CHAPTER 4INTERMOLECULAR FORCES AND CORRESPONDING STATES AND

OSMOTIC SYSTEMS

Tables of Contents

4.1 Potential-Energy Function4.2 Electrostatic Forces4.3 Polarizability and Induced

Dipoles4.4 Intermolecular Forces between

Nonpolar Molecules4.5 Mie's Potential-Energy Function

for Nonpolar Molecules4.6 Structural Effects4.7 Specific (Chemical) Forces4.8 Hydrogen Bonds

4.9 Electron Donor-Electron Acceptor Complexes

4.10 Hydrophobic Interactions4.11 Molecular Interactions in Dense

Fluid Media: Osmotic Pressure and Donnan Equilibria

4.12 Molecular Theory of Corresponding States

4.13 Extension of Corresponding- States Theory to More Complicated Molecules

4.14 Summary

Introduction

Thermodynamic properties are determined by intermolecular forces

The objective of this chapter is to give a brief introduction to the nature and variety of forces between molecules

In the classical approximation

the kinetic energy (Qkin) depends only on the temperature.

ZN, the configurational partition function t(r1rN) is the potential energy of the entire system of N

molecules whose positions are described by r1rN. depends on intermolecular forces. For an ideal gas (t = 0)

The complete canonical partitional function

Q Q N T Z N T VNtrans kin , , ,

Q

mkT

h N

N

kin

2 12

3 2!

Z

kTd dN

t N

V N

exp

r rr r

1

1

Z VNNid

VTNZTNQTNQQ N ,,,, kinint

Intermolecular forces considered

Electrostatic forces: charged particles (ions); permanent dipoles, quadrupoles, and higher multipoles.

Induction forces : a permanent dipole (or quadrupole) and an induced dipole.

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.

4.1 Potential-Energy Functions

Force F between molecules (spherically symmetric molecules)

Potential energy

Fd

dr

r

rFdrr)(

A general form for complicated molecules Force should be as a function of distance, angle of

orientation of molecules

F r r, , ,... , , ,.... (4-2)

4.2 Electrostatic Forces

Coulomb's relation (Inverse-square law)

qi , point electric charge

o , the dielectric permittivity of a vacuum

o = 8.85419 10-12 C2 J-1 m-1

24 r

qqF

o

ji

(4-3)

Potential energy

For qi = zi e, e = unit charge = 1.60218 × 10-19 C

r

qq

r

r

qqdr

r

qq

o

jiij

ij

o

ji

ro

jiij

4

0 ,Let

nintegratio ofconstant 44 2

JmmJC

C][

4 112

22

r

ezz

o

jiij (4-5)

For a medium other than vacuum

, the absolute permittivity, =o r

r=dielectric constant (permittivity relative to that of a vacuum)

r

ezz

r

ezz

r

jijiij 0

22

44

Comparison to physical intermolecular energies Coulomb energy is large and long range

Consider isolated ion (Cl- and Na+) in contact, the sum of the two ionic radii = 0.276 nm = 2.76 A

The same order of magnitude as typical covalent bond (200kT at room temperature)

When two ions are 560 A apart, Coulomb energy = kT

m)(N J1036.8

m mJC

C

10276.0108542.84

1060218.111

419

112

2

912

2192

rezz

o

jiij

J1028.8 200

J100414.0K300K J1038.119

19123

kT

kT

r = 2.76A

Electrostatic forces-long range

Have a much longer range than most other intermolecular forces that depend on higher powers of the reciprocal distance

Salt crystal, very high melting points of salt Long range nature of ionic forces is

responsible for the difficult in constructing a theory of electrolyte solutions

Dipole Moment

A particle has two electric charges of the same magnitude e but of opposite sign, held a distance d apart.

ed

de+ e-

(4-6)

Units and constants used in this chapter

1 D(ebye) =3.33569×10-30 C m ε0 = 8.85419×10-12 C2 J-1 m-1

k = 1.38066×10-23 J K-1

e = 1.60218×10-19C The dipole moment of a pair +e and –e separated b

y 0.1 nm (1 A)

Debye 4.8

m C1060218.1m101C1060218.1 291019

Table 4-1 Permanent dipole moments

Molecules (Debye) Molecules (Debye)

CO 0.10 H2O 1.84

HBr 0.80 HF 1.91

NH3 1.47 CH3CN 3.94

SO2 1.61 KBr 9.07

Potential energy of two permanent dipoles Considering the Coulombic forces between the fo

ur charges.

