Showing papers in "Journal of Chemical Physics in 1958"

Free Energy of a Nonuniform System. I. Interfacial Free Energy

Abstract: It is shown that the free energy of a volume V of an isotropic system of nonuniform composition or density is given by : NV∫V [f 0(c)+κ(▿c)2]dV, where NV is the number of molecules per unit volume, ▿c the composition or density gradient, f 0 the free energy per molecule of a homogeneous system, and κ a parameter which, in general, may be dependent on c and temperature, but for a regular solution is a constant which can be evaluated. This expression is used to determine the properties of a flat interface between two coexisting phases. In particular, we find that the thickness of the interface increases with increasing temperature and becomes infinite at the critical temperature Tc , and that at a temperature T just below Tc the interfacial free energy σ is proportional to (T c −T) 3 2 . The predicted interfacial free energy and its temperature dependence are found to be in agreement with existing experimental data. The possibility of using optical measurements of the interface thickness to provide an additional check of our treatment is briefly discussed.

Nature of the Glass Transition and the Glassy State

Abstract: The thermodynamic properties of amorphous phases of linear molecular chains are obtained from statistical mechanics by means of a form of the quasi‐lattice theory which allows for chain stiffness and the variation of volume with temperature. A second‐order transition is predicted for these systems.This second‐order transition has all the qualitative features of the glass transition observed experimentally. It occurs at a temperature which is an increasing function of both chain stiffness and chain length and a decreasing function of free volume.The molecular ``relaxation times'' are shown to increase rapidly as the second‐order transition temperature is approached from above.To permit quantitative application of the theory and determine the relationship between the second‐order transition and the glass transition observed in ``slow'' experiments these two transitions are tentatively identified. By this means quantitative predictions are made concerning the variations of (1) glass temperature with molecula...

Reaction Rates by Nuclear Magnetic Resonance

Abstract: The Bloch equations for nuclear magnetic resonance are modified to describe the magnetic resonance of a single nuclear species X which is transferred back and forth between two (or more) magnetic environments (A,B) by kinetic molecular processes. The modified Bloch equations involve the usual assumptions of the Bloch theory and, in addition, require (a) that the X nuclear relaxation times be independent of the molecular exchange rates, and (b) that the X nuclear magnetization in A relax independently of the X magnetization in B, and vice versa. The modified Bloch equations are easily solved in the slow passage case, with arbitrary rf saturation. Earlier relations between reaction rates, and resonance line shapes, which were developed by Gutowsky, McCall, and Slichter, and extended by a number of other investigators, are easily derived using the modified Bloch equations. In the present work the modified equations are used to show how rapid exchange rates can sometimes be measured in solutions where the X r...

Reactions of Gaseous Molecule Ions with Gaseous Molecules. V. Theory

Abstract: Ion‐molecule reactions of the sort observed as secondary reactions in mass spectrometers have been treated by the methods of the modern kinetic theory; that is, the rate of reaction is expressed in terms of the velocity distribution functions of the reactants and the cross section for the reaction. The cross section, which is calculated by means of the properties of the classical collision orbits, is found to have an inverse square root dependence on energy. The ion distribution function, which is far from Maxwellian, is found by means of an explicit solution of the Boltzmann equation. A simple relation is given which relates the mass spectrometric data to the specific rate of the same reaction under thermal conditions. For the simpler molecules, this rate may be calculated completely a priori, with excellent agreement with experiment.

Calculation of Nuclear Magnetic Resonance Spectra of Aromatic Hydrocarbons

Abstract: The free electron model of Pauling is used to calculate the magnetic field around a benzene ring which is rotating rapidly about all axes in an external magnetic field. It is assumed that the π electrons precess in two circular paths, one on each side of the ring, equal in radius to the C–C distance in the benzene ring. The separation of these rings is taken as 1.28 A, which gives a calculated value for the nuclear magnetic resonance shielding value for benzene protons equal to the observed value. The field thus calculated is employed to predict shielding values for other aromatic compounds. Agreement with experiment is in general good, but there are some exceptions.

Size Effect in Heterogeneous Nucleation

Abstract: Assuming a spherical nucleating particle, the effect of particle size and surface properties upon nucleation efficiency is investigated. A general result is derived which is then applied to the condensation, sublimation, and freezing of water on foreign nuclei. The size effect is found to become important in the range 100–1000 A of particle radius. For particles larger than this, nucleation efficiency is substantially independent of size, while for smaller particles the efficiency is very greatly reduced.

