Chapter 4
Intermolecular Forces, Corresponding States and Osmotic Systems

Thermodynamic properties of any pure substance are determined by intermolecular forces that operate between the molecules of that substance. Similarly, thermodynamic properties of a mixture depend on intermolecular forces that operate between the molecules of the mixture. The case of a mixture, however, is necessarily more complicated because consideration must be given not only to interaction between molecules belonging to the same component, but also to interaction between dissimilar molecules. To interpret and correlate thermodynamic properties of solutions, it is therefore necessary to have some understanding of the nature of intermolecular forces. The purpose of this chapter is to give a brief introduction to the nature and variety of forces between molecules.

We must recognize at the outset that our understanding of intermolecular forces is far from complete and that quantitative results have been obtained for only simple and idealized models of real matter. Further, we must point out that analytic relations that link intermolecular forces to macroscopic properties (i.e., statistical mechanics) are also at present limited to relatively simple and idealized cases.¹ It follows, therefore, that, for most cases, we can use our knowledge of intermolecular forces in only an approximate manner to interpret and generalize phase-equilibrium data. Molecular physics is always concerned with models and we must beware whenever we are tempted to substitute models for nature. Frequently, the theory of intermolecular forces gives us no more than a qualitative, or perhaps semiquantitative, basis for understanding phase behavior, but even such a limited basis can be useful for understanding and correlating experimental results.

¹ However, thanks to advances in molecular simulation with powerful computers, it is now possible to calculate physical properties of some bulk substances provided that we have quantitative knowledge of their intermolecular forces. There is good reason to believe that molecular simulation will become increasingly useful as computers become even more powerful.

While the separation between molecular physics and practical problems in phase behavior is large, every year new results tend to reduce that separation. There can be no doubt that future developments in applied thermodynamics will increasingly utilize and rely on statistical mechanics and the theory of intermolecular forces.

When a molecule is in the proximity of another, forces of attraction and repulsion strongly influence its behavior. If there were no forces of attraction, gases would not condense to form liquids, and in the absence of repulsive forces, condensed matter would not show resistance to compression. The configurational properties of matter can be considered as a compromise between those forces that pull molecules together and those that push them apart; by configurational properties we mean those properties that depend on interactions between molecules rather than on the characteristics of isolated molecules. For example, the energy of vaporization for a liquid is a configurational property, but the specific heat of a gas at low pressure is not.

There are many different types of intermolecular forces, but for our limited purposes here, only a few important ones are considered. These forces may be classified under the following arbitrary but convenient headings:

• Electrostatic forces between charged particles (ions) and between permanent dipoles, quadrupoles, and higher multipoles.

• Induction forces between a permanent dipole (or quadrupole) and an in duced dipole, that is, a dipole induced in a molecule with polarizable electrons.

• Forces of attraction (dispersion forces) and repulsion between nonpolar molecules.

• Specific (chemical) forces leading to association and solvation, i.e., to the formation of loose chemical bonds; hydrogen bonds and charge-transfer complexes are perhaps the best examples.

An introductory discussion of these forces is presented in most of this chapter, with special attention to those acting between nonpolar molecules and to the molecular theory of corresponding states. The chapter also includes some brief discussion of osmotic pressure and micelles.

4.1 Potential-Energy Functions

Molecules have kinetic energy as a result of their velocities relative to some fixed frame of reference; they also have potential energy as a result of their positions relative to one another. Consider two simple, spherically symmetric molecules separated by the distance r. The potential energy Γ shared by these two molecules is a function of r, the force F between the molecules is related to the potential energy by

(4-1)

The negative of the potential energy, i.e., Γ(r), is the work which must be done to separate two molecules from the intermolecular distance r to infinite separation. Intermolecular forces are usually expressed in terms of potential-energy functions. The common convention is that a force of attraction is negative and one of repulsion is positive.

In the simplified discussion above, we have assumed that the force acting between two molecules depends on their relative position as specified by only one coordinate, r. For two spherically symmetric molecules, such as argon atoms, this assumption is valid; but for more complicated molecules, other coordinates, such as the angles of orientation, may be required as additional independent variables of the potential-energy function. A more general form of Eq. (4-1) is

(4-2)

where is the gradient and θ, Φ,… designate whatever additional coordinates may be needed to specify the potential energy.

In the next sections, for simplicity, we assume that ions, atoms, or molecules are in free space (vacuum). For electrostatic forces, extension to a medium other than vacuum is made through introduction of the relative permittivity (or dielectric constant) of the medium. While we use the SI unit system throughout, in some cases we also give units in other systems that remain in common use.

4.2 Electrostatic Forces

Of all intermolecular forces, those due to point charges are the easiest to understand and the simplest to describe quantitatively. If we regard two point electric charges of magnitudes q_i and q_j, respectively, separated from one another in vacuo by distance r, then the force F between them is given by Coulomb’s relation, sometimes called the inverse-square law:²

² In the still-frequentiy-used cgs system of units, Coulomb’s law (in vacuo) is written

where F is in dynes, r is in centimeters, and q is in esu [or (erg cm)^1/2]. Note thai in Eq. (4-3) the proportional factor l/47xε_o implies that SI units are being used; this factor does not appear when cgs (or other) units are used.

(4-3)

where F is in newtons, q in coulombs, r in meters, and ε_o, the dielectric permittivity of a vacuum, is ε_o = 8.85419×10^-12 C² J^-1 m^-1.

Upon integrating, the potential energy is

(4-4)

The usual convention is that the potential energy is zero at infinite separation. Substituting Γ = 0 when r = ∞, the constant of integration in Eq. (4-4) vanishes.

For charged molecules (i.e., ions), q_i and q_j are integral multiples of the unit charge e; therefore, the potential energy between two ions is

(4-5)

where Z_i and Z_j are the ionic valences and e = 1.60218×10^-19 C.³

³ In the cgs unit system, the unit of charge is esu = 1 (erg cm)^1/2 = 3.33569xl0^-10C.

For a medium other than vacuum, Eq. (4-5) becomes

(4-5a)

where ε, the absolute permittivity (C² J^-1 m^-1), is defined by ε = ε₀ε_r;ε₀ is the permittivity of a vacuum, and ε_r is the (dimensionless) dielectric constant or permittivity relative⁴ to that of a vacuum; ε_r is unity in vacuo but greater than unity otherwise (for water at 25°C, ε_r = 78.41).⁵

⁴ The relative permittivity of a substance is easily measured by comparing capacitance C of a capacitor with the sample to that without the sample (C_o) and using ε_r = C/C_o. Note that the magnitude of ε_r can have a significant effect on the magnitude of the interaction between ions in solution. For example, water at 25°C reduces the long-range interionic coulombic interaction energy by about two orders of magnitude from its value in vacuo. When the dielectric constant is used in Eq. (4-5a) and similar equations, the medium is considered a structureless continuum.

⁵ According to the current international standard formulation (D. P. Fernández, A. R. H. Goodwin, E. W. Lemmon, J. M. H. Levelt-Sengers, and R. C. Williams, 1997, J. Phys. Chem. Ref. Data, 26: 1125).

Compared to others physical intermolecular energies, the magnitude of the Coulomb energy as given by Eq. (4-5) is large and long range. To illustrate, consider the isolated ions Cl^– and Na⁺ in contact. Distance r is given by the sum of the two ionic radii (r = 0.276 nm or 2.76 Å); the potential energy between these two ions in vacuo is

This energy is about 200kT, where k is Boltzmann’s constant; kT is the thermal energy [at room temperature, kT = (1.38 × 10^-23 J K^-1)×(300 K) = 0.0414×10^-19 J]. 200kT is of the same order of magnitude as typical covalent bonds. Only when the two ions are about 560 Å far apart is the Coulomb interaction equal to kT.

Electrostatic forces between ions are inversely proportional to the square of the separation and therefore they have a much longer range than most other intermolecular forces that depend on higher powers of the reciprocal distance. These electrostatic forces make the dominant contribution to the configurational energy of salt crystals and are therefore responsible for the very high melting points of salts. In addition, the iong-range nature of ionic forces is, at least in part, responsible for the difficulty in constructing a theory of electrolyte solutions.

Electrostatic forces can arise even for those particles that do not have a net electric charge. Consider a particle having two electric charges of the same magnitude e but of opposite sign, held a distance d apart. Such a particle has an electric couple or permanent dipole moment μ defined by

(4-6)

μ = e d

Asymmetric molecules possess permanent dipoles resulting from an uneven spatial distribution of electronic charges about the positively charged nuclei. Symmetric molecules, like argon or methane, have zero dipole moment and those molecules having very little asymmetry generally have small dipole moments. Table 4-1 gives a representative selection of molecules and their dipole moments. The common unit for dipole moment is the debye (D). The dipole moment of a pair of charges +e and -e, separated by 0.1 nm (or 1 Å) is μ = (1.60218×10^-19 C)×(10^-10 m) = 1.60218×10^-29 C m, corresponding to 4.8 D [1 D = 3.33569×10^-30 C m = 10^-18 esu cm = 10^-18 (erg cm³)^1/2].

Table 4-1 Permanent dipole moments.*

The potential energy of two permanent dipoles i and j is obtained by considering the coulombic forces between the four charges. The energy of interaction depends on the distance between dipole centers and on the relative orientations of the dipoles, as illustrated in Fig. 4-1, where angles θ and Φ give the orientation of the dipole axes.

Figure 4-1 Orientation of two dipoles.

If the distance r between the dipoles is large compared to d_i and d_j, the potential energy is

(4-7)

The orientation making the potential energy a maximum is that corresponding to when the dipoles are in the same straight line, the positive end of one facing the positive end of the other; the energy is a minimum when the dipoles are in a straight line, the positive end of one facing the negative end of the other.

In an assembly of polar molecules, the relative orientations of these molecules depend on the interplay of two factors: The presence of an electric field set up by the polar molecules tends to line up the dipoles, whereas the kinetic (thermal) energy of the molecules tends to toss them about in a random manner. We expect, therefore, that as the temperature rises, the orientations become more random until in the limit of very high temperature, the average potential energy due to polarity becomes vanishingly small. This expectation is confirmed by experimental evidence; whereas at low and moderate temperatures the behavior of polar gases is markedly different from that of nonpolar gases, this difference tends to disappear as the temperature increases. It was shown by Keesom (1922) that at moderate and high temperatures, orientations leading to negative potential energies are preferred statistically. The average potential energy Γ_ij between two dipoles i and j in vacuum at a fixed separation r is found by averaging over all orientations with each orientation weighted according to its Boltzmann factor (Hirschfelder et al., 1964). When the Boltzmann factors are expanded in powers of 1/kT, Γ_ij becomes

Equation (4-8) indicates that for a pure polar substance (i = j) the potential energy varies as the fourth power of the dipole moment. A small increase in the dipole

(4-8)⁶

⁶ This equation is based on a Boltzmann average

where dΩ is the element of the solid angle: dΩ = sin θ₁ sin θ₂ dθ₁ dθ₂ dΦ, where Φ = Φ₁) – Φ₂. However, it has been argued (J. S. Rowlinson, 1958, Mol. Phys., 1:414) that the proper average is

When that average is used, the numerical coefficient in Eq. (4-8) is -1/3 instead of -2/3. Equations (4-7) and (4-8) are for ideal dipoles, i.e. where r >> d. Therefore, these equations fail at small separations where r and d may be of similar magnitude. For real (non-ideal) dipoles, calculation of r and Γ and becomes more complicated. See Cohen et al., 1996, J. Coll. Interface Sci., 177:276.

moment can therefore produce a large change in the potential energy due to permanent dipole forces. Whereas the contribution of polar forces to the total potential energy is small for molecules having dipole moments of 1 debye or less, this contribution becomes increasingly significant for small molecules having larger dipole moments. Section 4.4 gives a quantitative comparison of the importance of polar forces relative to some other intermolecular forces.

In addition to dipole moments, it is possible for molecules to have quadrupole moments due to the concentration of electric charge at four separate points in the molecule. The difference between a molecule having a dipole moment and one having a linear quadrupole moment is shown schematically:⁷

⁷ Note that two dipoles shown are identical because if the top molecule is rotated 180°, we obtain the bottom molecule. However, the two quadrupoles are not identical; we cannot obtain one from the other by rotation.

For example, carbon dioxide, a linear molecule, has no dipole moment but its quadrupole moment is sufficiently strong to affect its thermodynamic properties that are different from those of other nonpolar molecules of similar size and molecular weight. For the simplest case, a linear molecule, quadrupole moment Q is defined by the sum of the second moments of the charges:

(4-9)

where the charge e_i is located at a point a distance d_i away from some arbitrary origin and where all charges are on the same straight line. Quadrupole moment Q is independent of the position of the origin provided that the molecule has no net charge and no dipole moment. For nonlinear quadrupoles or for molecules having permanent dipoles, the definition of the quadrupole moment is more complicated (Buckingham, 1959; Buckingham and Disch, 1963).