The potential energy of two dipoles

Maximum potential

Minimum potential

jijijio

jiij r

cossinsincoscos24 3

+ -+-

+ +- -

0

0θ ,180θ

ji

ji

0

180θ ,180θ

ji

ji

Average potential energy

The average potential energy ij between tow dipoles i and j in vacuum at a fixed separation r is found by averaging over all orientations with each orientation weighted according to Boltzmann factor (Hirschfelder et al., 1964)

62

22

2121/

2121/

/

/

43

2

sinsin

sinsin

kTr

ddde

ddde

de

de

o

ji

kT

kTij

kT

kTij

ijij

ij

ij

ij

(4-8)

Potential Energy for Dipole Moment

ij (distance)-6

For pure polar substance

ij (dipole moment)4

dipole moment < 1 debye, small contributiondipole moment > 1 debye, significant contribution

Quadrupole Moments

Molecules have quadrupole moments due to the concentration of electric charge at four separate points in the molecules

Quadrupole Moments For a linear molecule, quadrupole moment Q is defined by the sum of the

second moments of the charges

where the charge ei is located at a point at a distance di away from some arbitrary origin and where all charges are on the same straight line.

For nonlinear quadrupole or for molecules having permanent dipole, the

definition of the quadrupole moment is more complicated.

i

iideQ 2 (4-9)

Table 4-2 Quadrupole moments for selected molecules

Molecule Q1040(C m2)

Molecule Q1040(C m2)

H2 +2.2 C6H6 +12

C2H2 +10 N2 -5.0

C2H4 +5.0 O2 -1.3

C2H6 -2.2 N 2O -10

Potential energy between dipole and quadrupole or quadrupole and quadrupole The average potential energy is found by averaging over all orientations; each orie

ntation is weighted according to its Boltzmann factor (Hirschfelder et al., 1964). Upon expanding in powers of 1/kT,

For dipole i-quadrupole j

For quadrupole i-quadrupole j

...4 82

22

kTr

Q

o

jiij

...440

7102

22

kTr

QQ

o

jiij

(4-10)

(4-11)

Dependence of Potential Energy on Separation Distance Charged molecules (ions, Coulomb’s relation)

ij (distance)-1, long range effect Dipole moment,

ij (distance)-6, short range effect Dipole moment-Quadrupole moment

ij (distance)-8 , very short range effect Quadrupolemoment-Quadrupole moment

ij (distance)-10 , extremely short range effect

(4-11)

(4-10)

(4-8)

(4-5)

Remarks on Dipole(2)/Quadrupole(4)/Octopole(8)/

Hexadecapole(16) Moments Literature study

Dipole(extensive)>Quadrupole(less)>Octopole(little)>Hexadecapole (much less)

Effect on thermodynamic properties Dipole(large)>Quadrupole(less)>Octopole(negligible)>Hexadecapole

(negligible) due to short ranges

4.3 Polarizability and Induced Dipoles

A nonpolar molecule has no permanent 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.

The induced dipole moments is defined as

Where E is the field strength and is polarizability, a fundamental property of the substance.

i E

Polarizability

Indicates that how easily the molecules 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 direction of field.

Polarizability volume

Polarizability has the units C2J-1 m2, however, it is common practice to present polarizabilities in units of volume as

’ is called polarizability volume.