Determination of Molecular Structures from Ground State Rotational Constants

Abstract: Kraitchman has shown that a single isotopic substitution on an atom is sufficient to determine directly the coordinates of that atom with respect to the principal axes of the original molecule. Kraitchman's formulas represent exact solutions of the equations for the equilibrium moments of inertia. However, the effects of the zero‐point vibrations are such that the coordinates obtained by substitution from the ground state moments of inertia I0 are systematically less than r0. These coordinates have here been called r (substitution) or rs, and it is found that rs≃(r0+re)/2, and Is= ∑ imirsi2≃(I0+Ie)/2.In the usual method of solution, the coordinate of one atom is determined from the equation for I0, and therefore the difference I0—Is must be made up by this one coordinate. This introduces a large error in the structures normally determined from ground state constants, and results in variations of 0.01 A in structures determined from different sets of isotopic species. If instead, we obtain the structure on...

Theory of Isotropic Hyperfine Interactions in π‐Electron Radicals

Abstract: Indirect proton hyperfine interactions in π‐electron radicals are first discussed in terms of a hypothetical CH fragment which holds one unpaired π electron and two σ‐CH bonding electrons. Molecular orbital theory and valence bond theory yield almost identical results for the unpaired electron density at the proton due to exchange coupling between the π electron and the σ electrons. The unrestricted Hartree‐Fock approximation leads to qualitatively similar results. The unpaired electron spin density at the proton tends to be antiparallel to the average spin of the π electron, and this leads to a negative proton hyperfine coupling constant.The theory of indirect proton hyperfine interaction in the CH fragment is generalized to the case of polyatomic π‐electron radical systems; e.g., large planar aromatic radicals. In making this generalization there is introduced an unpaired π‐electron spin density operator, ρN, where N refers to carbon atom N. Expectation values of the spin density operator ρN are called ``spin densities,'' ρN, and can be positive or negative. In the simple one‐electron molecular orbital approximation a π‐electron radical always has a positive or zero spin density at carbon atom N, 0≤ρN≤1. In certain π‐electron radical systems; e.g., odd‐alternate hydrocarbon radicals, the spin densities at certain (unstarred) carbon atoms are negative when the effects of π—π configuration interaction are included in the π‐electron wave function.The previously proposed linear relation between the hyperfine splitting due to proton N, aN, and the unpaired spin density on carbon atom N, ρN, aN=QρN is derived under very general conditions. Two basic approximations are necessary in the derivation of this linear relation. First, it is necessary that σ—π exchange interaction can be treated as a first‐order perturbation in π‐electron systems. Second, it is necessary that the energy of the triplet antibonding state of the C–H bond be much larger than the excitation energies of certain doublet and quartet states of the π electrons. This derivation of the above linear relation makes no restrictive assumptions regarding the degree of π—π or σ—σ configuration interaction. The validity of the above approximations is discussed and illustrated by highly simplified calculations of the proton hyperfine splittings in the allyl radical, assuming the π—π configuration interaction—and hence the negative spin density on the central carbon atom—to be small.Isotropic hyperfine interactions in molecules in liquid solution can also arise from spin‐orbital interaction effects, and it is shown that these effects are negligible for proton hyperfine interactions in aromatic radicals.

Chemical Equilibrium in Complex Mixtures

Abstract: A new method for the determination of the equilibrium composition of complex mixtures is described. The general method, which is based on the minimization of free energy, states the problem with unusual simplicity, avoiding many of the usual difficulties of description and computation. Two specific computation procedures are shown, one using a steepest descent technique applied to a quadratic fit, the other making use of linear programing.

Adiabatic Rate Processes at Electrodes. I. Energy-Charge Relationships

Abstract: The distinction between adiabatic and nonadiabatic reaction mechanisms at an electrode/solution interface is emphasized; in general, only adiabatic or near‐adiabatic paths are important in thermally activated reactions. The role of the dielectric in determining the course of such reactions is discussed. A general interpretation of the transfer coefficient and an account of the activation process in simple systems are given in terms of the distribution of electron charge density in the transition state; these are shown to be in good accord with experiment. The conventional approach to these problems, in which an approximate potential‐energy path for the reaction is calculated from intersecting potential‐energy curves for the initial and final states of the system, is criticized, and an alternative representation in terms of charge variation is proposed.