Experimental determination of quadrupole moments is difficult and not many measurements have been made (Buckingham, 1959; Buckingham and Disch, 1963; Flygare and Benson, 1971; Sutter and Flygare, 1976). Table 4-2 gives quadrupole moments⁸ for some molecules whose symmetry is sufficient to allow a single scalar to specify the moment. (To distinguish between the two quadrupole structures shown above, it is necessary to assign a direction – plus or minus – to the magnitude of the quadrupole moment). Flygare (1971) and Sutter and Flygare (1976) have tabulated quadrupole moments for molecules of low symmetry.

Table 4-2 Quadrupole moments for selected molecules.*

⁸ The SI unit for quadrupole moment is C m²; 1 C m² = 0.29979×10¹⁴ esu cm² = 0.29979×10¹⁴ erg^1/2 cm^5/2. The unit buckingham (B) is also commonly used: 1 B = (1 debye)×(1 Å) = 10^-26 esu cm².

The potential energy between a quadrupole and a dipole, or between a quadrupole and another quadrupole, is a function of the distance of separation and the angles of mutual orientation. The average potential energy is found by averaging over all orientations; each orientation is weighted according to its Boltzmann factor (Hirschfelder et al., 1964). Upon expanding in powers of 1/kT, we obtain:

For dipole i-quadrupole j:

(4-10)

For quadrupole i-quadrupole j:

(4-11)

Whereas the scientific literature on dipole moments is extensive, considerably less is known about quadrupole moments and little work has been done on higher multipoles such as octapoles and hexadecapoles. The effect of quadrupole moments on thermodynamic properties is already much less than that of dipole moments and the effect of higher multipoles is usually negligible (Parsonage and Scott, 1962). This relative rank of importance follows because intermolecular forces due to multipoles higher than dipoles are extremely short range; for dipoles, the average potential energy is proportional to the sixth power of the inverse distance of separation and for quadrupoles the average potential energy depends on the tenth power of the reciprocal distance. For higher multipoles, the exponent is larger.

4.3 Polarizability and Induced Dipoles

A nonpolar molecule such as argon or methane has no permanent dipole moment but when such a molecule is subjected to an electric field, the electrons are displaced from their ordinary positions and a dipole is induced. In fields of moderate strength, the induced dipole moment μⁱ is proportional to the field strength E:

(4-12)

where the proportionality factor α is a fundamental property of the substance. It is called the polarizability, that is, how easily the molecule’s electrons can be displaced by an electric field. Polarizability can be calculated in several ways, most notably from dielectric properties and from index-of-refraction data. For asymmetric molecules, polarizability is not a constant but a function of the molecule’s orientation relative to the direction of the field. Table 4-3 gives average polarizabilities of some representative molecules. In SI units (electric field in V m^-1 and μ in C m), polarizability has the units C² J^-1 m². However, it is common practice to present polarizabilities in units of volume⁹ (such as cm³), using the relation

⁹ 1 C² J^-1 m² = 0.8988×10¹⁶ cm³.

Table 4-3 Average polarizabilities *.

The units of 4πε_o are C² J^-1 m^-1, and therefore α’ (also called the polarizability volume) has dimensions of volume.

When a nonpolar molecule i is situated in an electric field set up by the presence of a nearby polar molecule j, the resultant force between the permanent dipole and the induced dipole is always attractive. The mean potential energy was first calculated by Debye and is usually associated with his name. It is given by

(4-13)

Polar as well as nonpolar molecules can have dipoles induced in an electric field. The general Debye formula, therefore, for the mean potential energy due to induction by permanent dipoles, is

(4-14)

An electric field may also be caused by a permanent quadrupole moment. In that case, the average potential energy of induction between a quadrupole j and a nonpolar molecule i is again attractive; if both molecules i and j have permanent quadrupole moments,

(4-15)

For molecules with a permanent dipole moment, the potential energy due to induction is usually small when compared to the potential energy due to permanent dipoles; and similarly, for molecules with a permanent quadrupole moment, the induction energy is usually less than that due to quadrupole-quadrupole interactions.

4.4 Intermolecular Forces between Nonpolar Molecules

The concept of polarity has been known for a long time but until about 1930 there was no adequate explanation for the forces acting between nonpolar molecules. It was very puzzling, for example, why such an obviously nonpolar molecule as argon should nevertheless show serious deviations from the ideal-gas laws at moderate pressure. In 1930 it was shown by London that so-called nonpolar molecules are, in fact, nonpolar only when viewed over a period of time; if an instantaneous photograph of such a molecule were taken, it would show that, at a given instant, the oscillations of the electrons about the nucleus had resulted in distortion of the electron arrangement sufficient to cause a temporary dipole moment. This dipole moment, rapidly changing its magnitude and direction, averages zero over a short period of time; however, these quickly varying dipoles produce an electric field which then induces dipoles in the surrounding molecules. The result of this induction is an attractive force called the induced dipole-induced dipole force. Using quantum mechanics, London showed that, subject to certain simplifying assumptions, the potential energy between two simple, spherically symmetric molecules i and j at large distances is given by

(4-16)¹⁰

¹⁰ When molecules i and j are in a medium whose dielectric constant is s, it is tempting to use Eq. (4-16) with e replaced by s. However, it is incorrect to do so. The dielectric constant E can be used in potentials arising from electrostatics (as in the previous section) but not for a potential arising from interactions between fluctuating induced dipoles. Equation (4-16) is the first term of a series in (1/r). The next term is proportional to (1/r)⁸. Therefore, Eq. (4-16) is not valid for very small r, it is inappropriate especially when r ≤ σ, where σ is the molecular diameter.

where h is Planck’s constant and v₀ is a characteristic electronic frequency for each molecule in its unexcited state. This frequency is related to the variation of the index of refraction n with light frequency v by

(4-17)

where c is a constant. It is this relationship between index of refraction and characteristic frequency that is responsible for the name dispersion for the attractive force between nonpolar molecules.

For a molecule i, the product hv_{0 i} is very nearly equal to its first ionization potential I_i.¹¹ Equation (4-16) is therefore usually written in the form

¹¹ The first ionization potential is the work which must be done to remove one electron from an uncharged molecule M: M → e^- + M⁺. The second ionization potential is the work needed to remove the second electron according to: M⁺ → e^- + M²⁺.

If molecules i and j are of the same species, Eq. (4-18) reduces to

(4-18)

Equations (4-18) and (4-19) give the important result that the potential energy between nonpolar molecules is independent of temperature and varies inversely as the sixth power of the distance between them. The attractive force therefore varies as the reciprocal seventh power. This sharp decline in attractive force as distance increases explains why it is much easier to melt or vaporize a nonpolar substance than an ionic one where the dominant attractive force varies as the reciprocal second power of the distance of separation.¹²

(4-19)

¹² Various authors, in addition to London, have derived expressions for the attractive portion of the potential function of two, spherically symmetric, nonpolar molecules. These expressions all agree on a distance dependence of r^-6 but the coefficients differ considerably. A good review of this subject is presented in Advances in Chemical Physics, J. O. Hirschfelder, (Ed.), 1967, Vol. 12, New York: Wiley-Interscience. See also Stone (1996).

London’s formula is more sensitive to the polarizability than it is to the ionization potential, because, for typical molecules, α is (roughly) proportional to molecular size while I does not change much from one molecule to another. Table 4-4 gives a representative list of ionization potentials. Since the polarizabilities dominate, it can be shown that the attractive potential between two dissimilar molecules is approximately given by the geometric mean of the potentials between the like molecules at the same separation. We can rewrite Eqs. (4-18) and (4-19):

Table 4-4 First ionization potentials.*

(4-20)

where k′ is a constant that is approximately the same for the three types of interaction: i-i, j-j, and i-j. It then follows that

Equation (4-21) gives some theoretical basis for the frequently applied geometric-mean rule, which is so often used in equations of state for gas mixtures and in theories of liquid solutions. We have already used this rule in Sec. 3.5 [compare Eq.(3-69)].

(4-21)

To show the relative magnitude of dipole, induction, and dispersion forces in some representative cases, London (1937) has presented calculated potential energies for a few simple molecules. His results are given in the form

(4-22)

where B is calculated separately for each contribution due to dipole, induction, and dispersion effects. In these calculations Eqs. (4-8), (4-14), and (4-19) were used; Table 4-5 gives some results similar to those given by London. The computed values of B indicate that the contribution of induction forces is small and that even for strongly polar substances, like ammonia, water, or acetone, the contribution of dispersion forces is far from negligible.

Table 4-5 Relative magnitudes of intermoiecuiar forces between two identical molecules at 0°C.

Table 4-6 gives some calculated results for intermoiecuiar forces between two molecules that are not alike. In these calculations Eqs. (4-8), (4-14), and (4-18) were used. Again, we notice that polar forces are not important when the dipole moment is less than about 1 debye and induction forces always tend to be much smaller than dispersion forces.

Table 4-6 Relative magnitudes of intermoiecuiar forces between two different molecules at 0°C.

London’s formula does not hold at very small separations where the electron clouds overlap and the forces between molecules are repulsive rather than attractive. Repulsive forces between nonpolar molecules at small distances are not understood as well as attractive forces at larger distances. Theoretical considerations suggest that the repulsive potential should be an exponential function of intermolecular separation, but it is more convenient (Amdur et al., 1954) to represent the repulsive potential by an inverse-power law of the type

(4-23)

where A is a positive constant and n is a number usually taken to be between 8 and 16.

To take into account both repulsive and attractive forces between nonpolar molecules, it is customary to assume that the total potential energy is the sum of the two separate potentials:

(4-24)

where A, B, n, and m are positive constants and where n > m. This equation was first proposed by Mie (1903) and was extensively investigated by Lennard-Jones. Equation (4-24) forms the basis of a variety of physicochemical calculations; it has been used especially to calculate thermodynamic and transport properties of dilute nonpolar gases (Hirschfelder et al., 1964).

4.5 Mie’s Potential-Energy Function for Nonpolar Molecules

Equation (4-24) gives the potential energy of two molecules as a function of their separation and it is apparent that at some distance r_min, Γ is a minimum; this minimum energy is designated by Γ_min. By algebraic rearrangement, Mie’s potential can be rewritten

(4-25)¹³

¹³ Parameter ε in Mie’s equation should not be confused with dielectric constant ε.

where ε = -Γ_min and where σ is the intermolecular distance when Γ = 0.

London has shown from the theory of dispersion forces that m = 6 but we do not have a theoretical value for n. It is frequently convenient for calculation to let n = 12, in that case Eq. (4-25) becomes

(4-26)

Equation (4-26) is the Lennard-Jones potential. ¹⁴ It relates the potential energy of two molecules to their distance of separation in terms of two parameters: an energy parameter ε that, when multiplied by -1, gives the minimum energy corresponding to the equilibrium separation; and a distance parameter σ that is equal to the interrnolecular separation when the potential energy is zero. An illustration of Eqs. (4-26) and (4-26a) is given in Fig. 4-2.

¹⁴ Better agreement with experiment is often obtained by letting n be an adjustable parameter. In that case Eq. (4-25) becomes (Lichtenthaler and Schafer, 1969):

(4-26a)

Figure 4-2 Three forms of Mie’s potential for simple, nonpolar molecules.

Because of the steepness of the repulsion potential, the numerical values of r_min and σ are not far apart. For a Mie (n, 6) potential, we obtain

(4-27)

Constants ε, σ, and n can be estimated from a variety of physical properties as well as from spectroscopic and molecular-beam experiments.¹⁵ Subject to several simplifying assumptions, they can be computed, for example, from the compressibility of solids at low temperatures or from specific-heat data of solids or liquids. More commonly, they are obtained from the variation of viscosity or self-diffusivity with temperature at low pressures, and, most commonly, from gas-phase volumetric properties as expressed by second virial coefficients (Moelwyn-Hughes, 1961).

¹⁵ See G. Maidand, M. Rigby, and W. Wakehani, 1981, Intermolecuiar Forces: Their Origin and Determination, Oxford: Oxford University Press.

Mie’s potential applies to two nonpolar, spherically symmetric molecules that are completely isolated. In nondilute systems, and especially in condensed phases, two molecules are not isolated but have many other molecules in their vicinity. By introducing appropriate simplifying assumptions, it is possible to construct a simple theory of dense media using a form of Mie’s two-body potential such as that of Lennard-Jones (Moelwyn-Hughes, 1961).