31-1-2

2-12

mmJC

mJC][

4'

o

Table 4-3 Average Polarizabilities

Molecule ’1024(cm3) Molecule ’1024(cm3)

H2 0.81 SO2 3.89

N2 1.74 Cl2 4.61

CH4 2.60 CHCl3 8.50

HBr 3.61 Anthracene 35.2

Mean Potential Energy -nonpolar-polar molecules 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

Tf

ro

jiij

62

2

4

PolarNonpolar

+ -

+

-

j

i

(4-13)

Mean Potential Energy -polar-polar Polar as well as nonpolar can have dipole induced

in an electric field. The mean potential energy due to induction by

permanent dipoles is

Tfro

ijjiij

62

22

4

PolarPolar

-

+

i j

+

-

(4-14)

Mean Potential Energy -Quadrupole-Quadrupole The average potential energy of induction

between two quadrupole molecules

Tfr

QQ

o

ijjiij

82

22

42

3

(4-15)

Mean potential energy due to moments Due to Induced dipole moment is smaller

than that to permanent dipole-dipole interactions

Due to Induced quadrupole moment is smaller than that to permanent quadrupole- quadrupole interactions

4.4 Intermolecular Forces between Nonpolar Molecules

In 1930 it was shown by London that 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 has resulted in a distortion of 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.

Potential energy for nonpolar molecules Using quantum mechanics, London showed that subject to c

ertain simplifying assumptions, the potential energy between two simple, spherically symmetric molecules i and j at large distances is given by

Where h is Planck’s constant, and vo is a characteristic electronic frequency for each molecule in its unexcited state.

ji

ji

o

jiij hh

hh

r 00

00

6242

3(4-16)

First ionization potential I for hv0

For a molecule i, the product hv0 is very nearly equal to its ionization potential, Ii

Ihv 0

ji

ji

o

ji

ji

ji

o

jiij

II

II

r

hh

hh

r

62

00

00

62

42

3

42

3

For molecules i and j

For the same molecules, i = j

Potential energy f(T)

Potential energy r-6

62

2

44

3

r

I

o

iiii

ji

ji

o

jiij II

II

r 6242

3(4-18)

(4-19)

Table 4-4 First ionization potentialsMolecule I (eV) Molecule I (eV)

C6H5CH3 8.9 CCl4 11.0

C6H6 9.2 C2H2 11.4

N-C7H16 10.4 H2O 12.6

C2H5OH 10.7 CO 14.1

J 1060218.1eV 1 19

Polarizability dominate over ionization potential London’s formula is more sensitive to the

polarizability () than it is to the ionization potential (I)

For typical molecules, polarizability () is roughly proportional to molecular size while the ionization potential (I) does not change much form one molecule to another

Where k’ is a constant that is approximately the same for the three

types of interaction, i-i, i-j, and j-j. 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

The above equation gives some theoretical basis The above equation gives some theoretical basis for the"geometric-mean rule"for the"geometric-mean rule" . .

6

2

6

2

6' ;' ;'r

kr

kr

k jjj

iii

jiij

2/1jjiiij (4-21)

(4-20)

Comparison of dipole, induction, and dispersion forces

London has presented calculated potential energies: Two Identical Molecules

iiB

r 6

62

4

43

2

kTro

iii

62

2

4

2

ro

iiii

62

2

44

3

r

I

o

iiii

4-8

4-13

4-19

Table 4-5 Relative magnitudes

Molecule Dipole, debye

Bx1079Jm6, dipole

Bx1079Jm6,

Induction

Bx1079Jm6,Dispersion

CCl4 0 0 0 1460

CO 0.10 0.0018 0.0390 64.3

HBr 0.80 7.24 4.62 188

HCl 1.08 24.1 6.14 107

H2O 1.84 203 10.8 38.1

(CH3)CO 2.87 1200 104 486

Remarks on Table 4-5

The contribution of induction forces is small; The contribution of induction forces is small; eeven for strongly polar substances such as amven for strongly polar substances such as ammonia, water, or acetonemonia, water, or acetone

the contribution of dispersion forces is far from the contribution of dispersion forces is far from negligiblenegligible.

the contribution of dipolar moment is large for the contribution of dipolar moment is large for dipole moment > 1.0 debytedipole moment > 1.0 debyte.