Isotropic nuclear resonance shifts

Abstract: It is shown that isotropic (or average) nuclear resonance shifts for a nucleus in a paramagnetic molecule in solution, and in a polycrystalline solid, can be used to distinguish between Fermi contact and ``pseudocontact'' contributions to isotropic nuclear‐hyperfine interactions. The pseudocontact interaction is that isotropic hyperfine coupling which arises from the combined effects of (electron‐spin)‐(nuclear‐spin) coupling, (electron‐orbit)‐(nuclear‐spin) coupling, and electron spin‐orbit interaction. When the magnetic hyperfine interaction between the electronic moment and nuclear spin is approximated by a point dipolar interaction, and the isotropic hyperfine interaction is exclusively pseudocontact, then the isotropic nuclear shift in a polycrystalline solid exceeds the solution shift by the factor 3(g∥+g⊥)/(g∥+2g⊥) where g∥ and g⊥ are the spectroscopic splitting factors parallel and perpendicular to the molecular symmetry axis. Isotropic shifts due to the Fermi contact interaction are the same for both solid state and solution cases.

Additivity Rules for the Estimation of Molecular Properties. Thermodynamic Properties

Abstract: A general limiting law is proposed which can be used to systematize the various laws of additivity of molecular properties. This can be stated as follows: ``If φ is a molecular property, then for the disproportionation reaction: RNR+SNS⇄2RNS, Δφ→0 as the separation between R and S becomes large compared to their dimensions.It is shown that the zero‐order approximation is equivalent to the law of additivity of atomic properties, the first‐order approximation to the law of additivity of bond properties, the second‐order approximation to the law of additivity of group properties, and so forth. It is shown that for Cp and S° (ideal gases), the additivity of atomic properties works to about ±2 cal/mole‐°K, while the additivity of bond properties is usually good to about ±1 cal/mole‐°K. The latter also estimates ΔHf° to about ±3 kcal/mole. The group additivity relation is generally obeyed to within ±0.5 cal/mole‐°K for Cp and S° and about ±0.6 kcal/mole for ΔHf°. Tables are presented for each of the partial pro...

Electron Spin Resonance in Transition Metal Chelates. I. Copper (II) Bis-Acetylacetonate

Abstract: Electron spin resonance at 3.2 cm has been observed from copper(II) in single crystals of copper(II) bis‐acetylacetonate diluted to ½ mole % with palladium(II) bis‐acetylacetonate at 77°K and 300°K. The parameters of the spin Hamiltonian have been expressed in terms of the atomic orbital coefficients of molecular orbital wave functions describing the electron hole in the orbital ground state and two excited states. Experimental determination of the spin Hamiltonian then results in expressions for the molecular orbital wave functions. The results indicate that the in‐plane sigma and pi bonding are appreciably covalent, whereas the out‐of‐plane pi bonding with the acetylacetonate ring is ionic.The hyperfine interaction may be expressed by a relation of the type K2g2=A2g∥2cos2θ+B2g⊥2sin2θ+2C2g∥g⊥sinθcosθ, where C2 increases with decreasing temperature. The appearance of the term in sinθ·cosθ is attributed to low symmetry crystal fields due to the crystallographic array, the main symmetry being determined by ...

Normal Vibrations of N‐Methylacetamide

Abstract: The in‐plane normal vibrations of N‐methylacetamide and its deuterated compound were calculated as a six‐body problem The force constants were transferred from diformylhydrazine and other molecules with similar structures The normal modes as well as the distributions of the potential energy among symmetry coordinates were also calculated The result of these calculations allows quantitative discussions concerning the nature of the amide I, II, II′, III, III′, and IV vibrations and other normal vibrations of N‐methylacetamide

Ion Exchange Kinetics. A Nonlinear Diffusion Problem

Abstract: Ideal limiting laws are calculated for the kinetics of particle diffusion controlled ion exchange processes involving ions of different mobilities between spherical ion exchanger beads of uniform size and a well-stirred solution, The calculations are based on the nonlinear Nernst-Planck equations of ionic motion, which take into account the effect of the electric forces (diffusion potential) within the system. Numerical results for counter ions of equal valence and six different mobility ratios are presented. They were obtained by use of a digital computer. This approach contains the well-known solution to the corresponding linear problem as a limiting case. An explicit empirical formula approximating the numerical results is given.

