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Atkins and Paula, Physical Chemistry

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Atkins and Paula, Physical Chemistry

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    From Atkins and Paula, Physical Chemistry

    Chapter 18, Molecular Interactions

    (Material in sec.18.4 is the most important section; this is how macromolecules and aggregates stay together, and why things are sticky!)

18.1 Electric dipole moments

     Polar molecules

     Polarization

    18.2 Polarizabilities (omitted--used in a couple of sections of 18.4; don‘t worry about it)

    18.3 Relative permittivities (just skim--18.3 is only important to see how electrostatic interactions are ―muted‖ in water or other substances.)

18.4. Interactions between dipoles (8 parts)

     Follow the math on the subsections with stars only if you think you have the background--it might be rewarding for future use; read the rest in detail.

*a. The potential energy of interaction, p.13

*b. The electric field, p.18

    *c. Dipole-dipole interactions (Keesom interaction, p. 19)

d. Dipole-induced-dipole interactions (p.24)

    e. Induced-dipole-induced-dipole interactions (dispersion or London forces, p.26) [I may give you longer derivation and discussion of this.]

    f. Hydrogen bonding [p.28. I definitely will give you a separate reading oriented toward biological macromolecules and water.]

    g. Hydrophobic interaction [ p.30. Important for lipid bilayer structure of cell membrane as well as many properties of proteins (e.g. notice which amino acids are hydrophobic, hydrophilic), etc.]

h. Total attractive interactions (p.31)

18.5. Repulsive and total interactions (p.31)

    Molecular recognition and drug design (I 18.1, p.35) Questions and exercises (p.39).

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    QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture.

    Molecular interactions are responsible for the unique properties of substances as simple as water and as complex as polymers. We begin our

    examination of molecular interactions by describing the electric properties of molecules, which may be interpreted in terms of concepts in electronic structure. We shall see that small imbalances of charge distributions in molecules allow them to interact with one another and with externally applied fields. One result of this interaction is the cohesion of molecules to form the bulk phases of matter. These noncovalent molecular

    interactions are also important for understanding the shapes adopted by biological and synthetic macromolecules, as we shall see in Chapter 19. (I

    will distribute an edited version of that chapter, applying (easy) equilibrium statistical mechanics to biological polymers, separately).

    Electric properties of molecules

    Many of the electric properties of molecules can be traced to the competing influences of nuclei with different charges or the competition between the control exercised by a nucleus and the influence of an externally applied field. The former competition may result in an electric dipole moment. The latter may result in properties such as refractive index and optical activity. [We

    don‘t care about the latter, only that you can induce a dipole moment, for the calculation of London forces.]

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18.Electric dipole moments 1

An electric dipole consists of two electric charges +q and ?q separated by

    a distance R.

    This arrangement of charges is represented by a QuickTime?and avector m (1). The magnitude of m is µ = qR and, TIFF (Uncompressed) decompressorare needed to see this picture.although the SI unit of dipole moment is coulomb metre (C m), it is still commonly reported in the non-SI unit debye, D, named after Peter Debye, a pioneer in the study of dipole moments of molecules, where QuickTime?and aTIFF (Uncompressed) decompressor(18.1) are needed to see this picture.

    The dipole moment of a pair of charges +e and ?e separated by 100 pm is ?291.6 × 10 C m, corresponding to 4.8 D. Dipole moments of small molecules are typically about 1 D. The conversion factor in eqn 18.1 stems

    from the original definition of the debye in terms of c.g.s. units: 1 D is the

    dipole moment of two equal and opposite charges of magnitude 1 e.s.u. separated by 1 Å.

(a) Polar molecules

    A polar molecule is a molecule with a

    permanent electric dipole moment. The

    permanent dipole moment stems from

    the partial charges on the atoms in the

    molecule that arise from differences in

    electronegativity or other features of QuickTime?and abonding (Section 11-6). Nonpolar TIFF (Uncompressed) decompressor

    are needed to see this picture.molecules acquire an induced dipole

    moment in an electric field on account of

    the distortion the field causes in their

    electronic distributions and nuclear

    positions; however, this induced moment

    is only temporary, and disappears as

    soon as the perturbing field is removed. Polar molecules also have their existing dipole moments temporarily modified by an applied field.

In elementary chemistry, an electric dipole moment is represented by the

    arrow added to the Lewis structure for the molecule, with the + marking the

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    positive end. Note that the direction of the arrow is opposite to that of µ.