Table 4-6 Relative magnitudes

Molecule(1)

Molecule(2)

Dipole(1)

Dipole(2)

Bx1079Jm6, dipole

Bx1079Jm6, Induction

Bx1079Jm6,Dispersion

CCl4 c-C6H12 0 0 0 0 1510

CCl4 NH3 0 1.47 0 22.7 320

CO HCl 0.10 1.08 0.206 2.30 82.7

H2O HCl 1.84 1.08 69.8 10.8 63.7(CH3)2CO NH3 2.87 1.47 315 32.3 185(CH3)2CO H2O 2.87 1.84 493 34.5 135

Remarks on Table 4-6

Polar forces are not important when the dipole moment is less than about 1 debye

induction forces always tend to be much smaller than dispersion forces.

Intermolecular Forces between Nonpolar Molecules

London's formula does not hold at very small separations

Repulsive forces between nonpolar molecules at small distances are not understood

Theoretical considerations suggest that the repulsive potential should be an exponential function of intermolecular separation

Total potential energy for nonpolar molecules Attractive potential

(London, 1937)

Repulsive potential(Amdur et al., 1954)

Total potential energy (Mie, 1903)

6r

B

nr

A

mn r

B

r

A attractiverepulsivetotal

4.5 Mie's Potential-Energy Function for Nonpolar Molecules

Mie's potential (1903)

Lennard-Jones potential

n m

n m r r

n m n m n m//1

612

4rr

(4-25)

(4-26)

Parameters in potential function

Parameters: , , m, n For a Mie (n, 6) potential

Parameters can be computed (with simplifying assumptions) from the compressibility of solids at low temperatures or from specific heat data of solid or liquids.

Parameters can also be obtained from the variation of viscosity or self-diffusivity with temperature at low pressures, and from gas phase volumetric properties (second virial coefficients).

min

6/16

rn

n

Application of Mie’s potential

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.

Simple theory of dense media using Mie’s potential Consider a condensed system near the triple point. Assume total potential energy is due to primarily to interactions between nearest

neighbors. Let the number of nearest neighbors is in a molecular arrangement be designated

by z. In a system containing N molecules, the total potential energy t is then

approximately given by

Where is the potential energy of an isolated pair; ½ is needed to avoid counting each pair twice.

Nzt 2

1

Substituting Mie’s equation

To account for additional potential energy resulting from interaction of a molecule with all of those outside its nearest-neighbor shell, numerical constant sn and sm are introduced by

mnt r

B

r

ANzNz

2

1

2

1

mm

nn

t r

Bs

r

AsNz

2

1

Determine of sn and sm

When the condensed system is considered as a lattice such as that existing in a regularly spaced crystal, the constants sn and sm can be accurately determined from the lattice geometry.

For example, a molecule in a crystal of the simple-cubic type has 6 nearest neighbors (z = 6) at a distance r, 12 at a distance r(2)1/2, 8 at a distance r(3)1/2. The attractive energy of one molecule with respect to all of the others is given

mmm

mm

mmm

mmm

s

r

zBs

r

rrrB

3

4

2

21

3

4

2

21

3

8

2

126

2

6B

attractive t,

Table 4-7 Summation constant sn and sm (Moelwyn-Hughes, 1961)

n or m Simple cubicz=6

Body-centered cubic, z=8

Face-centered cubic, z=12

m=6 sm =1.4003 sm = 1.5317 sm = 1.2045

n=9 sn =1.1048 sn = 1.2368 sn = 1.0410

n= 12 sn = 1.0337 sn = 1.1394 sn = 1.0110

n= 15 sn = 1.0115 sn = 1.0854 sn = 1.0033

Relation of rmin(isolated pair) and rmint(pair in a condensed system) At equilibrium, the potential energy of the condensed system

is a minimum

mm

nn

t r

Bs

r

AsNz

2

10

min

trr

t

dr

d

mBs

nAsr

m

nmnt

minn

m

mn

t s

s

r

r

min

min

trr minmin

4.6 Structural Effects Intermolecular forces of nonspherical molecules depend not only on t

he center-to-center distance but also on the relative orientation of

the molecules. Branching lower the boiling point; the surface area per

molecule decreases

Mixing liquids of different degrees of order usually brings about a net decrease of order, and hence positive contributions to the enthalpy h and entropy s of mixing