Paramagnetic resonance absorption in naphthalene in its phosphorescent state

Abstract: Electron paramagnetic resonance absorption in phenanthrene in its lowest photoexcited triplet state has been investigated at the boiling point of N2. The measurements have been made on phenanthrene molecules, oriented in diphenyl host crystals, at the frequency ∼2.3×1010 cycle sec—1 and also at lower frequencies at almost zero field (76 to 3 G). Taking the long principal magnetic axis in the plane of the phenanthrene molecule to be the x axis, the short axis in the plane to be the y axis, and the normal to the plane to be the z axis, the results have been described by the spin Hamiltonian, H=|β|H·g·S+S·T·S, where S=1,Tzz/hc≡D/hc=±0.100430 cm−1, σ{D/hc}=0.000010 cm−1,−Tyy/hc=Txx/hc≡E/hc=∓0.046576 cm−1, σ{E/hc}=0.000009 cm−1,gzz=+2.00209, σ{gzz}=0.00005,gyy=+2.00279, σ{gyy}=0.00005,gxx=+2.0041., A preliminary investigation of the hyperfine structure has been made. These results have been compared with published calculations.

Proton Magnetic Resonance Chemical Shift of Free (Gaseous) and Associated (Liquid) Hydride Molecules

Abstract: In order to study the factors giving rise to proton resonance chemical shifts of free molecules, as well as the association shifts due to hydrogen bonding, proton resonance measurements were carried out for a variety of simple hydride molecules in both the liquid and gaseous states. It is found that the proton shifts in the gaseous state can be interpreted in terms of the combined effect of the electronegativity and magnetic anisotropy of the atom to which the proton is bonded. The proton signals measured in the liquid state near the melting point, which correspond to maximum association, show large shifts to lower magnetic field relative to the corresponding gas signals. It is suggested that the association, or hydrogen bond shifts, in a system Y ——— H — X can be interpreted largely in terms of the reduction of the diamagnetic circulation in the H — X bond by the electrostatic field of the Y donor. Further possible contributions in certain anomalous cases are also considered.

Chain Stiffness and the Lattice Theory of Polymer Phases

Abstract: General formulas for the thermodynamic properties of amorphous polymer phases are obtained from statistical mechanics, with the aid of the lattice model, in a manner which avoids the use of restrictive assumptions concerning the nature of the individual polymer chains.Certain results, such as prediction of a second‐order transition for systems of semiflexible chains and the Flory‐Huggins formula for the entropy of mixing with monomeric solvent, are thus shown to be independent of the precise nature of the model assumed for the polymer chains.More complete information may be obtained by application of the general formulas to models descriptive of the molecular chains in question. As an example, the results of Flory for semiflexible linear chains whose stiffness arises exclusively from intramolecular nearest neighbors are obtained as a special case. (The conventional thermodynamic properties of polydisperse systems of chains of this type are shown to depend on the number average molecular weight.)

Electron Spin Resonance in Transition Metal Chelates. II. Copper(II) Bis-Salicylaldehyde-Imine

Abstract: The electron spin resonance at 3.2 cm has been observed for single crystals of copper(II) bis‐salicylaldehyde‐imine diluted to ½ mole percent with the corresponding nickel chelate. Analysis of the spectrum by methods given in the preceding article indicate that the sigma bonding and in‐plane II‐bonding are appreciably covalent, whereas the out‐of‐plane π bonding is ionic. Hyperfine interactions in addition to that resulting from the copper nucleus were observed and shown to be attributable to interactions with the nitrogen nuclei adjacent to the copper ion and with the two protons bonded to the carbon atoms directly adjacent to the nitrogen atoms. The two protons bonded directly to the nitrogen atoms do not contribute to the hyperfine interactions. Bonding information obtained from the magnitude of the hyperfine interaction due to the nitrogen nucleus agrees with that obtained from the g tensor and hyperfine interaction of the copper nucleus.In addition structural information about nickel(II) bis‐salicyla...

