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CHAPTER 4 INTERMOLECULAR FORCES AND CORRESPONDING STATES AND OSMOTIC SYSTEMS Tables of Contents 4.1 Potential-Energy Function 4.2 Electrostatic Forces 4.3 Polarizability and Induced Dipoles 4.4 Intermolecular Forces between Nonpolar Molecules 4.5 Mie's Potential-Energy Function for Nonpolar Molecules 4.6 Structural Effects 4.7 Specific (Chemical) Forces 4.8 Hydrogen Bonds 4.9 Electron Donor-Electron Acceptor Complexes 4.10 Hydrophobic Interactions 4.11 Molecular Interactions in Dense Fluid Media: Osmotic Pressure and Donnan Equilibria 4.12 Molecular Theory of Corresponding States 4.13 Extension of CorrespondingStates 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 Qtrans Qkin N , T Z N N , T ,V Qkin 2mkT h2 3 2 N 1 N! ZN t r1 r N dr1 dr N exp V kT 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. Z Nid V N For an ideal gas (t = 0) The complete canonical partitional function Q Qint N , T Qkin N , T Z N N , T ,V 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) d F dr r Potential energy (r) Fdr r 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) F qi qi q j 4o r 2 , point electric charge o , the dielectric permittivity of a vacuum o = 8.85419 10-12 C2 J-1 m-1 (4-3) Potential energy ij r qi q j 4o r 2 dr Let r , ij 0 ij qi q j qi q j 4o r constant of integratio n 4o r For qi = zi e, e = unit charge = 1.60218 × 10-19 C zi z j e 2 ij 4o r C2 [] 2 1 1 J CJ m m (4-5) For a medium other than vacuum zi z j e 2 zi z j e 2 ij 4r 40 r r , the absolute permittivity, =o r r=dielectric constant (permittivity relative to that of a vacuum) Comparison to physical intermolecular energies Coulomb energy is large and long range isolated ion (Cl- and Na+) in contact, the sum of the two ionic radii = 0.276 nm = 2.76 A 2 zi z j e 2 1 11.60218 1019 C2 ij 4o r 4 8.8542 1012 0.276 10 9 C 2 J 1 m 1 m Consider 8.36 1019 J (N m) r = 2.76A The same order of magnitude as typical covalent bond (200kT at room temperature) kT 1.38 1023 J K 1 300K 0.0414 1019 J 200 kT 8.28 1019 J When two ions are 560 A apart, Coulomb energy = kT 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 e+ d (4-6) e- 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 by 0.1 nm (1 A) 1.60218 1019 C1 1010 m 1.60218 1029 C m 4.8 Debye 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 four charges. The potential energy of two dipoles ij i j 4o r 2 cos cos i j sin i sin j cosi j θi 180, θ j 180 Maximum potential + 3 - - + Minimum potential + - i j 0 θi 180, θ j 0 + - i j 0 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) ij ije e ij / kT ij / kT d i j 2 d ije e ij / kT ij / kT sin 1 sin 2 d1d2 d sin 1 sin 2 d1d 2 d 2 2 2 6 3 4o kTr (4-8) Potential Energy for Dipole Moment ij (distance)-6 For pure polar substance ij (dipole moment)4 dipole moment < 1 debye, small contribution dipole 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 Q ei d i2 (4-9) i 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. 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 orientation is weighted according to its Boltzmann factor (Hirschfelder et al., 1964). Upon expanding in powers of 1/kT, For dipole i-quadrupole j i 2Q 2j ij ... 2 8 4o kTr (4-10) For quadrupole i-quadrupole j Qi2Q 2j 7 ij ... 