At mole fraction x = 0.5, mix

h for the binary containing n-decane is nearly twice that for the binary containing isodecane

4.7 Specific (Chemical) Forces Chemical forces:specific forces of attraction

which lead to the formation of new molecular speciesAssociation: to form polymers, dimers, trimers

acetic acid consists primarily of dimers due to hydrogen bonding

Solvation: to form complexes, a solution of sulfur trioxide in water by formation of s

ulfuric acid

4.8 Hydrogen Bonds Hydrogen fluoride

Crystal structure of ice

The bond strength hydrogen bonds 8 to 40 kJ /mol covalent bond 200 to 400 kJ /mol

Hydrogen bond is broken rather easily

Characteristic properties of hydrogen bonds (see Figure 4-5) Distances between the neighboring atoms of the two functional group

s (X-H- - -Y) in hydrogen bonds are substantially smaller than the sum of their van der Waals radii.

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

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.

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

Solvent effect on hydrogen bondingSolvent effect on hydrogen bonding

The thermodynamic constants for hydrogen-bonding reactions

are generally dependent on the medium in which they occur.

1: 1 hydrogen-bonded complex of trifluoroethanol (TFE) with acetone in the vapor phase and in CCl4 solution (inert solvent).

Vertical transfer reactionHorizonal complex-formation reaction

Transfer energy for complex into CCl4/(separated isomers into CCl4) (-8.7)/(-5.7-4.75)=0.83; Gibbs energy of transfer for complex into CCl4/(separated isomers into CCl4) (-4.1)/(-3.2-2.0)=0.79

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.

For most hydrogen-bonded complexes, stabilities decrease as the solvent changes from aliphatic hydrocarbon to chlorinated (or aromatic) hydrocarbon, to highly polar liquid.

Strong effect of hydrogen bonding on

physical properties of pure fluids Dimethyl ether and ethyl alcohol (hydrogen bo

nding), both are C2H6O

Strong dependence of the extent of

polymerization on solute concentration

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 dimer

s, trimers, or higher aggregates.

Hydrogen-bond formation between

dissimilar molecules Acetone and

chloroform(with hydrogen bonding)

Acetone with carbon tetrachloride (no hydrogen bonding )

Freezing-point data

Enthalpy of mixing data 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.

4.9 Electron Donor-Electron Acceptor Complexes

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

Data Type

1 Frequencies of charge-transfer absorption bonds primary

2 Geometry of solid complexes primary

3 NMR studies of motion in solid complexes primary

4 Association constants secondary

5 Molar absorptivity or other measurement of absorption intensity

secondary

6 Enthalpy changes upon association secondary

7 Dipole moments secondary

8 Infrared frequency shifts secondary

Primary and secondary data

“Primary” indicates that the data can be interpreted using well-established theoretical principles.

“Secondary” indicates that data reduction requires simplifying assumptions that may be doubtful.

Charge-transfer complexes (loose complex formation) If a complex is formed between A and B, light absorption is

larger. At different temperatures, it is possible to calculate also the

enthalpy and entropy of complex formation

Table 4-10 Spectroscopic equilibrium constants and enthalpTable 4-10 Spectroscopic equilibrium constants and enthalpies of formation for ies of formation for s-trinitrobenzene(electron s-trinitrobenzene(electron acceptoracceptor))/ aro/ aromatic complexs (electron matic complexs (electron donordonor) dissolved in cyclohexane a) dissolved in cyclohexane at 20 t 20 ooCC

Aromatic Equilibrium constant

-h(kJ mol-1)

Benzene 0.88 6.16

Mesitylene 3.51 9.63

Durene 6.02 11.39

Pentamethylbenzene 10.45 14.86

Hexamethylbenzene 17.50 18.30

Remarks on Table 4-10

Complex stability rises with the number of methyl groups on the benzene ring, in agreement with various other measurements indicating that -electrons on the aromatic ring become more easily displaced when methyl groups are added.