Thermodynamic study of SiC utilizing a mass spectrometer

Abstract: The sublimation of hexagonal SiC has been studied under equilibrium conditions The predominant gaseous species above SiC are Si, SiC2, and Si2C By combining the heat of formation of gaseous Si from solid SiC and the known standard heat of formation of SiC, a value of 113 kcal/g atom is obtained for the heat of sublimation of Si at 298°K From the measured partial pressures, using estimated free energy functions, dissociation energies for Si2, Si3, SiC, SiC2, Si2C, Si2C2, Si2C3, and Si3C are calculated and compared with previously known dissociation energies for group IV B molecules

Kinetic Studies of Hydroxyl Radicals in Shock Waves. II. Induction Times in the Hydrogen-Oxygen Reaction

Abstract: The formation of OH in the shock wave induced combustion of H2 and O2 has been measured by oscillographically recording the absorption of ultraviolet OH line radiation. The main features of the reaction course are: (1) an induction period whose length, ti, varies inversely with [O2], (2) an increase in the product [O2] ti as ti becomes short compared to the vibrational relaxation time of O2, and (3) at the end of the induction period, a sigmoid rise of [OH] to a maximum, followed by a slow decrease. ti has been studied over the ranges: 1100°≤T≤2600°K, 1.3×10—5≤[O2]≤8.0×10—4 mole/1, 0.25≤[H2]/[O2]≤5., 0.004≤[O2]/[Ar]≤0.20, and 5≤ti≤500 μsec. Agreement between incident and reflected shock experiments has been demonstrated. According to the branching chain mechanism known from explosion limit studies, ti is governed by the rate of H+O2→ lim k1OH+O according to: 2 k1[O2]ti=2.303 n, where n is the number of decades by which [OH] increases between initiation and the end of the induction period. The values of [O2]ti, which is nearly proportional to 1/k1, are summarized by: log10([O2]ti) (mole 1—1 sec)= —10.647+(3966±625)/T. The value k1=1.4×109 deduced at 1650°K from this work is combined with data near 800°K to give: k1=3×1011 exp(—17.5±3. kcal/RT) (mole/1)—1 sec.—1. The relation of these results to detonation experiments is discussed.

Expression in Terms of Molecular Distribution Functions for the Entropy Density in an Infinite System

Abstract: An expression for the entropy in a finite subvolume of an infinite fluid, originally proposed by Strato‐novich on the basis of information theory, is rewritten in terms of functions related to the potentials of average force. The result exhibits a correction to the corresponding expression obtained by H. S. Green for a finite system, in the form of a series in powers of the concentration, of which the first terms have been investigated by Richardson. By neglect of correlations of three or more molecules and powers of the density higher than the third, the result is specialized to the case of a gas of low density, for which the configurational entropy per molecule is calculated with use of the radial distribution obtained by Kirkwood for hard‐sphere and hard‐core Lennard‐Jones potentials.

Nuclear Magnetic Resonance Absorption in Glass. I. Nuclear Quadrupole Effects in Boron Oxide, Soda‐Boric Oxide, and Borosilicate Glasses

Abstract: The nuclear magnetic resonance spectrum of B11 has been investigated in several glassy materials containing boron oxide. A value of 2.76±0.05 Mc for the nuclear quadrupole coupling constant of B11 has been calculated from the second‐order quadrupole perturbation of the m = ½→—½ resonance line. The same quadrupole interaction has been observed in pure boron oxide, B2O3, soda‐boric oxide binary glasses, and two commercial borosilicate glasses. The quadrupole coupling constant is interpreted as arising from the bonding electrons in BO3 groups which comprise the glass network in boron oxide. Addition of oxygen ions to the glass network induces a change in the boron configuration from BO3 planar groups to BO4 tetrahedra resulting in a reduction in the quadrupole interaction. This effect has been observed in the Na2O–B2O3 glasses and in Corning ``Pyrex'' and 707 borosilicate glasses.

Statistical Mechanics of Transport Processes. XI. Equations of Transport in Multicomponent Systems

Abstract: The equations of hydrodynamics including the equations of continuity, equations of motion of the individual components as well as the over‐all equation of motion, and the energy transport equation are derived from statistical mechanics. Introduction of perturbations to the singlet and pair space distribution functions linearized in the temperature gradient, diffusion velocities, and the local rate of shear leads in the stationary case to the linear relations of irreversible thermodynamics between the gradients of temperature and chemical potential and the fluxes of heat and matter.

Proton Magnetic Resonance of the CH3 Group. IV. Calculation of the Tunneling Frequency and of T1 in Solids

Abstract: The thirty lowest eigenvalues of the Mathieu equation have been calculated numerically to nine places for a hindered rotor with threefold symmetry attached to a rigid framework. The results are tabulated for eight values of IV0/2ℏ2 ranging from 56.25 to 115.3125. The splittings of the torsional states of the CH3 group have been used to calculate the average tunneling frequency, νt, as a function of temperature for barrier heights of 825 cm—1 to 2770 cm—1. Some useful relations are developed between the eigenvalues and torsional splittings for the 2‐ and 3‐nodal potential functions.The BPP theory is used to calculate the proton spin‐lattice relaxation time, T1, for CH3 groups undergoing hindered reorientations in solids and also over‐all molecular tumblings, assuming Debye spectral density functions for both motions. The manner in which the tunneling affects T1 is considered at some length and two ways of introducing it are described. The coupling among the three protons within a given CH3 group is discuss...