2 10 40 4o kTr (4-11) Dependence of Potential Energy on Separation Distance Charged molecules (ions, Coulomb’s relation) ij (distance)-1, long range effect (4-5) Dipole moment, ij (distance)-6, short range effect (4-8) Dipole moment-Quadrupole moment (4-10) Quadrupolemoment-Quadrupole moment (4-11) • ij (distance)-8 , very short range effect • ij (distance)-10 , extremely short range effect 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 i E Where E is the field strength and is polarizability, a fundamental property of the substance. 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 C2 J -1m 2 ' [] 2 -1 -1 m3 4o CJ m ’ is called polarizability volume. 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. j The mean potential energy was first calculated by Debye i j 2 ij f T 2 6 4o r i (4-13) + - + Nonpolar Polar 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 ij i 2 j j 4o r 2 6 2 i f T i j - + + - Polar Polar (4-14) Mean Potential Energy Quadrupole-Quadrupole The average potential energy of induction between two quadrupole molecules 2 2 3 iQ j j Qi ij f T 2 8 2 4o r (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 quadrupolequadrupole 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 certain simplifying assumptions, the potential energy between two simple, spherically symmetric molecules i and j at large distances is given by 3 i j h 0i h 0 j ij 2 4o 2 r 6 h 0i h 0 j (4-16) Where h is Planck’s constant, and vo is a characteristic electronic frequency for each molecule in its unexcited state. First ionization potential I for hv0 For a molecule i, the product hv0 is very nearly equal to its ionization potential, Ii hv0 I 3 i j h 0i h 0 j ij 2 4o 2 r 6 h 0i h 0 j 3 i j I i I j 2 4o 2 r 6 I i I j For molecules i and j 3 i j I i I j ij 2 4o 2 r 6 I i I j (4-18) For the same molecules, i = j 3 i I i ii 4 4o 2 r 6 2 Potential energy f(T) Potential energy r-6 (4-19) Table 4-4 First ionization potentials Molecule 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 1 eV 1.60218 10 19 J 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 ij k ' i j r 6 ; 2 (4-20) 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 ij iijj 1/ 2 j i ii k ' 6 ; jj k ' 6 r r 2 (4-21) The above equation gives some theoretical basis for the"geometric-mean rule" . Comparison of dipole, induction, and dispersion forces London has presented calculated potential energies: Two Identical Molecules i 2 ii 2 3 4o kTr6 4 ii 2 2 i i 2 6 o 4 r 3 i I i ii 4 4o 2 r 6 4-8 B ii 6 r 2 4-19 4-13 Table 4-5 Relative magnitudes Molecule CCl4 Dipole, debye 0 CO Bx1079Jm6, Bx1079Jm6, Bx1079Jm6, dipole Induction Dispersion 0 0 1460 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; even for strongly polar substances such as ammonia, water, or acetone the contribution of dispersion forces is far from negligible. the contribution of dipolar moment is large for dipole moment > 1.0 debyte. Table 4-6 Relative magnitudes Molecule Molecule (1) (2) Dipole (1) Dipole (2) Bx1079Jm6, dipole Bx1079Jm6, Bx1079Jm6, Induction 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) B 6 r Repulsive potential (Amdur et al., 1954) Total potential energy (Mie, 1903) A rn A B total repulsive attractive n m r r 4.5 Mie's Potential-Energy Function for Nonpolar Molecules Mie's potential (1903) n / m n m 1/ n m nm n m r r (4-25) Lennard-Jones potential 12 6 4 r r (4-26) Parameters in potential function Parameters: , , m, n For a Mie (n, 6) potential 1 / n 6 6 n rmin 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). 