Showing papers in "Journal of Chemical Physics in 1950"

On the Isotopic Chemistry of Carbonates and a Paleotemperature Scale

Abstract: The temperature variation of the fractionation of oxygen in exchange reactions between dissolved carbonate and water and between calcite and water and calculated on theoretical grounds, and checked experimentally. In the course of the experiments it was necessary to investigate several methods of decomposing calcium carbonate to carbon dioxide for mass spectrometer analysis. A method was developed for growing calcium carbonate from solution with the same isotopic composition as the carbonate shells of organisms produced at the same temperature from water of the same isotopic composition, and the results of these experiments at various temperatures are expressed in an equation relating the temperature of formation with the isotopic composition of the calcium carbonate and of the water.

On the Non‐Orthogonality Problem Connected with the Use of Atomic Wave Functions in the Theory of Molecules and Crystals

Abstract: The treatment of molecules and crystals by the Heitler‐London method or by the collective electron model can be based on the atomic orbitals φμ of the system. These orbitals are in general overlapping, and the corresponding overlap integrals Sμν, given by (1), have almost universally been neglected in the literature as causing undesirable complications. Here we will take these overlap integrals into consideration and show that they, instead of being negligible, are of essential importance in molecules and in crystals. The problem is simply solved by considering the orthonormalized functions [open phi]μ, given by (21), as the real atomic orbitals. The solution is worked out in detail for (I) the molecular orbital method of treating molecules, (II) the Bloch orbital method of treating crystals, and (III) the Heitler‐London method of treating both these systems in some simple spin cases. Some numerical applications are given for ionic crystals, showing that the overlap effects are responsible for all the rep...

A Viscosity Equation for Gas Mixtures

Abstract: By application of the kinetic theory, with several simplifying assumptions, the previous equation of Buddenberg and the author has been modified to give a general equation for viscosity as a function of molecular weights and viscosities of the pure components of the mixture. Agreement of the equation with experimental data is demonstrated for a number of highly irregular binary gas systems and mixtures of three to seven components.

The Statistical Mechanical Theory of Transport Processes. IV. The Equations of Hydrodynamics

Abstract: The equations of hydrodynamics—continuity equation, equation of motion, and equation of energy transport—are derived by means of the classical statistical mechanics. Thereby, expressions are obtained for the stress tensor and heat current density in terms of molecular variables. In addition to the familiar terms occurring in the kinetic theory of gases, there are terms depending upon intermolecular force. The contributions of intermolecular force to the stress tensor and heat current density are expressed, respectively, as quadratures of the density and current density in the configuration space of a pair of molecules.

On the stability of gas bubbles in liquid-gas solutions

Abstract: With the neglect of the translational motion of the bubble, approximate solutions may be found for the rate of solution by diffusion of a gas bubble in an undersaturated liquid‐gas solution; approximate solutions are also presented for the rate of growth of a bubble in an oversaturated liquid‐gas solution. The effect of surface tension on the diffusion process is also considered.

Intramolecular Reaction in Polycondensations. I. The Theory of Linear Systems

Abstract: For three types of linear polycondensing systems, equilibrium molecular size distributions, including rings as allowable species, are derived. Average molecular weights and amounts of ring and chain fractions are calculated therefrom. The fractions of rings are shown to increase with dilution, and with molecular weight. It is shown that beyond a critical dilution it is sometimes possible to obtain 100 percent yield of rings by driving the condensation to completion. Detailed calculations are made for two important cases corresponding to condensations of the decamethylene glycol‐adipic acid type: (1) for equimolar amounts of the two monomers, and (2) unequal amounts, with one monomer type completely reacted.

Statistical Mechanics of Swelling of Network Structures

Abstract: Inaccuracies in previously published theories dealing with the swelling of network polymers in solvents are pointed out. The treatment of Flory and Rehner, based on the tetrahedral model, is overly restrictive and leads to an entropy of swelling which is too small. A more recent treatment of Kuhn, Pasternak, and Kuhn introduces an error in the opposite direction. In the present paper the entropy of swelling a network of any degree of connectivity, or functionality f, is derived in a manner which avoids these inaccuracies. Corrected relationships are given for the partial molal‐free energy of dilution, swelling equilibrium and the influence of deformation on swelling equilibrium. These differ appreciably from those previously given when the degree of interlinking (i.e., concentration of cross linkages if f=4) is large and the equilibrium degree of swelling therefore is small. Previous expressions for elastic deformation at constant volume are unaffected.

