where IC is the transmission coefficient and F* is the partition function of the activated complex. The X's are the absolute activities ~f the respective species and must be obtained from the liquid theory. Thus, if we are able to compute the activities, we will be able to calculate reaction rates and solubilities. The partition function we are using for binary mixtures is the following Figure 8.
V i ~ o s i t yof liquid argon in centipoirer rerrvr temperature.
where
The factorial term is the usual mixing factor which arises from assuming random mixing of the two different species. fi and f2 are the solid- and gas-lie portions of the partition functions for the respective species. The main problem with mixtures is to determine how the parameters E,, V,, n, and a vary with concentration. Many years ago van der Waals suggested that the energy and volume should vary in the following way: MO
XO
700
900
I100
T IPKI
~i~~~~ 9. Viscosity of liquid sodium in centipoirer verrvr temperature.
where XI and X z are the respective mole fractions, and the E,'s and V,'s are the parameters for the pure components. It is wishful thinking to assume that these equations will work for liquids. They are much more applicable to gases. Averaging formulas developed by Prigogine ( I S ) seem more promising. We are presently developing procedures to determine the parameters as functions of concentration. Wafer
Figure 10.
Self-diffusion coefficient of molten sodium versus temperoture.
Current Investigations
At this time we are applying significant structure theory to water, oxygen, and binary mixtures. We are also doing some further work on molten metals. The results of these investigations will appear in the
An interesting problem is presented by water due to the strong hydrogen bonding and the presence of diierent species. We assume that water is composed primarily of two species: "structured" water molecules which are hound in a rigid lattice and thus do not rotate and "unstructured" water molecules which are assumed to be freely rotating. At low temperatures the structural water molecules are the predominant species. As the temperature increases, more and more molecules begin to rotate until several hundred degrees above the boiling point they are all freely rotating. The ratio of the two species is controlled by an equilibrium constant. The partition function is Volume 40, Number 1 1 , November 1963
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571
where
At low temperatures f, reduces to a partition function for the structured water molecules which are treated as 6degree Einstein oscillators. At higher temperatures f, becomes a partition function for freely rotating water molecules. Since the strong hydrogen bonds will cause clustering in the gas phase, f, includes a term for dimer water molecules. E, and V, are related by an equilibrium constant to the sublimation energies and solid volumes of the structured and unstructured species. Preliminary calculations indicate that the results for water will be nearly as good as has been obtained by significant structure theory for other pure liquids. Conclusions
This theory can be applied to a tremendous range of liquids, and the degree of success has ranged from excellent to good. There is no other theory which will work as well for so many differentclasses of liquids. The theory rests on a model of the liquid state which is reasonable and successful. Its usefulness cannot be doubted. Given some properties of the solid (the sublimation energy, the volume of the solid, and the Einstein characteristic temperature) all the thermodynamic properties may be calculated from equations (12) and (13) and similar expressions. To determine n and a, the liquid volume, Vm, a t the melting point must be known. Using an initial guess of V , and a method of successive approximations, we can obtain consistent values for n and a. In those cases when the properties of the solid are unknown or for some special liquids, the parameters E,, 8, and V, may he determined from the experimental pressure, entropy, and liquid volume a t the melting point. The usefulness of the model is clear; the meaning of the model needs careful elaboration. Since the model can he applied to so many different liquids, can be used to calculate transport properties and surface tensions, and can be used to compute the radial distribution
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Journal of Chemical Education
function, the basic tenents of the theory, while undoubtedly needing refinement, are none the less good approximations. Certainly the idea that a molecule in the liquid state sometimes possesses external vibrational degree of freedom and a t other times translational degrees of freedom cannot be wrong. The division of these degrees of freedom into their respective fractions V,/V and (V - V,)/V is a good first approximation and the estiiatition of the degeneracy factor needs elaboration. A criticism frequently raised against this theory is that it has a number of adjustable parameters. We have tried to point out in this article that there are no adjustable parameters. The parameters E,, e, and V, are obtained from the solid state. If there were a theory of solids, we could calculate these quantities; since there is not, we must obtain them by experimentation. The parameters a and n maybe ohtained from equations (7) and (10). If all spherical molecules exactly obeyed a reduced equation of state, the same dimensionless values for n and a could be used for all liquids. In terms of a reduced equation of state an analogy can be drawn between T,and a, and V , and n. Since this theory is applicable to a greater degree of accuracy than a reduced equation, it is necessary to use accurate values of n and a. Acknowledgment
The authors wish to express appreciation to the U. S. Army Ordnance under contract DA-ARO(D)-31-124G298 for partial financial support of this research, and one of them (R.P.M.) expresses appreciation to the National Institutes of Health for an NIH Postdoctoral Fellowship. Literature Cited
EISENSTEIN, A.
AND
GINGRICH, N., Phys. Reu., 62, 261
(1942).
WALTER J. AND ERYING, H., J . Chm. Phvs., 9,393 (1941). EYEING, H., J. Chem. Phys., 4, 283 (1936). FULLER, E. J., REE,T., AND EYRING, H., P ~ cNal. . Acad. Sei., 45, 1594 (1959). THOMSON, T. R., EYRING, H.,AND REE, T., PTOC. Nal. A d . Sei., 46, 336 (1960). CHANG. S. AND PARK.H. S.. J . Korean Chm. Soc.,. 7.. July (1963).
HENDERSON, D., EYRING,H., AND FELIX,D., J. Phya Chem., 66, 1128 (1962). McLAuGn~rn,D., To be puhlished. CARLSON, C. M., EYRING,H., AND REE, T., P m . Nat Acad. Sci., 46, 649 (1960). CARLSON, C. M., EYRING,H.,AND REE. T., Proc. Nat. Acad. Sci., 46, 333 (1960). REE, T. S., REE, T., AND EYRING, H.,PTOC. Nat. Acad. Sci., 48, 501 (1962). CHANG, S., ef al., Second Symposium on Thermophysied Properties, January, 1962. Princeton University, Princeton, New Jersey. PRIGOGINE, I., "The Molecular Theory of Solutions," Interscience Publishers, Inc., New York, 1957, chap. 9. BATSCHINSKI, A. J., 2. physik. Chem., 84, 643 (1913).