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 1 t Nz 2 Where is the potential energy of an isolated pair; ½ is needed to avoid counting each pair twice. Substituting Mie’s equation 1 1 A B t Nz Nz n m 2 2 r r To account for additional potential energy resulting from interaction of a molecule with all of those outside its nearestneighbor shell, numerical constant sn and sm are introduced by 1 sn A sm B t Nz n m 2 r r 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 6 12 8 t, attractive B m m m r 2r 3r zBs 6B 2 4 m 1 m m m m2 r r 2 3 2 4 sm 1 m m 2 3 Table 4-7 Summation constant sn and sm (Moelwyn-Hughes, 1961) n or m m=6 Simple cubic Body-centered Face-centered cubic, z=8 cubic, z=12 z=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 dt 0 dr r rmin t rmin t nm sn nA sm mB rmin rmin t 1 sn A sm B t Nz n m 2 r r rmin rmin t nm sm sn 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. 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, mixh 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 species Association: acetic acid consists primarily of dimers due to hydrogen bonding Solvation: to form polymers, dimers, trimers to form complexes, a solution of sulfur trioxide in water by formation of sulfuric 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 groups (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 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 reaction Horizonal 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 bonding), 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 dimers, 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 enthalpies of formation for s-trinitrobenzene(electron acceptor)/ aromatic complexs (electron donor) dissolved in cyclohexane at 20 oC Equilibrium constant 0.88 -h(kJ mol-1) Mesitylene 3.51 9.63 Durene 6.02 11.39 Pentamethylbenzene 10.45 14.86 Hexamethylbenzene 17.50 18.30 Aromatic Benzene 6.16 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 2nitropropane, 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-donating Power of the hydrocarbon Evidence for the existence of a donor-acceptor Interaction between Tricholrobenzene And aromatic hydrocarbons 4.10 Hydrophobic Interaction Some molecules have a dual nature One part of molecule is soluble in water, hydrophilic, water-loving part 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 (hydrogen bond) between water molecules in highly structured liquid water. These attractive forces must be disrupted (使分裂) or distorted (扭曲) 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. 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 twobody 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 + . 1 1 1 pure1 T, P 1 pure1 T , P RT ln a1 (4-38) (4-38a) (4-39) For a pure fluid, v P T pure1 P pure1 P vpure1 (4-40) pure1 (T , P) 1 1 pure1 (T , P ) RT ln a1 pure1 (T , P) vpure1 RT ln a1 0 vpure1 RT ln a1 ln a1 vpure1 RT (4-41) If the solution in phase is dilute, x1 is close to unity, and 1 is also close to unity. a1 = 1 x1 ln a1 ln x1 vpure1 (4-42) RT When x2 1( very dilute ), ln x1 ln 1 - x2 x2 vpure1 x2 RT (4-43) Because x2 1, n2 n1 and x2 n2 / n1 vpure1 x2 RT n2 n RT 2 RT n2 n1 n1 V n1vpure1 n2 RT V n2 RT Van’f Hoff equation for osmotic pressure (4-44) Van’f Hoff equation Assumptions The The solution is very dilute. solution is incompressible. Application , T, and mass concentration of solute, the solute’s molecular weight can be calculated. A standard procedure for measuring molecular weights of large 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) Measure 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), 1 2 RT B * c2 C * c2 c2 M2 Where M2 is the molar mass of solute B*, the osmotic