Consider a condensed system at conditions not far removed from those prevailing at the triple point. We assume that the total potential energy is due primarily to interactions between nearest neighbors. Let the number of nearest neighbors in a molecular arrangement be designated by z. In a system containing N molecules, the total potential energy Γ_t is then approximately given by

(4-28)

where Γ is the potential energy of an isolated pair. Factor 1/2 is needed to avoid counting each pair twice. Substituting Mie’s equation into Eq. (4-28), we have

(4-29)¹⁶

¹⁶ Even if we neglect the effect of Bonnearest neighbors, Eq. (4-28) is not exact because it assumes additivity of two-body potentials to give the potential energy of a muitibody system

where r is the distance between two adjacent molecules.

Equation (4-29) considers only interactions between nearest-neighbor molecules. To account for additional potential energy resulting from interaction of a molecule with all of those outside its nearest-neighbor shell, numerical constants s_n and s_m (that are near unity) are introduced by rewriting the total potential energy:

(4-30)

When the condensed system is considered as a lattice such as that existing in a regularly spaced crystal, the constants s_n and s_m can be accurately determined from the lattice geometry. For example, a molecule in a crystal of the simple-cubic type has 6 nearest neighbors at a distance r, 12 at a distance r, 8 at a distance r, etc. The attractive energy of one molecule with respect to all of the others is then given by

(4-31)

and

(4-32)

Similarly, s_n can be calculated for the repulsive potential. Table 4-7 shows summation constants for several geometrical arrangements.

Table 4-7 Summation constants snar, d sm for cubic lattices (Moelwyn-Hughes, 1961).

Having obtained numerical values for constants s_n and s_m, it is now possible to obtain a relation between the equilibrium distance of separation r_min of an isolated pair of molecules and the equilibrium distance of separation r_min between a molecule and its nearest neighbors in a condensed system. At equilibrium, the potential energy of the condensed system is a minimum; therefore,

(4-33)

From Eq. (4-30) we obtain

(4-34)

and comparing this result with that obtained for an isolated pair of molecules, we have

(4-35)

Since m is equal to 6, and assuming that n is between 8 and 16, the values in Table 4-7 show that the equilibrium distance in an isolated pair is always a few percent larger than that in a condensed system. This leads to the interesting result that for a pair of adjacent molecules in a condensed system, the absolute value of the average potential energy is roughly 50% smaller than that corresponding to the equilibrium separation between a pair of isolated molecules.

Equation (4-35) can also be used to estimate r_min for an isolated pair of molecules from data for r_mint of the condensed system. At low temperatures, the condensed system is a crystal and the average distance between two neighboring molecules is approximately r_mint. This approximation becomes increasingly better as the temperature falls and becomes exact at T = 0 K. Therefore, r_mint can be obtained from low-temperature molar-volume data if the lattice geometry is known. Another possibility is to determine r_mint directly from low-temperature X-ray scattering data.

By algebraic rearrangement of Eq. (4-30) it is possible to obtain a relation between the equilibrium potential energy ε of an isolated pair of molecules and the equilibrium potential energy Γ_t(r = r_mint) of the condensed system:

(4-36)

where Γ_t(r = r_mint), the lattice energy at T = 0 K, is given by

(4-37)

In this equation, Δ_subh₀ is the enthalpy of sublimation at 0 K and the second term is the zero-point energy. The Debye temperature θ_D is obtained from the temperature dependence of the specific heat that, at very low temperatures, follows Debye’s T law. Although an appreciable number of Δ_subh₀ values are tabulated (D’Ans-Lax, 1967), enthalpy-of-sublimation data are available only at temperatures above zero. However, Δ_subh₀ can be calculated from these data in connection with the temperature dependence of the specific heat. In this way Γ_t(r = r_mint) can be estimated and Eq. (4-36) then yields an estimated value for ε (Mie, 1903). Since m is equal to 6, n must be estimated from at least one other physical property.

Mie’s potential, as well as other similar potentials for nonpolar, spherically symmetrical molecules, contains one independent variable r and one dependent variable Γ. When these variables are nondimensionalized with characteristic molecular constants, the resulting potential function leads to a useful generalization known as the molecular theory of corresponding states, as discussed in Sec. 4.12.

4.6 Structural Effects

Intermolecular forces of nonspherical molecules depend not only on the center-to-center distance but also on the relative orientation of the molecules. The effect of molecular shape is most significant at low temperatures and when the intermolecular distances are small, i.e., especially in the condensed state. For example, there are significant differences among the boiling points of isomeric alkanes that have the same carbon number; a branched isomer has a lower boiling point than a straight chain, and the more numerous the branches, the lower the boiling point. To illustrate, Fig. 4-3 shows results for some pentanes and hexanes.

Figure 4-3 Boiling points (in °C) of some alkane isomers.

A similar effect of branching on boiling point is observed within many families of organic compounds. It is reasonable that branching should lower the boiling point; with branching, the shape of a molecule tends to approach that of a sphere and the surface area per molecule decreases. Therefore, intermolecular attraction per pair of molecules becomes weaker and a lower kinetic energy kT is sufficient to overcome that attraction.

Boiling-point differences for isomers could also be due to different interactions between methyl and methylene groups. However, to explain differences in the boiling points of linear and branched alkanes only in terms of the methyl-methylene interaction would require an unreasonably large force-field difference between methyl and methylene groups. Differences in molecular shape provide a more likely explanation.

Similarly, other thermodynamic properties are affected by branching. Spectroscopic results (Te Lam et al., 1974; Delmas and Purves, 1977; Tancrède et al., 1977; Heintz and Lichtenthaler, 1977, 1984), for example, show the presence of orientational order among long chains of pure n-alkanes, which does not exist among more nearly spherical branched alkanes. Short-range orientational order in systems containing anisotropic molecules can be detected by studying the thermodynamic properties of a solution wherein the substance with orientational order is dissolved in a relatively inert solvent with little or no orientational order. Mixing liquids of different degrees of order usually brings about a net decrease of order, and hence positive contributions to the enthalpy Δ_mixh and entropy Δ_mixs of mixing. This effect is illustrated in Fig. 4-4, that shows the composition dependence of Δ_mixh for linear and branched decane mixed with cyclohexane at 25 and 40°C. The difference in Δ_mixh for the two isomers is surprisingly large. At mole fraction x = 0.5, Δ_mixh for the binary containing n-decane is nearly twice that for the binary containing isodecane. Further, the temperature dependence of Δ_mixh for the n-decane system is much stronger than that for the isodecane system. Upon mixing with cyclohexane (a globular molecule), short-range orientational order is destroyed and hence the mixing process requires more energy with the linear alkane than with the branched alkane. At higher temperatures, orientational order in the pure n-alkanes is already partially destroyed by thermal motion of the molecules; mixing isothermically at higher temperature requires less energy than at lower temperature. The branched alkanes show only little orientational order and therefore the temperature dependence of Δ_mixh for branched alkane/cyclohexane mixtures is weak.

4.7 Specific (Chemical) Forces

In addition to physical intermolecular forces briefly described in previous sections, there are specific forces of attraction which lead to the formation of new molecular species; such forces are called chemical forces. A good example of such a force is that between ammonia and hydrogen chloride; in this case, a new species, ammonium chloride, is formed. Such forces, in effect, constitute the basis of the entire science of chemistry and it is impossible to discuss them adequately in a few pages. However, it is important to recognize that chemical forces can, in many cases, be of major importance in determining thermodynamic properties of solutions. Whereas in the previous sections we were able to write some simple formulas for the potential energies of physically interacting molecules, we cannot give simple quantitative relations that describe on a microscopic level interaction between chemically reactive molecules. Instead, we briefly discuss in a qualitative manner some relations between chemical forces and properties of solutions.

Figure 4-4 Effect of molecular structure on enthalpy of mixing: cyclohexane/n-decane and cyclohexane/isodecane (2,6-dimeihyl octane).

Numerous types of specific chemical effects are of importance in the thermodynamics of solutions. For example, the solubility of silver chloride in water is very small; however, if some ammonia is added to the solution, the solubility rises by several orders of magnitude due to the formation of a silver/ammonia complex. Acetone is a much better solvent for acetylene than for ethylene because acetylene, unlike ethylene, can form a hydrogen bond with the carbonyl oxygen in the solvent. Because of an electron donor-electron acceptor interaction, iodine is more soluble in aromatic solvents like toluene and xylene than in paraffinic solvents like heptane or octane. Finally, an example is provided by a well-known industrial process for the absorption of carbon dioxide in ethanol amine; carbon dioxide is readily soluble in this solvent because of specific chemical interaction between (acidic) solute and (basic) solvent.

The main difference between a physical and a chemical force lies in the criterion of saturation: Chemical forces are saturated but physical forces are not. The saturated nature of chemical forces is intimately connected with the theory of the covalent bond and also with the law of multiple proportions that says the ratio of atoms in a molecule is a small, integral number. If two hydrogen atoms meet, they have a strong tendency to form a hydrogen molecule H₂, but once having done so, they have no appreciable further tendency to form a molecule H₃. Hence, the attractive force between hydrogen atoms is “satisfied” (or saturated) once the stable H₂ molecule is formed. On the other hand, the purely physical force between, say, two argon atoms knows no such “satisfaction.” Two argon atoms that are attracted to form a doublet still have a tendency to attract a third argon atom, and a triplet has a further tendency to attract a fourth. It is true that in the gaseous state, doublets are much more frequent than triplets but that is because in the dilute state, a two-body collision is much more probable than a three-body collision. In the condensed or highly concentrated state, there are aggregates of many argon atoms.

Chemical effects in solution are conveniently classified in terms of association or salvation. By the former, we mean the tendency of some molecules to form polymers; for example, acetic acid consists primarily of dimers due to hydrogen bonding. By solvation, we mean the tendency of molecules of different species to form complexes; for example, a solution of sulfur trioxide in water shows strong solvation by formation of sulfuric acid. This particular example illustrates a severe degree of solvation but there are many cases where the solvation is much weaker; for example, there is a tendency for chloroform to solvate with acetone due to hydrogen bonding between the primary hydrogen of the chloroform and the carbonyl oxygen of the acetone; this tendency has a profound effect on the properties of chloroform/acetone solutions. Chloroform also forms hydrogen bonds with diisobutyl ketone, but in this case the extent of complexing is much smaller because of steric hindrance and, as a result, mixtures of chloroform with diisobutyl ketone behave more ideally than do mixtures of chloroform with acetone. Solvation effects in solution are very common. When these effects are strong, they often produce negative deviations from Raoult’s law since they necessarily decrease the volatilities of the original components.

It is easy to see that whenever solvation occurs in solution it has a marked effect on the thermodynamic properties of that solution. It is, perhaps, not as obvious that association effects are also of major importance. The reason is that the extent of association is a strong function of the composition, especially in the range dilute with respect to the associating component. Pure methanol, for example, exists primarily as dimer, trimer and tetramer, but when methanol is dissolved in a large excess of hexane, it exists primarily as a monomer. As the methanol concentration rises, more polymers are formed; the fraction of methanol molecules that exists in the associated form is strongly dependent on the number of methanol molecules present per unit volume of solution; as a result, the fugacity of methanol is a highly nonlinear function of its mole fraction.

The ability of a molecule to solvate or associate is closely related to its electronic structure. For example, if we want to compare the properties of aluminum trichloride and antimony trichloride, we note immediately an important difference in their electronic structures:

In the trichloride, antimony has its octet of electrons, and therefore its chemical forces are saturated. Aluminum, however, has only six electrons and has a strong tendency to add two more. Consequently, aluminum trichloride solvates easily with any molecule that can act as an electron donor whereas antimony trichloride does not. This difference explains, at least in part, why aluminum trichloride, unlike antimony trichloride, is an excellent catalyst for some organic reactions, for example, the Friedel-Crafts reaction.

4.8 Hydrogen Bonds

The most common chemical effect encountered in the thermodynamics of solutions is that due to the hydrogen bond. While the “normal” valence of hydrogen is unity, many hydrogen-containing compounds behave as if hydrogen were bivalent. Studies of hydrogen fluoride vapor, for example, show that the correct formula is (HF)_n, where n depends on temperature and pressure, and may be as much as 6. The only reasonable explanation this is to write the structure of hydrogen fluoride in this manner:

where the solid line indicates the “normal” bond and the dashed line, an “auxiliary” bond. Similarly, studies on the crystal structure of ice show that each hydrogen is “normally” bonded to one oxygen atom and additionally attached to another oxygen atom:

It appears that two sufficiently negative atoms X and Y (that may be identical) may, in suitable circumstances, be united with hydrogen according to X--H---Y. Consequently, molecules containing hydrogen linked to an electronegative atom (as in acids, alcohols, and amines) show strong tendencies to associate with each other and to solvate with other molecules possessing accessible electronegative atoms.