Table 4-11 Spectroscopic equilibrium constants for formation for polar solvent/p-xylene complexes dissolved in n-hexane at 25 oC

Polar solvent Equilibrium constant

Acetone 0.25

Cyclohexanone 0.15

Propionitrile 0.07

Nitropropane 0.05

2-Nitro-2-methylpropane 0.03

Remarks on Table 4-11

No complex formation with saturated hydrocarbons (such as 2-nitro-2-methylpropane, 0.03, and 2-nitropropane, 0.05) 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 naphthalenes.

Evidence for complex formation from thermodynamic measurements

Electron-donatingPower of the hydrocarbon

Evidence for the existence of a donor-acceptorInteraction betweenTricholrobenzeneAnd aromatichydrocarbons

4.10 Hydrophobic Interaction

Some molecules have a dual nature One part of molecule is soluble in water, hydrophilic, water-loving p

art While another part is not water-soluble, hydrophobic, water-fearing

part

Have a unique orientation in an aqueous medium; to form suitably organized structures.

Such molecules called “amphiphiles”. The organized structure called “micelles”

Hydrophobic part (a long-chain hydrocarbon) is kept away from water

Hydrophilic terminal groups at the surface of the aggregates are water solvated and keep the aggregations in solution.

Reverse miscelles, by addition of a small amount of water to a surfactant containing organic nonpolar phase

hydrophobic effect 斥水性 The hydrophobic effect arises mainly from the strong attractive forces (hyd

rogen bond) between water molecules in highly structured liquid water. These attractive forces must be disrupted ( 使分裂 ) or distorted ( 扭曲 ) w

hen a solute is dissolved in water. Upon solubilization of a solute, hydrogen bonds in water are often not brok

en but they are maintained in distorted form. Water molecules reorient, or rearrange, themselves such that they can par

ticipate 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.

Table 4-12 Change in standard molar Gibbs energy (go), enthalpy (ho), and entropy

(Tso) for the transfer of hydrocarbons from their pure liquids into water at 25 oC

Hydrocarbon go ho Tso

Ethane 16.3 -10.5 -26.8

Propane 20.5 -7.1 -27.6

N-Butane 24.7 -3.3 -28.0

N-Hexane 32.4 0 -32.4

Benzene 19.2 +2.1 -17.1

Toluene 22.6 +1.7 -20.9

Remarks on Table 4-12

The standard entropy of transfer is strongly negative, due to the reorientation 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 hydrophobic effect.

This effect is, in part, also responsible for the immiscibility of nonpolar substances (hydrocarbons, fluorocarbons, etc) with water.

Closely related to the hydrophobic effect is the hydrophobic interaction. This interaction is mainly entropic and refers to the unusually strong attraction between hydrophobic molecules in water, in many cases, this attraction is stronger than in vacuo.

Energy of interaction of two contacting methane in vacuo is -2.5 x 10-21 J.

Energy of interaction of two contacting methane in water is -14 x 10-21 J.

4.11 Molecular interactions in dense fluid media Intermolecular forces

In the low pressure gas phase, interact in a “free” medium (i.e., in a vacuum), described by a potential function (e.g. Lennard-Jones)

In the liquid solvent, interact in a solvent medium, described by the potential of mean force

The essential difference is that the interaction between two molecules in a solvent is influenced by the molecular nature of the solvent but there is no corresponding influence on the interaction of two molecules in (nearly) free space.

In the low pressure gas phase

Two solute molecules in a solvent

10/31/2006

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 (擾亂 ) 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 solvation force between them.

The molecular nature of the solvent can produce potentials of mean force that are much different for the corresponding two-body potential in vacuo.

The potential of mean force is a measure of the intermolecular interaction of solute molecules in liquid solution.

Solution theories, such as 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 ( 滲透 ) virial coefficients are obtained through osmotic-pressure measurements.