Theory of Dilute Polymer Solution. II. Osmotic Pressure and Frictional Properties

Abstract: The osmotic second virial coefficient of flexible polymer solutions is calculated on the basis of the pearl‐necklace model with continuous medium approximation. Proper account is taken not only of the intermolecular interaction but also of the intramolecular interaction of segments or the so‐called ``excluded volume effect.'' Evaluation is carried out to the triple contact approximation. The results show that the intramolecular interaction has a marked influence on the molecular weight dependence of the coefficient.Then, theories of the intrinsic viscosity and frictional coefficient are developed on the same model as above, following the Kirkwood and Riseman scheme. The results show that, if α is the linear expansion factor of the polymer coil, the intrinsic viscosity is asymptotically proportional to α2 for the free‐draining case and α2.43 for the impermeable case. This implies that even in the limit for impermeable molecule, the hydrodynamic radius of the polymer coil is not proportional to its statisti...

Study of the Compressions of Several High Molecular Weight Hydrocarbons

Abstract: Isothermal compressions were measured for thirteen high‐purity liquid hydrocarbons and two binary mixtures of liquid hydrocarbons. These hydrocarbons have a molecular weight range of 170 to 351 and included normal paraffins, cycloparaffins, aromatics, and fused ring compounds. The pressure range for these measurements was from atmospheric to as high as 10 000 bars, being limited to lower values for some compounds to avoid possible solidification of the liquid. The volume changes due to pressure were measured at six temperatures spaced about equally in the range 37.8°C to 135.0°C. The volume changes and pressures were measured by methods similar to those of P. W. Bridgman.Pressure‐volume isotherms can be described adequately by the Tait equation, v0—v=C log(1+P/B), or for pressures above a certain minimum, whose value depends on the compound, by the Hudleston equation log[v2/3P/(v01/3−v1/3)]=A+B(v01/3−v1/3). For the Tait equation the parameter C can be predicted for hydrocarbon liquids from the relation C=0.2058 v0.Compressibility for a given hydrocarbon decreases with increasing pressure at constant temperature and increases with increasing temperature at constant pressure. The compression, and the compressibility, of liquid hydrocarbons are strongly dependent on molecular structure. Cyclization introduces a rigidity of molecular shape which decreases the compressibility markedly. Furthermore, fused ring cyclization as exemplified by naphthyl and decalyl structures has a considerably greater effect in decreasing compressibility than cyclization to nonfused rings such as cyclopentyl, cyclohexyl, or phenyl, even at equivalent carbon atom in ring percentages.Isobars and isochores were drawn and studied over the full range of temperature and pressure. The coefficient of thermal expansion, (1/v0) (δv/δT) P, for a given hydrocarbon, decreases with increasing pressure at constant temperature. (δ2v/δT2) P undergoes a sign change at a certain pressure, whose value depends on the compound; (δv/δT) P increases with increasing temperature below this pressure and decreases with increasing temperature above this pressure. The pressure coefficient, (δP/δT) v, is not a function of volume alone but is also dependent on the temperature and pressure. (δE/δv) T and (δE/δP) T go to zero and then reverse sign for compounds that can be studied to sufficiently high pressures.

General Theory of Bimolecular Reaction Rates in Solids and Liquids

Abstract: An argument is presented to show that the theory of kinetics of diffusion limited reactions developed previously may be generalized to a theory of bimolecular reaction rates in condensed phases. The rate at which particles arrive at a separation suitable for reaction is given by differential equations describing the random walks of the particles. The probability of reaction of particles separated by this collision diameter (i.e., near neighbors in the condensed system) is given by a steric and activation energy factor. The resulting boundary value problem is solved in detail. Explicit expressions for the reaction rates are presented. The reaction follows second‐order kinetics except that the most general rate ``constant'' is time dependent. Reaction rates are discussed in terms of the diffusion parameters and forces between the reactant particles. Physical interpretations of the rate constant are suggested. Reaction rates in liquid solvents are related to the viscosity of the solvent phase. An effective capture radius is derived for particles with long‐range forces.