The major difference between a hydrogen bond and a normal covalent bond is the former’s relative weakness. The bond strength of most hydrogen bonds lies in the neighborhood 8 to 40 kJ mol^-1, whereas the usual covalent bond strength is in the region 200 to 400 kJ mol^-1. Therefore, the hydrogen bond is broken rather easily and it is for this reason that hydrogen-bonding effects usually decrease at higher temperatures where the kinetic energy of the molecules is sufficient to break these loose bonds.

Before we discuss the effect of hydrogen bonding on physical properties, we review a few characteristic properties of hydrogen bonds that have been observed experimentally (compare Fig. 4-5):

I. Distances between the neighboring atoms of the two functional groups (X---H---Y) in hydrogen bonds are substantially smaller than the sum of their van der Waals radii.

II. X—H stretching modes are shifted toward lower frequencies (lower wave numbers) upon hydrogen-bond formation.

III. Polarities of X—H bonds increase upon hydrogen-bond formation, often leading to complexes whose dipole moments are larger than those expected from vectorial addition.

IV. Nuclear-magnetic-resonance (NMR) chemical shifts of protons in hydrogen bonds are substantially smaller than those observed in the corresponding isolated molecules. The observed deshielding follows from reduced electron densities at protons participating in hydrogen bonding.

All effects (I to IV) are less pronounced in isolated hydrogen-bonded dimers than in hydrogen-bonded liquids or crystals, indicating the importance of long-range interactions through chains or networks of hydrogen bonds.

The thermodynamic constants for hydrogen-bonding reactions are generally dependent on the medium in which they occur (for a review see Christian and Lane, 1975). Tucker and Christian (1976), for example, report data for the 1:1 hydrogen-bonded complex of trifluoroethanol (TFE) with acetone in the vapor phase and in CCl₄ solution. To relate the vapor-phase results to the liquid-phase results, Tucker and Christian constructed the thermodynamic cycle shown in Fig. 4-6 for the TFE/acetone complex, including values of energy and Gibbs energy for both the vertical transfer reactions and the horizontal complex-formation reactions. Figure 4-6 shows that the transfer energies and Gibbs energies for each of the individual components are not small compared with the values for the association reaction. The energy of transfer of the complex into CCl₄ is 83% of that of the separated monomers; the corresponding Gibbs energy of transfer is 79% that of the monomers. The results indicate that the transfer energy and Gibbs energy of the complex are not even approximately canceled by the transfer energies and Gibbs energies of the constituent molecules. The complex is less stable in the relatively inert solvent than in the vapor phase, indicating an important solvent effect on hydrogen-bond formation. For most hydrogen-bonded complexes, stabilities decrease as the solvent changes from aliphatic hydrocarbon to chlorinated (or aromatic) hydrocarbon, to highly polar liquid.

Figure 4-5 Characteristic properties of hydrogen-bonded systems: (i) intermoiecular geometries: (A) ordinary, (B) strong hydrogen bonds; (II) vibrational spectra, XH stretching frequencies; (III) increase in polarity on complex formation; (IV) NMR deshieiding effect observed at protons participating in hydrogen bonds. (P. Schuster, 1978, The Fine Structure of the Hydrogen Bond in Intermoiecular Interactions from Diatomics to Biopolymers, B. Pullman, Ed., New York: Wiley & Sons).

Figure 4-6 Thermodynamic data for complex-formation reactions and for transfer reactions: Trifluoroetnanol (TFE)/acetone system at 25°C. Numbers in parentheses are Δg° (kj mol^-1) based on the unit molarity, ideai-dilute-solution standard state. Remaining quantities are standard energies, Δu° (kJ mol^-1), for the various steps.

The strong effect of hydrogen bonding on physical properties is best illustrated by comparing some thermodynamic properties of two isomers: dimethyl ether and ethyl alcohol. These molecules both have the formula C₂H₆O but strong hydrogen bonding occurs only in the alcohol. Table 4-8 shows some of the properties. Due to the additional cohesive forces in the hydrogen-bonded alcohol, the boiling point, enthalpy of vaporization, and Trouton’s constant are appreciably larger than those of the ether. Also, because ethanol can readily solvate with water, it is infinitely soluble in water whereas dimethyl ether is only partially soluble.

Table 4-8 Some properties of the isomers ethanol and dimethyl ether.

Hydrogen bonding between molecules of the same component can frequently be detected by studying the thermodynamic properties of a solution wherein the hydrogen-bonded substance is dissolved in a nonpolar, relatively inert solvent.

When a strongly hydrogen-bonded substance such as ethanol is dissolved in an excess of a nonpolar solvent (such as hexane or cyclohexane), hydrogen bonds are broken until, in the limit of infinite dilution, all the alcohol molecules exist as monomers rather than as dimers, trimers, or higher aggregates. This follows simply from the law of mass action: In the equilibrium nA A_n (where n is an integer greater than one) the fraction of A molecules that are monomeric (i.e., not polymerized) increases with falling total concentration of all A molecules, polymerized or not. As the total concentration of A molecules in the solvent approaches zero, the fraction of all A molecules that are monomers approaches unity.

The strong dependence of the extent of polymerization on solute concentration results in characteristic thermodynamic behavior as shown in Figs. 4-7 and 4-8.

Figure 4-7 gives the enthalpy of mixing per mole of solute as a function of solute mole fraction at constant temperature and pressure. The behavior of strongly hydrogen-bonded ethanol is contrasted with that of nonpolar benzene; the qualitative difference indicates that the effect of the solvent on the ethanol molecules is markedly different from that on the benzene molecules. When benzene is mixed isothermally with a paraffinic or naphthenic solvent, there is a small absorption of heat and a small expansion due to physical (essentially dispersion) forces. However, when ethanol is dissolved in an “inert” solvent, hydrogen bonds are broken and, because such breaking requires energy, much heat is absorbed; further, because a hydrogen-bonded network of molecules tends to occupy somewhat less space than that corresponding to the sum of the individual nonbonded molecules, there is appreciable expansion in the volume of the mixture, as shown in Fig. 4-8.

Figure 4-7 Hydrogen bonding in solution. Enthalpic effects for two chemically different solutes in cyclohexane at 20°C.

Figure 4-8 Hydrogen bonding in solution. Volumetric effects for two chemically different solutes in n-hexane at 6°C.

Evidence for hydrogen-bond formation between dissimilar molecules can be obtained in a variety of ways; good surveys are given by Pimentel and McClellan (1960), Schuster et al. (1976), and Huyskens et al. (1991). We shall not discuss them here but only discuss briefly two types of thermodynamic evidence best illustrated by an example, i.e., hydrogen bonding between acetone and chloroform.

Dolezalek (1908) observed that the partial pressures for liquid mixtures of acetone and chloroform were lower than those calculated from Raoult’s law and he interpreted this negative deviation as a consequence of complex formation between the two dissimilar species. However, negative deviations from ideality can result from causes other than complex formation;¹⁷ further, a binary liquid mixture that forms weak complexes between the two components may nevertheless have partial pressures slightly larger than those calculated from Raoult’s law (Booth et al, 1959). Thus Dolezalek’s evidence, while pertinent, is not completely convincing.

¹⁷ For example, solutions of nonpolar polymers in nonpolar liquid solvents show strong negative deviations from Raoult’s law, as discussed in Sec. 8.2.

More direct evidence for hydrogen-bond formation has been obtained by Campbell and Kartzmark (1960), who measured freezing points and enthalpies of mixing for mixtures of acetone with chloroform and with carbon tetrachloride. By comparing the properties of these two systems, it becomes evident that there is a large difference between the interaction of acetone with chloroform and the interaction of acetone with carbon tetrachloride, a molecule similar to chloroform except that the latter possesses an electron-accepting hydrogen atom that is capable of interacting with the electron-donating oxygen atom in acetone.

Figure 4-9 Evidence for hydrogen bonds: Freezing-point data.

Figure 4-9 gives freezing-point data for the two systems. Mixtures containing carbon tetrachloride exhibit simple behavior with a eutectic formed at -105°C and 87.5 mol % acetone. However, mixtures containing chloroform show more complicated behavior: Two eutectics are formed, one at -106°C and 31 mol % acetone and the other at -115°C and 74 mol % acetone, and there is a convex central section whose maximum is at 50 mol %. This maximum indicates that the compound (CH₃)₂CO---HCCl₃ exists in the solid state, although it is readily dissociated in the liquid state. The existence of such a compound is excellent evidence for strong interaction between the two dissimilar molecules. Since the maximum in the diagram occurs at the midpoint on the composition axis, we conclude that the complex has a 1:1 stoichiometric ratio, as we would expect from the structure of the molecules.

Figure 4-10 shows enthalpies of mixing for the two binary systems. Unfortunately, the original reference does not mention the temperature for these data but it is probably 25°C or some temperature nearby. The enthalpy of mixing of acetone with carbon tetrachloride is positive (heat is absorbed), whereas the enthalpy of mixing of acetone with chloroform is negative (heat is evolved), and it is almost one order of magnitude larger. These data provide strong support for a hydrogen bond formed between acetone and chloroform. The effect of physical intermolecular forces (dipole, induction, dispersion) causes a small amount of heat to be absorbed, as shown by the data for the carbon tetrachloride mixtures; in the chloroform mixtures, however, there is a chemical heat effect that not only cancels the physical contribution to the observed enthalpy of mixing but, because it is much larger, causes heat to be evolved. Because energy is needed to break hydrogen bonds, it necessarily follows that heat is liberated when hydrogen bonds are formed.

Figure 4-10 Evidence for hydrogen bonds: Enthalpy-of-mixing data.

The freezing-point data and the calorimetric results indicate that in the liquid phase an equilibrium exists:

It can be shown from the law of mass action that when acetone is very dilute in chloroform, all of the acetone in solution is complexed with chloroform. For this reason the enthalpy data shown in Fig. 4-10 were replotted by Campbell and Kartzmark on coordinates of enthalpy of mixing per mole of acetone versus mole fraction of acetone. From the intercepts of this plot (0 mol % acetone), Δh = -8.75 kJ mol^-1 for the system containing chloroform and Δh = 2.62 kJ mol^-1 for the system containing carbon tetrachloride. From these results we can calculate that the enthalpy of complex formation, i.e., the enthalpy of the hydrogen bond, is (-8.75 - 2.62) =-11.37 kJ mol^-1.

In view of the experimental uncertainties that are magnified by this particular method of data reduction, Campbell and Kart/mark give Δh = (11.3 ± 0.4) kJ mol^-1 as the enthalpy of hydrogen-bond formation. This result is in fair agreement with enthalpy data for similar hydrogen bonds determined by different measurements.

4.9 Electron Donor-Electron Acceptor Complexes

While the consequences of hydrogen bonding are probably the most common chemical effect in solution thermodynamics, chemical effects may also result from other kinds of bonding forces leading to loose complex formation between electron donors and electron acceptors (Andrews and Keefer, 1964; Foster 1973, 1974; Kuznetsov, 1995), sometimes called charge-transfer complexes.

The existence of donor-acceptor complexes can be established by a variety of experimental methods (Foster, 1973, 1974). Several types of data for donor-acceptor complexes are listed in Table 4-9 that also gives the designation “primary” or “secondary” for each type. Primary data are those derived from measurements by an interpretation requiring theoretical assumptions which are well established in fields other than donor-acceptor-complex studies. All data classified as secondary depend on a particular method for determining the concentration of complex in the sample.

Table 4-9 Sources of experimental data for donor-acceptor complexes (Gutman, 1978).

Ultraviolet spectroscopy is the method used most frequently to study donor-acceptor complexes; there are extensive tabulations of charge-transfer-frequency data in the literature (Briegleb, 1961; Andrews and Keefer, 1964; Rose, 1967; Mulliken and Person, 1969). These data provide a valuable source of information about complexes that exhibit a charge-transfer (CT) band. Spectroscopic data may be used to give a quantitative measure of complex stability (Rossotti and Rossotti, 1961).