Osmotic Pressure

Van’t Hoff (1890) Fig 4-17

Fig 4-17

The semi-permeable membrane is permeable to the solvent (1) but impermeable to the solute (2). The pressure on phase is P, while that on phase is P + .

11

PT ,1 pure1

11 pure1 ln, aRTPT

(4-38)

(4-38a)

(4-39)

1 pure1 pure1 pure vPP

vPT

For a pure fluid,

RT

va

aRTv

aRTvPT

aRTPTPT

1 pure1

11 pure

11 pure1 pure

11 pure111 pure

ln

ln0

ln),(

ln),(),(

(4-40)

(4-41)

If the solution in phase is dilute, x1 is close to unity, and 1 is also close to unity. a1 = 1 x1

RTnV

RTnvnV

RTn

nRT

nn

nRTxv

nnxnnx

RTxv

xxxxRT

vxa

2

21 pure1

1

2

12

221 pure

122122

21 pure

2212

1 pure11

/ and ,1 Because

-1lnln ),dilutevery (1When

lnln

Van’f Hoff equation for osmotic pressure

(4-42)

(4-43)

(4-44)

Van’f Hoff equation

Assumptions The solution is very dilute. The solution is incompressible.

Application Measure , T, and mass concentration of solute, the solute

’s molecular weight can be calculated. A standard procedure for measuring molecular weights of l

arge molecules (polymer or biomacromolecules such as proteins) whose molecular weight cannot be accurately determined from other colligative property measurements (boiling point elevation or freezing point depression)

For finite concentration

Van’t Hoff’s equation is a limiting law for the concentration of solute goes to zero.

For finite concentration, a series expression is used

Osmotic virial expansion

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

Where M2 is the molar mass of solute B*, the osmotic second virial coefficient C*, the osmotic third virial coefficient

2

2222

**1

cCcBM

RTc

(4-54)

For dilute solution, we can neglect three-body interaction (C*). Thus, a plot of /c2 against c2

is linear, with intercept equal to RT/M2 and slope equal to RTB*.

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

Attractive force

Repulsive force

Macroscopic and microscopic

Osmotic second virial coefficients ( a macroscopic property) are related to intermolecular forces (microscopic property) between two solute molecules.

B22* can provide useful information on interaction between polymer of protein molecules in solution.

Donnan Equilibria

The osmotic-pressure relation given by van’t Hoff was derived for solutions for nonelectrolytes or for solutions of electrolytes where the membrane’s

permeability did not distinguish between cations and anions. Consider a chamber divide 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.

Donnan Equilibria

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 αand phase β, by an ion-selective membrane. The membrane is permeable to water, Na+ and Cl- but it is impermeable to R-.

Fig. 4-19

Electroneutrality before equilibrium is attained

0

Cl

0

Na

0

R

0

Na -- and cccc

Let represent the change in Na+ concentration in phase. BecauseR- cannot move from one side to the other, the change in Cl- Concentration in phase is –.

0 :In

:In f

R

0

Cl

f

Cl

0

Na

f

Na

0

R

f

R

f

Cl

0

Na

f

Na

---

---

ccccc

ccccc

Na+

Cl-

(4-46)

(4-47)

(4-48)

Calculate δ from know initial concentration For the solvent, we write

pressure. zeroat and re temperatusystemat solvent liquid pure is

lnln*

**

s

ssssssss

ss

aRTvPaRTvP

PPa

a

v

RT

aRTvPaRTvP

aRTvPaRTvP

s

s

s

ssss

ssssss

ln

lnln

lnln **

(4-49)

(4-50)

(4-51)solvent

-- ClNaClNa

-

NaClNaCl

Cl and Na into ddissociate totalis NaCl

have also We

---

---

ClCl

*

ClNaNa

*

Na

ClCl

*

ClNaNa

*

Na

lnln

lnln

aRTvPaRTvP

aRTvPaRTvP

(4-52)

(4-53)