The Stark effect (Section 13-5) is used to measure the electric dipole

    moments of molecules for which a rotational spectrum

    can be observed. In many cases microwave

    spectroscopy cannot be used because the sample is not

    volatile, decomposes on vaporization, or consists of

    molecules that are so complex that their rotational

    QuickTime?and aspectra cannot be interpreted. In such cases the dipole TIFF (Uncompressed) decompressorare needed to see this picture.moment may be obtained by measurements on a liquid

    or solid bulk sample using a method explained later.

    Computational software is now widely available, and

    typically computes electric dipole moments by

    assessing the electron density at each point in the

    molecule and its coordinates relative to the centroid of

    the molecule; however, it is still important to be able

    to formulate simple models of the origin of these moments and to understand how they arise. The following paragraphs focus on this aspect.

All heteronuclear diatomic molecules are polar, and typical values of µ

    include 1.08 D for HCl and 0.42 D for HI (Table 18-1). Molecular symmetry

    is of the greatest importance in deciding whether a polyatomic molecule is polar or not. Indeed, molecular symmetry is more important than the question of whether or not the atoms in the molecule belong to the same element. Homonuclear polyatomic molecules may be polar if they have low

    symmetry and the atoms are in inequivalent positions. For instance, the angular molecule ozone, O (2), is homonuclear; however, it is polar 3

    because the central O atom is different from the outer two (it is bonded to two atoms, they are bonded only to one); moreover, the dipole moments associated with each bond make an angle to each other and do not cancel. Heteronuclear polyatomic molecules may be nonpolar if they have high symmetry, because individual bond dipoles may then cancel. The heteronuclear linear triatomic molecule CO, for example, is nonpolar 2

    because, although there are partial charges on all three atoms, the dipole moment associated with the OC bond points in the opposite direction to the dipole moment associated with the CO bond, and the two cancel (3).

    To a first approximation, it is possible to resolve the dipole moment of a polyatomic molecule into contributions from various groups of atoms in the molecule and the directions in which these individual contributions lie (Fig. 18.1).

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    Fig. 18.1 The resultant dipole moments (pale yellow) of

    the dichlorobenzene isomers (b to d) can be obtained

    approximately by vectorial addition of two chlorobenzene

    dipole moments (1.57 D), purple.

    Thus, 1,4-dichlorobenzene is nonpolar by symmetry on

    account of the cancellation of two equal but opposing C

    Cl moments (exactly as in carbon dioxide). 1,2-QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture.Dichlorobenzene, however, has a dipole moment which is

    approximately the resultant of two chlorobenzene dipole

    moments arranged at 60? to each other. This technique of

    ‗vector addition‘ can be applied with fair success to other

    series of related molecules, and the resultant µ of two res

    dipole moments µ and µ that make an angle θ to each 12

    other (4)

    QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture.

is approximately QuickTime?and aTIFF (Uncompressed) decompressor(18.2a) are needed to see this picture.

    When the two dipole moments have the same magnitude (as in the dichlorobenzenes), this equation simplifies to

    QuickTime?and aTIFF (Uncompressed) decompressor(18.2b) are needed to see this picture.

    Self Test 18.1 Estimate the ratio of the electric dipole moments of ortho

    (1,2-) and meta (1,3-) disubstituted benzenes.

Correct Answer

    µ(ortho)/µ(meta) = 1.7

    A better approach to the calculation of dipole moments is to take into account the locations and magnitudes of the partial charges on all the atoms. These partial charges are included in the output of many molecular structure software packages. To calculate the x-component, for instance, we need to know the partial charge on each atom and the atom‘s x-coordinate relative

    to a point in the molecule and form the sum

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    QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture.(18.3a)

Here q is the partial charge of atom J,x is the x-coordinate of atom J, and JJ

    the sum is over all the atoms in the molecule. Analogous expressions are used for the y- and z-components. For an electrically neutral molecule, the origin of the coordinates is arbitrary, so it is best chosen to simplify the

    measurements. In common with all vectors, the magnitude of m is related to

    the three components µ,µ, and µ by xyzQuickTime?and aTIFF (Uncompressed) decompressor(18.3b) are needed to see this picture.

    Example 18.1 Calculating a molecular dipole moment

Estimate the electric dipole moment of the amide group shown in (5) by using

    the partial charges (as multiples of e) in Table 18-2

    and the locations of the atoms shown.

    QuickTime?and a

    TIFF (Uncompressed) decompressor

    are needed to see this picture.

    QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture.

    Method We use eqn 18.3a to calculate each of the components of the dipole moment and then eqn 18.3b to assemble the three components into the magnitude of the dipole moment. Note that the partial charges are multiples of ?19the fundamental charge, e = 1.609 × 10 C.