The general basis of such measurements is given schematically in Fig. 4-11 showing two optical cells; in the upper cell we have in series two separate, dilute solutions of components A and B in some “inert” solvent. In the lower cell there is a single solution of A and B dissolved in the same “inert” solvent. We put equal numbers of A molecules in the top and bottom cells and we do the same with B. Monochromatic light of equal intensity is passed through both cells. If a complex is formed between A and B, and if the light frequency is in the absorption range of the complex, then light absorption is larger in the lower cell. On the other hand, if no complex is formed, light absorption is the same in both cells. The quantitative difference in light absorption provides a basis for subsequent calculations of complex stability and, when such spectroscopic measurements are performed at different temperatures, it is possible to calculate also the enthalpy and entropy of complex formation.

Figure 4-11 Spectroscopic measurement of acid-base complex.

Table 4-10 gives some results for complex formation between s-trinitrobenzene (an electron acceptor) and aromatic hydrocarbons (electron donors). The results show that complex stability rises with the number of methyl groups on the benzene ring, in agreement with various other measurements indicating that π-electrons on the aromatic ring become more easily displaced when methyl groups are added; for example, the ionization potentials of methyl-substituted benzenes fall with increasing methyl substitution.¹⁸ For s-trinitrobenzene with aromatics, complex stability is strong. For complexes of aromatics with other, more common polar organic molecules, complex stability is much weaker but not negligible, as indicated by experimental results of Weimer (1966) shown in Table 4-11. For the polar solvents listed, Weimer found no complex formation with saturated hydrocarbons and as a result we may expect the thermodynamic properties of solutions of these polar solvents with aromatics to be significantly different from those of solutions of the same solvents with paraffins and naphthenes, as has been observed (Orye et al, 1965, 1965a). The tendency of polar solvents to form complexes with unsaturated hydrocarbons, but not with saturated hydrocarbons, supplies a basis for various commercial separation processes in the petroleum industry, including the Edeleanu and Udex processes.

¹⁸ A lower ionization potential means that an electron can be removed more easily.

Table 4-10 Spectroscopic equilibrium constants and enthalpies of formation for s-trinitrobenzene/aromatic complexes dissolved in cyclohexane at 20°C.*

Table 4-11 Spectroscopic equilibrium constants of formation for polar solvent/p-xylene complexes dissolved in n-hexane at 25°C (Weimer, 1966).

Evidence for complex formation is often obtained from thermodynamic measurements. Agarwal (1978), for example, has measured volumes of mixing for 1,2,4-trichlorobenzene with benzene, toluene, p-xylene, and mesitylene at 30°C as a function of composition. The results, shown in Fig. 4-12, are negative for the entire composition range for all four mixtures and the trend indicates that the interaction between unlike molecules rises with increasing electron-donating power of the hydrocarbon.

Figure 4-12 Effect of complex formation on volume of mixing. Binary systems containing 1,2,4-trichlorobenzene at 30°C.

A quantitative correlation exists between the volume of mixing and the ionization potential of the hydrocarbons, as shown in Fig. 4-13, where the volume of mixing (at equimolar composition) is a straight-line function of the ionization potential. The results in Fig. 4-13 give evidence for the existence of a donor-acceptor interaction between trichlorobenzene and aromatic hydrocarbons.

Figure 4-13 Correlation of volume of mixing with the ionization potential of the hydrocarbon. Equimolar 1,2,4-triclorobenzene/hydrocarbon mixtures at 30°C.

Sometimes, however, the interpretation of thermodynamic data does not provide such an obvious conclusion. Mahl et al. (1978) for example, have measured enthalpy and volume of mixing for tetrahydrofuran (THF) with benzene, toluene, and xylenes at 25°C as a function of composition. The results, given in Figs. 4-14 and 4-15, are negative for both properties over the entire composition range for all five mixtures, suggesting complex formation between unlike molecules. However, the magnitude of the enthalpy of mixing at equimolar composition follows the sequence benzene > toluene > xylenes. At first glance, this sequence is surprising because the ionization potential of benzene is larger than that of toluene, and that, in turn, is larger than that of xylene. One might, therefore, have expected a reverse sequence. However, the observed enthalpy of mixing is determined not only by chemical forces but also by physical forces.

Figure 4-14 Effect of complex formation on enthalpy of mixing. Binary systems containing tetrahydrofuran (THF) at 25°C.

For these mixtures, chemical and physical forces are of the same order of magnitude, and therefore, the sequence of enthalpy effects is not necessarily the same as the sequence of complex stability. Exothermic mixing, together with volumetric contraction, provides good evidence of moderately strong interactions between unlike molecules in the liquid state. It seems that the lone pair of electrons on the oxygen atom of the THF forms charge-transfer complexes with benzene, toluene, and xylenes. However, precise quantitative information on complex stability cannot be determined from the enthalpic and volumetric data because of unknown solvent effects. For such information, spectroscopic data are more useful.

Figure 4-15 Effect of complex formation on volume of mixing. Binary systems containing tetrahydrofuran (THF) at 25°C.

Because most published data on molecular complexes are based on measurements in the liquid phase, solvent effects on molecular-complex properties must be taken into account in data interpretation. There is no solvent that may be regarded as completely inert; therefore, solute properties in solution are not identical to those in the vapor phase (Tamres, 1973). Reliable results on complex formation in the vapor phase are necessary for understanding the effect of a liquid solvent on solution equilibria, particularly since the theory (Mulliken, 1952) of charge-transfer complexes is based on the concept of the isolated donor, acceptor, and complex, a situation attained only in the low-pressure gas phase. A review of solvent effects is given by Davis (1975) and, in particular for biological systems, by Gutmann et al. (1997).

The examples shown above illustrate how specific (chemical) intermolecular forces can profoundly influence thermodynamic properties. At present, with rare exceptions, the theory of chemical bonding is not sufficiently developed to establish fundamental quantitative relations between specific intermolecular forces and thermodynamic properties, but in many cases it is possible to interpret thermodynamic behavior in terms of pertinent chemical phenomena. Such interpretation can then facilitate meaningful correlation of thermodynamic properties, based on molecular concepts.

4.10 Hydrophobic Interaction

It has been known for many years that some molecules are soluble in alcohol, ether, and many other solvents, but not in water; often these molecules have a dual nature: one part of molecule is soluble in water (the hydrophilic, i.e. water-loving, part), while another part is not water-soluble (the hydrophobic, i.e. water-fearing, part).

Molecules that are partly hydrophilic and partly hydrophobic¹⁹ are forced by their dual nature to adopt unique orientations in an aqueous medium, that is, to form suitably organized structures. Such molecules, called amphiphiles, play important roles in living organisms at the cell level, and find potentially useful applications in biotechnology and in chemical industry (Tanford, 1980; Chen and Rajagoplan, 1990). The organized structures are called micelles (Hoffmann and Uibricht, 1986). As schematically represented in Fig. 4-16(a) for a surfactant in aqueous solution, in micelles (spherical, ellipsoidal or in form of bilayers) the hydrophobic part (usually a long-chained hydrocarbon) is kept away from water, while the hydrophilic terminal groups (uncharged, cationic, or anionic) at the surface of the aggregates are water-solvated and keep the aggregates in solution. However, addition of a small amount of water to a surfactant-containing organic nonpolar phase may induce the formation of thermodynamically stable aggregates designated by reverse micelles, schematically represented in Fig. 4-16(b). In this case, the surfactant orients itself such that its terminal hydrophilic groups point inward, surrounding the water core, while the long hydrocarbon hydrophobic chains point outward into the bulk organic phase. An example is provided by the double-chained anionic surfactant AOT²⁰ [shown in Fig. 4-16(c)] in solutions of isooctane with small amounts of water. We do not discuss further this large subject but mention it only to indicate the tremendous importance of structural effects.²¹

Figure 4-16 Schematic representation of a cross-section of micelles (a) and reverse micelles (b). in reverse micelles, the polar terminal group of the surfactant’s molecules are directed towards the interior of the aggregate (forming an inner water core), and the hydrophobia chains are exposed to the organic solvent, as illustrated for the doubSe-chained surfactant AOT {c}.

¹⁹ Typical examples are surfactant molecules (the main components of detergents), such as sodium-n-dodecyl-1-sulfate, C₁₂H₂₅SO₄Na, that contains a long (hydrophobic) hydrocarbon chain and a terminal (hydrophilic) polar or ionic group.

²⁰ Surfactant AOT is sodium-di-2-ethyihexyl sulfosuccinate, with a molecular weight of 444 g mol^-1.

²¹ For a review of applications of reversed micelles in biotechnology see M. J. Pires, M. R. Aires-Barros, and J. M. S. Cabral, 1996, Biotechology Progress, 12: 290; T. Ono and T. M. Goto, 1997, Current Opinion in Colloid & Interface Science, 1: 397.

The origin of the hydrophobic effect is drastically different from those described in the previous sections. It is mainly an entropic phenomenon often observed in nature. The hydrophobic effect arises mainly from the strong attractive forces (hydrogen bonds) between H₂O molecules in highly structured liquid water. These attractive forces must be disrupted or distorted when a solute is dissolved in water. Upon solubilization of a solute, hydrogen bonds in water are often not broken but they are maintained in distorted form. Water molecules reorient, or rearrange, themselves such that they can participate in hydrogen-bond formation, more or less as in bulk pure liquid water. In doing so, they create a higher degree of local order than that in pure liquid water, thereby producing a decrease in entropy. It is this loss of entropy (rather than enthalpy) that leads to an unfavorable Gibbs energy change for solubilization of nonpolar solutes in water (Tanford, 1980). For example, hydrocarbons are only sparingly soluble in water, i.e. they have a highly unfavorable Gibbs energy of solubilization whose most important contribution is entropic.

For example, as shown in Table 4-12, at 25°C the standard Gibbs energy for the transfer (Δg°) of n-butane from its pare liquid to water is about +24.7 kJ mol^-1. This quantity is the sum of an enthalpic contribution (Δh°) and an entropic contribution (TΔs°). For n-butane, these are, respectively, -3.3 kJ mol^-1 and -28.0 kJ mol^-1 at 25°C; the large decrease in entropy accounts for 85% of the Gibbs energy of solubilization. For other hydrocarbons (e.g. n-hexane), the entropic contribution to Δg° is even larger.

As summarized in Table 4-12, the standard entropy of transfer is strongly negative, due to the reorieatation of the water molecules around the hydrocarbon. The poor solubility of hydrocarbons in water is not due to a large positive enthalpy of solution but rather to a large entropy decrease caused by what is called the hydrophoblc effect. This effect is, in part, also responsible for the immiscibility of nonpolar substances (hydrocarbons, fluorocarbons, etc.) with water.

Closely related to the hydrophobia effect is the hydrophobic interaction (Israelachvili, 1992). This interaction is mainly entropic and refers to the unusually strong attraction between hydrophobia molecules (and surfaces) in water;²² in many cases, this attraction is stronger than in vacuo. For example, from Table 4-5 and using Eq. (4-22), we can calculate that the energy of interaction of two contacting methane molecules (with a molecular diameter of 4 Å) in vacuo is -2.5 × 10^-21 J. In water, the same interaction energy is -14 × 10^-21 J.

²² A molecular-thermodynaraic model for hydrophobic hydration of small nonpolar molecules and of extended hydrophobia surfaces is given by N. A. M. Besseling and J. LyMema, 1997, J. Phys. Chem. B, 101: 7604.

Table 4-12 Change in standard molar Gibbs energy (Δg°), enthalpy (Δh°), and entropy (TΔs°), all in kJ mol^-1, for the transfer of hydrocarbons from their pure liquids into water at 25°C (Tanford, 1980).

4.11 Molecular Interactions in Dense Fluid Media

In previous sections we described the intermolecular forces that arise between molecules in the low-pressure gas phase. However, there is a significant difference in the physical interpretation of intermolecular forces between gaseous molecules and those between solute molecules in a liquid solvent. Molecules in the low-pressure gas phase interact in a “free” medium (i.e. a vacuum), but dissolved solutes interact in a solvent medium. Interactions between two molecules in a vacuum are described by a potential function (e.g. Lennard-Jones) but interactions between two molecules in a solvent medium are described by what is called the potential of mean force that plays a major role in colloid science and in the physical chemistry of protein solutions. The essential difference is that the interaction between two molecules in a solvent is influenced by the molecular nature of the solvent b.ut there is no corresponding influence on the interaction of two molecules in (nearly) free space.

For example, for two solute molecules in a solvent, their intermolecular pair potential includes not only the direct solute-solute interaction energy, but also any changes in the solute-solvent and solvent-solvent interaction energies as the two solute molecules approach each other. A solute molecule can approach another solute molecule only by displacing solvent molecules from its path. Thus, at some fixed separation, while two molecules may attract each other in free space, they may repel each other in a solvent medium if the work that must be done to displace the solvent molecules exceeds that gained by the approaching solute molecules. Further, solute molecules often perturb the local ordering of solvent molecules. If the energy associated with this perturbation depends on the distance between the two dissolved molecules, it produces an additional salvation force between them.