-

-

-

-

-

-

----

--

--

ClNa

ClNa

ClNa

ClNa

ClNaClNa

ClNaClNaClNaClNa

ClClNaNa

ClClNaNa

ln

ln

lnln

lnln

lnln

aa

aa

vv

RTPP

aa

aaRTvvPP

aaRTvvPaaRTvvP

aRTvPaRTvP

aRTvPaRTvP

(4-54)solute

sv

vv

s

s

s

s

s

a

a

aa

aa

aa

aa

vv

RT

a

a

v

RT

-ClNa

-

-

-

-

-

ClNa

ClNa

ClNa

ClNa

ClNa

lnln

In a very dilute solutions, 1 ss aa

Activity of solute ii ca

-- ClNaClNacccc

solvent solute

(4-55)

(4-56)

0

Na

0

Na

20

Na

20

Na

0

Na

0

Na

0

Na

20

Na

0

Na

20

Na

20

Na

20

Na

20

Na

0

Cl

0

Na

0

Na

ClNaClNa

2

2

2

2

-

--

cc

c

ccc

ccc

ccc

cccc

cccc

can be calculated and the final equilibrium concentration can be Calculated for eqs (4.47) and (4-48)

(4-58)

The fraction of original sodium chloride in β that has moved to α

1

0

Na

0

Na

0

Na

0

Na

0

Na

0

Na0

Na

0

Na

0

Na

0

Na0

Na

0

Na

0

Na

20

Na

2

2

1

21

1

2

2

c

c

c

c

ccc

cc

c

c

cc

c

The osmotic pressure

22 0

Cl

0

Na -ccRT

The difference in electrical potential

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.

The difference in electric potential is given by the Nerst equation

Upon setting activities equal to concentration

Na

Nalna

a

ezN

RT

NaA

0

Na

0

Na

0

Nalncc

c

ezN

RT

NaA

4.12 Molecular Theory of Corresponding States

iii i

r

Q Q N T Q N T Vtrans int , , ,

Q

mkT

h N

r r

kTdr drtrans

Nt N

N

2 12

32

11

!

... exp...

...

N

NtN drdr

kT

rrZ ...

...exp... 1

1

(4-64)

(4-65)

(4-66)

(4-67)

Equation of state

P kTQ

VkT

Z

VT N

N

T N

ln ln

, ,

t ij iji j

r

ZkT

rd

rdr

NN ij

i j

N

3 1

3 3... exp ...

Z ZkT V

NNN

N

3

3* , ,

A kT ZconfN ln

(4-68)

(4-69)

(4-70)

(4-71)

(4-72)

A N T vconf ,

Z zkT V

NN

N

3

3* ,

P

NkT

z

V T N

ln *

,

~, ~ ,

~T

kTv

V

NP

P

3

3

~*

~,~P F T v

4.13 Extension of corresponding states theory to more complicated molecules

10 zzz

Conclusions

Physical and chemical forces determines the properties of systems

Intermolecular forces responsible for molecules behavior

Homework-5, Prob 4-17

225

waterpure

22

22

10K

waterof oleprotein /m of mole protein offraction mole

cm3/molin water of volumefic1000/speci cc) (1000 water of mole

.cm3/mol.......... cm3/g 1/0.997 water of volumespecific

lysozyne) of mwg/liter)/( (5 protein of mole

waterof cc1000:

1lnln

A

A

A

A

AAAAw

x

x

a

a

basisRT

vxxxxx

Homework-3, Prob 4-19

2

2222

1c*Cc*B

MRT

c

020

cc*RTBM

RT

cc

0cc

Plot of 0cc vs

Determine Intercept and slope

Ans: M2= 12439 g/mol, B* = 2.93x10-7 L mol g-2

Relation of and t(rmint)

mnm

m

n

mtt s

s

zsrr

N

/

min

21

Dt Rhrrt 8

90submin

c c c zc

c cc1 2 31

2 3

0 77 0 75 7 422

30 290 . , . , . , .

ZkT

dr drNt

A

N

B

NA B

... exp ...

ij i j 1

2 2/1jiij

cA vcN 23

32 cTc

k1