second virial coefficient C*, the osmotic third virial coefficient (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 Clbut it is impermeable to R-. Fig. 4-19 Electroneutrality before equilibrium is attained 0 0 cNa c R and 0 0 cNa c Cl (4-46) Let represent the change in Na+ concentration in phase. Because R- cannot move from one side to the other, the change in ClConcentration in phase is –. In : f 0 cNa c Na cClf- cRf- cR0- (4-47) In : f 0 cNa c Na cClf- cCl0- cRf- 0 (4-48) Na+ Cl- Calculate δ from know initial concentration For the solvent, we write s s (4-49) *s P vs RT ln as s s *s P vs RT ln as (4-50) *s is pure liquid solvent at system temperatu re and at zero pressure. *s P vs RT ln as *s P vs RT ln as P vs RT ln as P vs RT ln as RT as ln P P vs as solvent (4-51) We also have NaCl NaCl (4-52) NaCl is total dissociate d into Na and Cl Na Cl Na Cl - - (4-53) * Na * P vNa RT ln aNa - P v - RT ln a Cl Cl Cl * Na P vNa RT ln aNa * Cl - P vCl - RT ln aCl - P v P v P v Na Na Na RT ln a P v RT ln a v RT ln a a P v v RT ln a RT ln aNa P v RT ln a Cl Cl Na Cl - Cl Na Cl - - Cl - Na a a - Na Cl P P vNa vCl - RT ln a a - Na Cl a a - RT Na Cl P P ln vNa vCl - aNa aCl - Cl - solute (4-54) Na aCl - solvent solute aNa a RT as RT ln ln Cl vs as vNa vCl - aNa aCl v aNa a Cl aNa a Cl - as as Na v Cl- vs (4-55) as as 1 ai ci In a very dilute solutions, Activity of solute cNa c - c c Cl Na Cl (4-56) cNa c c c Cl Na Cl - c c c c 2c c c 2c c 2c c c c 2c 0 Na 0 Na 0 Cl - 0 2 Na 0 Na c 0 Na 2 0 Na 0 Na 0 2 Na 0 Na 0 Na 0 2 Na c 0 Na 2 2 0 2 Na 0 Na 0 Na (4-58) can be calculated and the final equilibrium concentration can be Calculated for eqs (4.47) and (4-48) The fraction of original sodium chloride in β that has moved to α c c 0 2 Na 0 Na c c 0 Na 1 0 cNa 0 cNa c 0 2cNa 0 Na 0 Na 0 2cNa c 2 c 2 0 Na 0 Na 1 c c 1 0 Na 1 0 Na 0 2cNa The osmotic pressure 0 0 2RT cNa c - 2 Cl 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 aNa RT ln N Aez Na aNa Upon setting activities equal to concentration 0 cNa RT ln 0 0 N Aez Na cNa cNa 4.12 Molecular Theory of Corresponding States r i i ii (4-64) Q Qint N , T Qtrans N , T ,V (4-65) t r1 ... rN dr1 ... drN ... exp N! kT 3 N 2mkT 2 1 Qtrans h 2 (4-66) t r1...rN Z N ... exp dr1...drN kT (4-67) Equation of state ln Q ln Z N P kT kT V T , N V T , N t ij rij i j ZN 3N rij r1 rN d 3 ... d 3 ... exp kT i j ZN 3N kT V Z , 3 , N * N Aconf kT ln Z N (4-68) (4-69) (4-70) (4-71) (4-72) Aconf N T , v kT V 3N Z N z * , 3 N P ln z * NkT V T , N V ~ P 3 ~ kT ~ T ,v 3 ,P N ~ ~ P F * T , v~ 4.13 Extension of corresponding states theory to more complicated molecules z z z 0 1 Conclusions Physical and chemical forces determines the properties of systems Intermolecular forces responsible for molecules behavior Homework-5, Prob 4-17 ln xw ln 1 x A x A2 x A x A2 vpure water RT basis : 1000cc of water mole of protein (5 g/liter)/( mw of lysozyne) specific volume of water 1/0.997 cm3/g ...........cm3/mol mole of water (1000 cc) 1000/speci fic volume of water in cm3/mol mole fraction of protein mole of protein /m ole of water a A2 x A2 5 K 10 2 2 aA xA Homework-3, Prob 4-19 1 2 RT B * c2 C * c2 c2 M2 RT RTB * c c0 c c0 M 2 Plot of c c0 vs c c0 Determine Intercept and slope Ans: M2= 12439 g/mol, B* = 2.93x10-7 L mol g-2 Relation of and t(rmint) 1 2 sn t r rmin t N zsm sm m / n m 9 t r rmin t subh0 R D 8 c 1T k c 2 N 3 c1 0.77, c2 0.75, c3 7.42, zc A c2vc 3 2 c1 0.290 3 c2 c3 NA NB t Z N ... exp dr drB ... kT A ij i 1 / 2 j ij 1 2 i j