The molecular nature of the solvent can produce potentials of mean force that are much different from the corresponding two-body potential in vacuo²³. The potential of mean force is a measure of the intermolecular interaction of solute molecules in liquid solution. Solution theories, such as the McMillan-Mayer theory (1945), provide a direct quantitative relation between the potential of mean force and macroscopic thermodynamic properties (the osmotic virial coefficients) accessible to experiment. Osmotic virial coefficients are obtained through osmotic-pressure measurements.

²³ For a description of how solvent effects influence intermolecular and interparticle interaction potentials, see Israelachvili (1992).

Osmotic Pressure

Osmotic pressure is a phenomenon frequently encountered in nature, especially in biological systems. The first systematic quantitative studies were made in the late 19th century. The physical chemistry of osmotic pressure was developed by van’t Hoff, one of the founders of physical chemistry (about 1890) and a pioneer in applying thermodynamics to the study of liquid solutions. We briefly summarize the main concepts of osmotic pressure.

Consider a system divided into two parts, α and β, by a semi-permeable membrane, as shown in Fig. 4-17.

Figure 4-17 Schematic diagram of an osmotic-pressure measurement apparatus. The membrane is permeable to the solvent (1) but not to the solute (2).

The semi-permeable membrane is permeable to the solvent (1) but impermeable to the solute (2). Phases α and β are at the same temperature. The pressure on phase α is P, while that on phase β is P + π. The equation of chemical equilibrium is,

(4-38)

where μ, is the chemical potential, given by

(4-38a)

(4-39)

where a is the activity, related to composition through α₁ = γ₁x₁, where γ is the activity coefficient and x is the mole fraction.

For a pure fluid, (∂ μ∂ P)_T = υ. Assuming that the molar volume does not vary with pressure (incompressible fluid), we have

(4-40)

Equation (4-38) can be rewritten as

(4-41)

If the solution in phase β is dilute, x₁ is close to unity; in that event, γ₁ is also close to unity and Eq. (4-41) becomes

(4-42)

When x₂ << 1, In x₁ = In (1 – x₂) ≈ -x₂. Equation (4-42) becomes

(4-43)

Because x₂ << 1, n₂ << n₁ and x₂ ≈ n₂/n₁. Equation (4-43) then becomes

(4-44)

where V = n₁υ_{pure 1} is the total volume available to n₂ moles of solute.

Equation (4-44) is the van’t Hoff equation for osmotic pressure π, analogous to the ideal-gas equation.

The only important assumptions for Eq. (4-44) are:

• The solution is very dilute.

• The solution is incompressible.

An important application of Eq. (4-44) was recognized many years ago: If we measure π and T, and if we know the mass concentration of the solute (g/liter), we can then calculate the solute’s molecular weight.

Osmometry provides a standard procedure for measuring molecular weights of large molecules (polymers or biomacromolecules such as proteins) whose molecular weights cannot be accurately determined from other colligative-property measurements such as boiling-point elevation or freezing-point depression.

Van’t Hoff s equation is a limiting law; Eq. (4-44) is an asymptote that is approached as the concentration of solute goes to zero.

For finite concentrations, it is useful to write a series expansion in the mass concentration c₂, typically expressed in (g/liter):

(4-45)²⁴

²⁴ The osmotic virial expansion is analogous to the virial expansion for gases (to be presented in Chap. 5),

(4-45a)

where ρ is the molar density and B and C are the second and third virial coefficients. The molar concentration ρ is related to the mass soiute concentration c₂ by ρ = c₂/M₂. Substituting in the above equation, we obtain

(4-45b)

Equations (4-45a) and (4-45b) are essentially the same but the dimensions of B and C are not the same as those of B* and C*.

where M₂ is the molar mass of solute. The osmotic second virial coefficient is designated by B*; the third, by C*, etc.

In Eq. (4-45), if we set B* = C* =… = 0, we recover the van’t Hoff equation.

For dilute solutions, we can neglect three-body (and higher) interactions in Eq. (4-44). Thus, a plot of π/c₂ against c₂ is linear (for small values of c₂), with intercept equal to RT/M₂ and slope equal to RTB*₂₂. To illustrate, Fig. 4-18 shows osmotic-pressure data (McCarty and Adams, 1987; Haynes et al., 1992) measured by membrane osmometry²⁵ for aqueous protein solutions: α-chymotrypsin in a 0.1 M potassium sulfate buffer, at pH 5 and 25°C; and lysozyme and ovalbumin in a 0.06 M cacodylate buffer (aqueous solutions of dimethylarsinic acid sodium salt, (CH₃)₂) ASO₂Na·3H₂O, at pH 5.8 and 37°C. Table 4-13 lists the proteins’ osmotic second virial coefficients and number-average molecular weights, regressed from data shown in Fig. 4-18.

Figure 4-18 Osmotic-pressure data for a-chymotrypsin (•)(Haynes etal., 1992) in 0.1 M potassium suifate buffer, at pH 5 and 25°C, and for lysozyme () and ovalbumin (♦) (McCarty and Adams, 1987) in 0.06 M cacodylate buffer, at pH 5.8 and 37°C.

²⁵ A more accurate technique for experimentally determining osmotic second virial coefficients is provided by low-angle laser-tight scattering (LALLS).

Figure 4-18 shows that, contrary to what is observed for lysozyme and ovalbumin, π/c₂ for α-chymotrypsin falls with increasing c₂, giving a negative osmotic second virial coefficient. This negative value indicates that, at the prevailing experimental conditions, dilute α-chymotrypsin molecules are slightly attracted to each other.

Table 4-13 Osmotic second virial coefficients and number-average molecular weights for α-chymotrypsin, lysozyme, and ovalbumin in aqueous buffer solutions, regressed from the data shown in Fig. 4-18.

As discussed elsewhere (Hill, 1959, 1986), osmotic second virial coefficients (a macroscopic property) are related to (microscopic) intermolecular forces between two solute molecules. Quantitative data for B^*₂₂ can provide useful information on interactions between, for example, polymer or protein molecules in solution.

Donnan Equilibria

The osmotic-pressure relation given by van’t Hoff was derived for solutions of nonelectrolytes or else for solutions of electrolytes where the membrane’s permeability did not distinguish between cations and anions. But now consider a chamber divided into two parts by a membrane that exhibits ion selectivity, i.e. some ions can flow through the membrane while others cannot. In this case, the equilibrium conditions become more complex because, in addition to the usual Gibbs equations for equality of chemical potentials, it is now also necessary to satisfy an additional criterion: electrical neutrality for each of the two phases in the chamber. The thermodynamics of equilibrium in systems containing an ion-selective membrane was first discussed by Donnan early in the 20^th century.

To illustrate the essentials,²⁶ we consider an aqueous system containing three ionic species: Na⁺, Cl^- and R^-, where R^- is some anion much larger than Cl^-. Water is in excess; all ionic concentrations are small. The chamber is divided into two equisized parts, phase a and phase p, by an ion-selective membrane as indicated in Fig. 4-19. This membrane is permeable to water, Na⁺ and Cl^- but it is impermeable to R^-.

²⁶ For a clear and simple discussion, see E. A. Moeiwyn-Hughes, 1961, Physical Chemistry, 2^nd Ed., New York: Pergamon Press. For a comprehensive discussion, see M. P. Tombs and A. R. Peacocke, 1974, The Osmotic Pressure of Biological Macromolecules, Oxford: Clarendon Press.

Figure 4-19 Schematic representation of Donnan equilibrium.

Initially (before equilibrium is attained), the left side contains only water, Na⁺ and R^- at molar concentrations . The right side initially contains only water and Na⁺ and Cl^- at molar concentrations . Electroneutrality requires that

(4-46)

We now allow the initial state to attain equilibrium. Let δ represent the change in Na⁺ concentration in α. Because R^- cannot move from one side to the other, the change in Cl^- concentration in β is-δ.

At equilibrium, the final (f) concentrations are

(4-47)

(4-48)

Our task now is to calculate δ from the known original concentrations.

For the solvent (s), we write

(4-49)²⁷

²⁷ It is tempting to write the (erroneous) equilibrium equations These equations are incorrect because the electric potential of phase α is not equal to that of phase β. See Chap. 9.

We relate the chemical potential to the pressure and to the activity by

(4-50)

and we use a similar relation for u^β_s. Here υ is the molar volume and α is the activity. The standard state (*) is pure liquid solvent at system temperature and at zero pressure. Substitution in Eq. (4-49) gives

(4-51)

where π is osmotic pressure. We also have

(4-52)

Because sodium chloride is totally dissociated into sodium ions and chloride ions, Eq. (4-52) can be rewritten

(4-53)

Substituting relations similar to Eq. (4-50) into Eq. (4-53), we obtain

(4-54)

where _i is the partial molar volume of i when all solutes are at vanishingly low concentrations.

Upon equating π from Eq. (4-51) to π from Eq. (4-54), we have

(4-55)

In a very dilute solution, a^α_s=a^β_s of = 1 and the activity of a solute i is equal to its molar concentration, a_i = c_i. Equation (4-55) then becomes

(4-56)

From the definition of δ, Eq. (4-56) becomes

(4-57)

giving

(4-58)

When Eq. (4-58) is used to obtain δ, Eqs. (4-47) and (4-48) give the final equilibrium concentrations as a function of the initial concentrations.

Rearranging Eq. (4-58), we obtain the fraction of the original sodium chloride in β that has moved to α

(4-59)

Also, the osmotic pressure is given by

(4-60)

Because the equilibrium concentration of Na⁺ is not the same in both sides, we have a concentration cell (battery) with a difference in electric potential across the membrane. This difference in electric potential ΔΦ is given by the Nernst equation,

(4-61)

Upon setting activities equal to concentrations and using Eq. (4-58), Nernst’s equation becomes

(4-61a)

where Z_Na+ is the valence of the sodium ion (+1), e is the charge of an electron, and N_A is Avogadro’s constant. These results (and others based on similar arguments) are of much importance in biology and medicine because animals (and man) contain numerous semipermeable membranes that exhibit ion selectivity. Further, these results are of importance in industrial separation processes using charged membranes as, for example, electrodialysis and reverse osmosis, as discussed in Sec. 8.4.

4.12 Molecular Theory of Corresponding States

Classical or macroscopic theory of corresponding states was derived by van der Waals based on his well-known equation of state. It can be shown, however, that van der Waals’ derivation is not tied to a particular equation but can be applied to any equation of state containing two arbitrary constants in addition to gas constant R.

From the principle of continuity of the gaseous and liquid phases, van der Waals showed that at the critical point

(4-62)

These relations led van der Waals to the general result that for variables v (volume), T (temperature), and P (pressure) there exists a universal function such that²⁸ that is valid for all substances; subscript c refers to the critical point. Another way of stating this result is to say that, if the equation of state for any one fluid is written in reduced coordinates (i.e., υ/υ _C, T/T _C, P/P _C), that equation is also valid for any other fluid.

²⁸ For example, van der Waals’ equation in reduced variables is

(4-63)

Classical theory of corresponding states is based on mathematical properties of the macroscopic equation of state. Molecular or microscopic theory of corresponding states, however, is based on mathematical properties of the potential-energy function.

Intermolecular forces of a number of substances are closely approximated by the inverse-power potential function given by Eq. (4-24). The independent variable in this potential function is the distance between molecules. When this variable is made diraensionless, the potential function can be rewritten in a general way such that the dimensionless potential is a universal function F of the dimensionless distance of separation between molecules:

(4-64)

where ε_i is an energy parameter and δ_i is a distance parameter characteristic of the interaction between two molecules of species i. For example, if function F is given by the Lennard-Jones potential, then ε_i is the energy (times minus one) at the potential-energy minimum, and δ_i is the distance corresponding to zero potential energy. However, Eq. (4-64) is not restricted to the Lennard-Jones potential, nor is it restricted to an inverse-power function as given by Eq. (4-24). Equation (4-64) merely states that the reduced potential energy (Γ_ii/ε_i) is some universal function of the reduced distance (r/δ_i).

Once the potential-energy function of a substance is known, it is possible, at least in principle, to compute the macroscopic configurational properties of that substance by the techniques of statistical mechanics. Hence a universal potential-energy function, Eq. (4-64), leads to a universal equation of state and to universal values for all reduced configurational properties.

To obtain macroscopic thermodynamic properties from statistical mechanics, it is useful to calculate the canonical partition function of a system depending on temperature, volume, and number of molecules. For fluids containing small molecules, the partition function Q is expressed as a product of two factors,²⁹

where P_R = P/P_C, T_R = T/T_C, and υ_R = υ/υ_C,

²⁹ A more detailed discussion is given in App.B.