Answer The expression for µx is

    QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture.

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    corresponding to µ = 0.42 D. The expression for µ is: xy

    QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture.

It follows that µ = ?2.7 D. Therefore,because µ = 0, yzQuickTime?and a

    TIFF (Uncompressed) decompressor

    are needed to see this picture.

    We can find the orientation of the dipole moment by

    arranging an arrow of length 2.7 units of length to

    have x, y, and z components of 0.42, ?2.7, and 0 units;

    the orientation is superimposed on (6).

    Self Test 18.2 Calculate the electric dipole moment of

    formaldehyde, using the information in (7).

     QuickTime?and aTIFF (Uncompressed) decompressor are needed to see this picture.

    Correct Answer: - 3.2D.

    (b) Polarization

    The polarization, P, of a sample is the electric

    dipole moment density, the mean electric dipole

    moment of the molecules, µ, multiplied by the

    number density, :

    QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture.(18.4)

    In the following pages we refer to the sample as a dielectric, by which is

    meant a polarizable, nonconducting medium.

    The polarization of an isotropic fluid sample is zero in the absence of an applied field because the molecules adopt random orientations, so µ = 0. In

    the presence of a field, the dipoles become partially aligned because some orientations have lower energies than others. As a result, the electric dipole moment density is nonzero. We show in the Justification below that, at a

    temperature T

    QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture.(18.5)

where z is the direction of the applied field . Moreover, as we shall see, there

    is an additional contribution from the dipole moment induced by the field.

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Justification 18.1 The thermally averaged dipole moment

    The probability dp that a dipole has an orientation in the range θ to θ + dθ is given by the Boltzmann distribution (Section 16-1b), which in this case is

    QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture.

    where E(θ) is the energy of the dipole in the field: E(θ) = ?µ cos θ, with 0 ? θ ? π. The average value of the component of the dipole moment parallel to

    the applied electric field is therefore

    QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture.

    with x = µ/kT. The integral takes on a simpler appearance when we write y = cos θ and note that dy = ?sin θ dθ.

    QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture.

At this point we use

    QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture.

It is now straightforward algebra to combine these two results and to obtain

    QuickTime?and a

    TIFF (Uncompressed) decompressor(18.6) are needed to see this picture.

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L(x) is called the Langevin function.

Under most circumstances, x is very small (for example, if µ = 1 D and T = ?1300 K, then x exceeds 0.01 only if the field strength exceeds 100 kV cm,

    and most measurements are done at much lower strengths). When x 1, the

    exponentials in the Langevin function can be expanded, and the largest term that survives is

    When x is small, it is possible to simplify expressions by using the expansion x1213e = 1 + x + 2x + 6x + ? ? ? ; it is important when developing approximations that all terms of the same order are retained because low-order terms might cancel.

    QuickTime?and aTIFF (Uncompressed) decompressor(18.7) are needed to see this picture.

Therefore, the average molecular dipole moment is given by eqn 18.6.

    QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture.

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18-3 Relative permittivities

When two charges q and q are separated by a distance r in a vacuum, the 12

    potential energy of their interaction is (see Appendix 3):

    QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture.(18.12a)

    When the same two charges are immersed in a medium (such as air or a liquid), their potential energy is reduced to

    QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture.(18.12b)

where ε is the permittivity of the medium. The permittivity is normally

    expressed in terms of the dimensionless relative permittivity, ε, (formerly r

    and still widely called the dielectric constant) of the medium:

    QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture.(18.13)

    The relative permittivity can have a very significant effect on the strength of the interactions between ions in solution. For instance, water has a relative

    permittivity of 78 at 25?C, so the interionic Coulombic interaction energy is

    reduced by nearly two orders of magnitude from its vacuum value. Some of the consequences of this reduction for electrolyte solutions were explored in Chapter 5.

    The relative permittivity of a substance is large if its molecules are polar or

    highly polarizable. The quantitative relation between the relative permittivity

    and the electric properties of the molecules is obtained by considering the polarization of a medium, and is expressed by the Debye equation (for the

    derivation of this and the following equations, see Further reading):

    QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture.(18.14)

where ρ is the mass density of the sample, M is the molar mass of the

    molecules, and P is the molar polarization, which is defined as m

    QuickTime?and aTIFF (Uncompressed) decompressor(18.15) are needed to see this picture.

     2The term µ/3kT stems from the thermal averaging of the electric dipole moment in the presence of the applied field (eqn 18.5). The corresponding

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