(4-65)

where the translational contributions to the energy of the system are separated from all others, due to other degrees of freedom such as rotation and vibration. It is assumed that contributions from rotation and vibration depend only on temperature. These contributions are called internal because (by assumption) they are independent of the presence of other near-by molecules.

In the classical approximation, the translational partition function, Q_trans. splits into a product of two factors, one arising from the kinetic energy and the other from the potential energy. For a one-component system of N molecules, Q_trans is given by

(4-66)

where m is the molecular mass, k is Boltzmann’s constant, h is Planck’s constant, and Γ_t(r₁,…,r_N) is the potential energy of the entire system of N molecules whose positions are described by vectors r₁,…, r_N For a given number of molecules and known molecular mass, the first factor depends only on the temperature. The second factor, called the configurational integral, Z_N, depends on temperature and volume:

(4-67)

Hence the configurational part provides the only contribution that depends on intermolecular forces. However, Z_N is not unity for an ideal gas (Γ_t = 0). For an ideal gas,

The equation of state is obtained from Q (see App. B):

(4-68)

The equation of state depends only on Z_N when Eqs. (4-65) and (4-66) are valid. Therefore, the main problem in applying statistical mechanics to real fluids lies in the evaluation of the configurational partition function.

There are four assumptions that lead to the molecular theorem of corresponding states, clearly stated by Pitzer (1939) and Guggenheim (1945). They are:

1. The partition function is factored according to Eq. (4-65), where Q_int is independent of the volume per molecule.

2. The classical approximation Eq. (4-66) is used for Q_trans.³⁰

³⁰ This assumption is not valid for low-molar-mass fluids like H9, He, and Ne. For these fluids, quantum effects play an important role at low temperatures. However, these fluids can be included in corresponding-states correlations by using effective redaction parameters, as discussed in Chap. 5.

3. The potential energy Γ_t is represented as the sura of the interactions Γ_ij (r_ij) of all possible pairs of molecules. For a given ij pair, Γ_ij depends only on the distance r_ij between them:

(4-69)

4. The potential energy of a pair of molecules, reduced by a characteristic energy, is represented as a universal function of the intermolecular distance, reduced by a characteristic length, i.e., Eq.(4-64).

Assumptions 3 and 4 are substituted into the configurational partitioo function. Further, we use reduced coordinates obtained by dividing the three-dimensional position vectors r₁,…, r_N by the scale factor σ³. Thus

(4-70)

Apart from the factor σ^3N, the configurational integral depends only on N, on the reduced temperature kT/ε, and on the reduced volume V/σ³ (through the limits of the integral):

(4-71)

where is a universal function.

Because configurational Helmholtz energy is given by

(4-72)

and because A^conf is an extensive property (proportional to N), we have

(4-73)

where the function Ψ depends only on the intensive variables T and υ = V/N.

Equations (4-72) and (4-73) imply that the configurational integral must be of the form

(4-74)

where function z* depends only on intensive variables. Substitution of Eq. (4-74) into Eq. (4-68) gives the equation of state

(4-75)

Introducing the reduced variables

(4-76)

we find that z* is a function of and [Eq. (4-74)]; the equation of state becomes

(4-77)

where F* is a universal function. The nature of this function depends only on the nature of the potential function r_ij in Eq.(4-69).

Equation (4-77) expresses the molecular (or microscopic) theory of corresponding states. This theory is analogous to the macroscopic theory of corresponding states expressed by Eq. (4-63); the difference lies in the reducing parameters.

The reduced quantities [Eq. (4-76)] are defined in terms of macroscopic variables T, V, P, N and molecular parameters ε and σ. The use of molecular parameters is important in the extension of the theorem of corresponding states to mixtures.

To relate the macroscopic and the microscopic theories of corresponding states, it is desirable to establish a connection between the parameters of one theory and those of the other. In the microscopic theory, there are two independent parameters: an energy parameter and a distance parameter. In the macroscopic theory, there appear to be three – v_c, T_c, and P_c – but only two of these are independent because, according to the theory, the compressibility factor at the critical point is the same for all fluids.

The connection between the macroscopic and the microscopic theories of corresponding states can be established by substituting Eq. (4-77) into the relations given by Eq. (4-62). It then follows that the macroscopic critical properties υ_c, T_c, and P_c are related to the molecular parameters ε and σ by

(4-78)

(4-79)

(4-80)

where N_A is Avogadro’s constant (υ_c is per mole) and c₁, c₂, and c₃ are universal constants. For simple nonpolar molecules, i.e., those nonpolar molecules having a small number of atoms per molecule, these relations have been found empirically for the case where the generalized function F is replaced by the Lennard-Jones (12-6) potential (Hirschfelder et al., 1964). For that particular case, we have, approximately,

(4-81)

Since the critical temperature is a measure of the kinetic energy of the fluid at a characteristic state (where the liquid and gaseous states become identical), the simple proportionality between energy parameter ε and critical temperature T_c is reasonable. Similarly, the critical volume reflects the size of the molecules; hence, the proportionality between distance parameter σ³ and the critical volume is also reasonable. The proportionality of the critical pressure to the ratio ε/σ³ follows because, according to the theory, the compressibility factor z_c is the same for all fluids.

While parameters ε_i and σ_i are directly related to the macroscopic properties of substance i, the macroscopic properties used to estimate ε_i and σ_i need not be critical properties. For example, ε_i can also be evaluated from the Boyle temperature and σ_i from the molar volume at the normal boiling point. Nor is it necessary, in principle, that the molecular parameters be determined from therrnodynamic data since these parameters are also related to transport properties like viscosity and diffusivity (Hirschfelder et al., 1964). However, potential parameters obtained from different properties of the same fluid tend to be different because the assumed potential function (e.g., Lennard-Jones) is not the “true” potential function but only an approximation. Farther, the four assumptions cited above are not strictly true for most substances. If a particular potential-energy function is to be used for calculating equilibrium properties, it is best to evaluate the parameters from a property similar to the one being investigated.

An important advantage of the molecular theory, relative to the classical theory of corresponding states, is that the former permits calculation of other macroscopic properties (e.g. transport properties) in addition to those which may be calculated by classical thermodynamics from an equation of state. For the purposes of phase-equilibrium thermodynamics, however, the main advantage of the molecular theory is that it can be meaningfully extended to mixtures, thereby providing some aid in typical phase-equilibrium problems.

For mixtures, Eq. (4-67) can be written as

(4-82)

where (dr_i)^Ni is an abbreviated notation for the coordinates of the centers of the N_i molecules of kind i. Again, Γ_t is given by Eq. (4-69), but in addition to the pair interactions between similar molecuSes, Eq. (4-69) now includes also those between dissimilar molecules. In view of the physical significance of the parameters ε_i and σ_i it is possible to make reasonable predictions of what these parameters are for the interaction between dissimilar molecules. Thus, as a first approximation, the London theory suggests that, for the interaction of two unlike molecules i and j having nearly the same size and ionization potential,

(4-83)

and on the basis of a hard-sphere model for molecular interaction,

(4-84)

Equations (4-83) and (4-84) provide a basis for obtaining some properties of a variety of mixtures. Some applications of the molecular theory of corresponding states to phase-equilibrium calculations are discussed in later chapters.

4.13 Extension of Corresponding-States Theory to More Complicated Molecules

The theory of corresponding states as expressed by the generalized potential function [Eq. (4-64)] is a two-parameter theory and is therefore limited to those molecules whose pair-wise energies of interaction can be adequately described in terms of a function using only two parameters. Such molecules are called simple molecules; strictly speaking, only the heavier noble gases (argon, krypton, xenon) are “simple” but the properties of several others are closely approximated by Eq. (4-64). A simple molecule is one whose force field has a high degree of symmetry; that is equivalent to saying that the potential energy is determined only by the distance of separation and not by the relative orientation between two molecules. Noepolar (or slightly polar) molecules like methane, oxygen, nitrogen, and carbon monoxide are therefore nearly simple molecules. For more complex molecules, however, It is necessary to introduce at least one additional parameter in the potential function and thereby to construct a three-parameter theory of corresponding states. This can be done in several ways but the most convenient for practical purposes is to divide molecules into different classes, each class corresponding to a particular extent of deviation from simple-molecule behavior (Pitzer et ai, 1955, 1957, 1958). This extension of the theory of corresponding states relaxes the fourth assumption of the four listed in the preceding section. Assumptions 1 and 2 are unaffected. Assumption 3 is relaxed somewhat in the sense that we now assume that it is sufficient to use an average potential function wherein we have averaged out all effects of asymmetry in the intermolecular forces. Extension of corresponding-states theory is mostly concerned with assumption 4.

In the three-parameter theory of corresponding states, Eq. (4-64) still applies but the generalized function F is now different for each class. The class must be designated by some third parameter; for practical purposes, a convenient parameter is one that is easily calculated from readily available data. Different parameters have been proposed but Pitzer’s proposal is, perhaps, the most useful because his third parameter is calculated from experimental data that tend to be accurate as well as accessible. Pitzer defines an acentric factor ω that is a measure of the acentricity, i.e., the noncentral nature of intermolecular forces.

The definition of the acentric factor is arbitrary and chosen for convenience. According to two-parameter (simple-fluid) corresponding-states, the reduced saturation pressures of all liquids should be a universal function of their reduced temperatures; in fact, however, they are not, and Pitzer uses this empirical result as a measure of deviation from simple-fluid behavior. For simple fluids, it lias been observed that at a temperature equal to 7/10 of the critical, the saturation pressure P^s divided by the critical pressure P_c is given very closely by

4-85

Pitzer therefore defines the acentric factor by

4-86

For simple fluids, ω ≈ 0 and for more complex fluids, ω > 0.³¹ Acentric factors for some typical fluids are given in Table 4-14. (Appendix J presents a more extensive list). The acentric factor is easily determined from a minimum of experimental information; the data required are the critical temperature, the critical pressure, and the vapor pressure at a reduced temperature of 0.7. This reduced temperature is usually close to the normal boiling point where vapor-pressure data are most likely to be available.³²

³¹ For quantum gases (He,H2, Ne), ω is slightly negative. Since Pitzer’s theory of corresponding states is not applicable to quantum gases, these negative acentric factors are not significant.

³² Pitzer (1995) also discuss methods to determine co in case vapor-pressure data are not available.

Table 4-14 Acentric fzctors

The three-parameter corresponding-states theory asserts that all fluids having the same acentric factor have the same reduced configurational properties at the same reduced temperature T_R and reduced pressure P_R. Pitzer and Brewer have tabulated these reduced configurational properties as determined from experimental data for representative fluids (Pitzer, 1995) as functions of T_R and P_R. Any property of a fluid, in reduced form, is assumed to be given by a function of the three variables: P, t_R, and ω.

For example, for the compressibility factor z, Pitzer used a truncated Taylor series in ω:

4-87

where z⁽⁰⁾ is the compressibility factor for a simple fluid (e.g., argon or methane for which ω = 0) and z⁽¹⁾ represents the deviation of the real fluid from z⁽⁰⁾. With this approach, Pitzer correlated volumetric and thermodynamic properties of normal fluids³³ and their mixtures, using functions z⁽⁰⁾ and z⁽¹⁾) in tabular form, over the range of reduced temperatures, 0.8 to 4.0, and reduced pressures, 0 to 9.0.

³³ Fluids that have strong hydrogen bonds, large dipole moments or quantum effects are excluded from the normal category.

However, the original three-parameter Pitzer correlation is inadequate for calculations performed in the critical region and for liquids at low temperatures. To overcome these limitations, several modifications as well as extensions of Pitzer’s work to wider ranges of T_R and P_R have been published (Lu et al, 1973; Schreiber and Pitzer, 1989), as briefly mentioned in Sec. 5.7. A comprehensive compilation was presented by Lee and Kesler (1975).

In the generalized three-parameter correlation of Lee and Kesler, the volumetric and thermodynamic functions (e.g., densities, fugacity coefficients, second virial coefficients, etc.) of fluids are analytically represented by a modified Benedict-Webb-Rubin (BWR) equation of state. This equation is written first for a simple fluid (referred to by superscript 0), such as argon or methane, and then for a reference fluid (superscript r), chosen to be n-octane. The compressibility factor z for a normal fluid at reduced temperature T_R, reduced volume V_R, and acentric factor co is written in the form

4-88

In Eq.(4-88), the compressibility factors of both the simple fluid, z⁽⁰⁾, and the reference fluid, z^(r), are represented by the following reduced form of a modified BWR equation of state:

4-89

where constants b^*, c^*, and d^* are functions of the reduced temperature. These constants, as well as constants c₄, β, and γ, are determined for the simple fluid from data for argon and methane, and for the reference fluid from data for n-octane. Lee and Kesler give the constants of Eq.(4-89) and also expressions for several thermodynamic functions derived from the same equation. To extend their correlation to mixtures, Lee and Kesler provide the necessary mixing rules.

Inclusion of a third parameter very much improves the accuracy of corresponding-states correlations. For normal fluids (other than those that are highly polar or those that associate strongly by hydrogen bonding), the accuracy of the gas-phase compressibility factors given in the tables of Lee and Kesler is 2% or better, and for many fluids it is much better. The tables are not applicable to strongly polar fluids, although they are often so used with surprising accuracy except at low temperatures near the saturated-vapor region.

4.14 Summary

Both physical and chemical forces play an important role in determining properties of solutions. In some cases chemical forces can be neglected, thereby leading to purely “physical” solutions, but in other cases chemical forces predominate. The dissolved condition, therefore, represents a state of wide versatility where in one extreme the solvent is merely a diluent with respect to the solute, while in the other extreme it is a chemical reactant.

The aim in applying theraiodynamic methods to phase-equilibrium problems is to order, interpret, correlate, and finally, to predict properties of solutions. The extent to which this aim can be fulfilled depends in large measure on the degree of our understanding of intermolecular forces that are responsible for the molecules’ behavior.

Interrnolecular forces can be roughly classified into three categories. First, there are those that are purely electrostatic arising from the Coulomb force between charges. The interactions between charges, permanent dipoles, and quadrupoles, presented in Sec. 4.2, fall into this category. Second, there are polarization forces (Sec. 4.3) that arise from the dipole moments induced in the atoms and molecules by the electric fields of nearby charges of permanent dipoles. Finally, there are forces that are quantum-mechanical in nature. Such forces give rise to covalent bonding (including charge-transfer interactions) and to the repulsive interactions (due to the Pauli exclusion principle) that balance the attractive forces at very short distances.

These three categories are not rigid nor exhaustive, since for certain types of forces (e.g. van der Waals forces) an unambiguous classification is not possible, while other intermolecular interactions (such as magnetic forces) were not considered since they are always very weak for the systems of interest here.

Our quantitative knowledge of two-body intermolecular forces is limited to simple systems under ideal conditions, i.e. when the two molecules under consideration are isolated from all others. For non-simple systems, the effect of molecular structure and shape is often large but we have no adequate tools for describing such effects in a truly fundamental way; the best we can do is to estimate such effects by molecular simulation on computers.

Chemical forces (formation of weakly bonded dimers, trirners, etc.) are often dominant in determining thermodynamic properties. Numerous experimental methods (especially spectroscopy) can be used to augment and support thermodynamic measurements but as yet, our fundamental understanding of chemical forces is not satisfying.

For large molecules in solution, osmometry and light scattering provide information on intermolecular forces through the potential of mean force. Structural effects, as the hydrophobic effect, can exert a large influence.

For normal fluids, the theorem of corresponding states provides a powerful tool for estimating thermodynamic properties when experimental data are scarce.

Classical and statistical thermodynamics can define useful functions and derive relationships between them, but the numerical values of these functions cannot be determined by thermodynamics alone. Determination of numerical values, by either theory or experiment is, strictly speaking, outside the realm of thermodynamics; such values depend directly on the microscopic physics and chemistry of the molecules in the mixture. Future progress in phase-equilibrium thermodynamics depends, in part, on progress in statistical mechanics. However, progress in applications of phase-equilibrium thermodynamics is possible only with increased knowledge of intermoiecular forces.

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Problems

1. Consider one molecule of nitrogen and one molecule of ammonia 25 Å apart and at a temperature well above room temperature. Compute the force acting between these molecules. Assume that the potential energies due to various causes are additive and use the simple, spherically symmetric formulas for these potentials.

2. Experimental studies show that when two molecules of methane are 1 nm apart, the force of attraction between them is 2x10^-8 dyne. Using this result, we now want to estimate the force of attraction between two molecules of some substance B. We know little about substance B, but we know that its molecules are small and nonpolar. Further, we have critical data:

Using the theory of intermolecular forces and corresponding-states theory, estimate the force of attraction between two molecules of substance B when they are 2 nm apart.

3. Experimental studies for a simple, nonpolar gas A indicate that when two molecules of species A are 2 molecular diameters apart, the potential energy is -8xl0^-16 erg. Consider now two simple, nonpolar molecules of species B. When these two molecules are 2 molecular diameters apart, what is the potential energy?

Critical data for A and B are:

4. The dipole moment of HCI is 1.08 debye and its (mean) polarizability is (a/4πε_ο)=2.60x10^-30 m³ For a center of mass separation of 0.5 nm, calculate the contributions to the intermolecular energy from dipole-dipole and from dipole-induced dipole interactions for the two relative orientations, →→ and →↑.

5. Consider a spherical, nonpolarizable molecule of radius 3 Å having at its center a dipole moment of 2 debye. This molecule is dissolved in a nonpolar liquid having a dielectric constant equal to 3.5. Calculate the energy that is required to remove this molecule from the solution.

6. (a) Consider a gaseous mixture of N₂ and CO and compare it with a gaseous mixture of N₂ and Ar. Which mixture is more likely to follow Amagat’s law (approximately)? Why?

(b) Consider a diatomic molecule such as carbon monoxide. What is meant by the force constant of the C—O bond? How is it measured? Why is the heat capacity of CO significantly larger than that of argon?

7. What is meant by the terms “electron affinity” and “ionization potential”? How are these concepts related to the Lewis definition of acids and bases? In the separation of aromatics from paraffins by extraction, why is liquid SO₂ a better solvent than liquid NH₃?

8. Suppose that you want to measure the dipole moment of chlorobenzene in an inert solvent such as n-heptane. What experimental measurements would you make and how would you use them to compute the dipole moment?

9. Consider a binary solution containing components 1 and 2. The Lennard-Jones parameters ε₁₁, ε₂₂, and σ₁₁ and σ₂₂ are known. Assuming that σ₁₂ = 1/2(σ₁₁ + σ₂₂), express ε₁₂ in terms of pure-component parameters. Also find under which conditions ε₁₂ = (ε₁₁ε₂₂)^1/2,(Consider the attractive part only.)

10. What is a hydrogen bond? Cite all the experimental evidence that supports the conclusion that phenol is a hydrogen-bonded substance.

11. Qualitatively compare the activity coefficient of acetone when dissolved in carbon tetrachloride with that when dissolved in chloroform.

12. (a) The polymer polypropylene oxide, a polyether, is to be dissolved in a solvent at am-bient temperature. Consider these solvents: chlorobenzene, cyclohexane, and chloro-form. Which of these is likely to be the best solvent? The worst solvent? Explain. Would n-butanol be a good solvent? Would tertiary butanol be better? Explain. (The normal boiling point of rc-butanol is 117.5°C; that of c-butano! is 82.9°C.)

(b) Consider a polymer like cellulose nitrate. Explain why (as observed) a mixture of two polar solvents is frequently more effective in dissolving this polymer than either polar solvent by itself.

(c) Suppose that you wished to evaporate hydrogen cyanide from a solution in carbon tetrachloride or in octane. Which case is likely to require more heat?

13. Sketch qualitatively a plot of compressibility factor z versus mole fraction (from zero to unity) at constant temperature and pressure, for the following mixtures at 25 bar:

(a) Dimethylamine/hydrogen at !70°C.

(b) Dimethylamine/hydrogen chloride at 170°C.

(c) Argon/hydrogen chloride at 80°C.

In the sketch indicate where z = 1.

14. Using your knowledge of intermolecular forces, explain the following observations:

(a) At 30°C, the solubilities of ethane and acetylene in n-octane are about the same. However, at the same temperature, the solubility of acetylene in dimethylforrnamide is very much larger than that of ethane.

(b) At 10°C and 40 bar total pressure, the K factor for benzene in the methane/benzene system is much larger than that in the hydrogen/benzene system, at the same temperature and pressure (K = y/x). However, at 10°C and 3 bar the two K factors are nearly the same.

(c) Hydrogen gas at 0°C contains 1 mol % carbon dioxide. When this gas is compressed isothermally to pressure P, carbon dioxide condenses. However, when methane gas, also at 0°C and also containing 1 mol % carbon dioxide, is isothermally compressed to the same pressure P, carbon dioxide does not condense.

(d) At 100°C and 50 bar, the compressibility factor z for ethane gas is less than unity. However, at the same temperature and pressure the compressibility factor for helium gas is larger than unity.

(e) Which among the following liquids would be the best solvent for poly(vinyl chlo-ride), -(-CH₂CHCl-)_n-: i) _n-Heptane; ii) Ethanol; iii) Cyclohexanone; iv) Chioro-benzene.

(f) State what is the lowest nonvanishing muitipole moment (e.g. dipole, quadrupole, octopole, etc.) for each of the following molecules: i) Dichlorodifluoromethane; ii) Carbon tetrafluoride; iii) Carbon dioxide; iv) Mesitylene (1,3,5-trimethylbenzene).

(g) An absorber operating at 0°C and 600 psia uses heptane to absorb ethane and pro-pane from natural gas. It is found that heptane losses due to evaporation constitute a significative economic cost and it is therefore decided to lower the operating temperature to -20°C. However, after this is done, it is found that heptane losses have in-creased rather than decreased. Explain.

15. (a) What are the assumptions of the molecular theory of corresponding states?

(b) Does the Kihara potential (see Chap. 5) necessarily violate any of these assumptions? Explain.

(c) Which of these assumptions, if any, are not obeyed by hydrogen?

(d) Does the molecular theory of corresponding states have anything at all to say about c_p (specific heat at constant pressure)? If so, what?

16. A small drop of water at 25°C contains 0.01 molal potassium nitrate. The drop is completely surrounded by a semipermeable membrane; it is placed in an aqueous solution containing 0.01 M sodium nitrate and 2 g L^-1 lysozyme at 25°C. The electric charge on lysozyme is -2. The semi-permeable membrane has a cut-off molecular weight of 8,000. The molar mass of lysozyme is 14,000 g mol^-1. What is the osmotic pressure in the drop?

17. A container at 300 K is divided into two parts, α and β, separated by a semipermeable membrane which is permeable to the solvent (water) but not to the solute, a protein A at its isoelectric point. The molar mass of the protein is 5,000 g mol^-1.

Part β contains pure water. Part α contains protein A in aqueous solution; its concentration is 5 g/Iiter. Protein A has a tendency to dimerize according to

The equilibrium constant for dimerization is 10⁵. The mass density of water at 300 K is0.997 g cm^-3.

What is the osmotic pressure in part α?

Assume that at this low protein concentration, the liquid in part α behaves as an ideal dilute solution. In this case, ideal dilute solution is defined by relating the chemical potential of a solute i to its mole fraction x_i, according to

18. At 25°C osmotic-pressure data for aqueous solutions of the protein bovine serum albumin (BSA) in 0.15 M NaCl at pH 5.37 and 7.00 is given below. The isoelectric point of BSA is 5.37 in 0.15 M sodium chloride aqueous solutions.

(a) Estimate the molecular weight and the specific volume of BSA in the solution at pH 5.37.

(b) The specific volume of the unsolvated protein (which is considered spherical) is about 0.75 cm³ g^-1, and the excluded volume of the molecule is estimated to be 1.18x10^-24 m³ molecule^-1. At pH 5.37, does the solute appear to be solvated?

(c) For a charged particle in the presence of a low-molecular weight salt, the contribution of charge to the second osmotic coefficient is given by

where z is the charge of the particle of molecular weight M₂, p_l is the density (in g cm ^-3) of the solution, and m_MX is the concentration (molality) of the salt. From the osmotic-pressure data above, estimate the charge of BSA.

19. By membrane osmometry, osmotic pressures of aqueous solutions of the nonionic surfactant n-dodecylhexaoxyethylene monoether, C₁₂H₂₅(OC₂H₄)₄OC₂H₅, were measured at 25^°C. At concentrations below c₀ = 0.038 g L^-1, no osmotic pressure develops, indicating membrane permeation by the micelar species. Above this concentration, an osmotic pressure is measured, indicating the presence of impermeable aggregate species. The following table gives osmotic pressure data for various c - c₀ values:

(a) Obtain the second virial osmotic coefficient and the molecular weight for the species responsible for the osmotic pressure.

(b) Determine the number of molecules in the aggregate. Assuming they are spherical, estimate its molar volume and radius.