Some Studies in Molecular Orbital Theory I. Resonance Structures and Molecular Orbitals in Unsaturated Hydrocarbons

Abstract: The predictions of qualitative resonance theory, as to the electronic structures of unsaturated hydrocarbons, are analyzed in terms of LCAO molecular orbital theory. The following results are proved for ``alternant'' hydrocarbons, that is, hydrocarbons containing no odd‐membered unsaturated rings:(1) The number of unpaired electrons present in the ground state is at least as great as the number of carbon atoms having a deficiency of valence bonds in any principal resonance structure.(2) With a few special exceptions, these odd electrons are distributed over just those atoms which have a deficiency of valence bonds in one or more of the principal resonance structures.(3) In singly charged hydrocarbon anions or cations the ionic charge is located on just those atoms which bear charges in the various principal resonance structures.It follows that enumeration of the principal resonance structures of a hydrocarbon molecule or ion gives much valuable information as to its electronic structure; and in particular, an alternant hydrocarbon possessing no normal Kekule‐type structure must have a paramagnetic ground state, and behave as a radical or multiradical.These results do not depend on the neglect of overlap, or the assumption of a constant carbon‐carbon resonance integral.

Structural Investigations by Means of Nuclear Magnetism. II. Hindered Rotation in Solids

Abstract: The experimental absorption line widths, for nuclei with spin 1/2, at nuclear magnetic resonance are given as a function of temperature for a number of molecular crystals. Temperatures ranged from 90°K to the melting points of the compounds. In some cases it has been possible to relate observed line structure and transitions in the line width to the existence and frequency of certain types of hindered rotational motion in the solid state. These deductions are based on mathematical considerations of the quantitative effect of such motions on the structure and second moment of an absorption line. It is emphasized that relatively low frequency motion of the order 105 cycles/second suffices to narrow the width of an absorption line from its value in the absence of that motion.1,2‐dichloroethane, 1,1,1‐trichloroethane, and perfluoroethane were found to have line‐width transitions coinciding with changes in crystal form and anomalies in the heat capacity. In 1,2‐dichloroethane and perfluoroethane these transiti...

Kinetics of Heterogeneous Nucleation

Abstract: Consider a phase transition α⇄β. Stable β‐embryos can be retained in the temperature range of α‐stability in cavities of suitable extraneous solids present in the system. Retained β‐embryos may nucleate the α→β‐transformation in the range of β‐stability.Using these concepts a theory for the effect of thermal history upon the kinetics of phase transformations is proposed. The kinetics of heterogeneous nucleation in the absence of retained embryos is discussed.

The Kinetics of Phase Transitions in Binary Systems

Abstract: A kinetic theory of phase transitions in two‐component systems is presented which is applicable to the transitions, vapor to liquid and liquid to liquid. The rate of the transition is characterized by a two‐dimensional vector which can be used in conjunction with the equation of continuity to develop a partial differential equation which determines the kinetics of the process.A new type of energy surface is introduced, and an investigation of its salient features enables one to simplify the solution of the differential equation mentioned above. This purely kinetic treatment makes possible the evaluation of the unknown coefficient found in the older quasi thermodynamic treatment of transition rates in two component systems.

Statistical Mechanics of Dilute Polymer Solutions. II

Abstract: New thermodynamic relationships applicable to dilute solutions of heterogeneous high polymers are derived. The function previously given to express the interaction between two polymer molecules of equal size is modified to include chemically similar polymer molecules differing in size. With the aid of this more general pair partition function the mutually excluded volume for a pair of polymer molecules is easily computed. The osmotic pressure for a dilute solution of a heterogeneous polymer may be expressed in virial form through the application of conventional methods for treating imperfect gases. The second virial coefficient in the series expansion for the osmotic pressure is readily obtained but it is necessary to resort to the hard sphere approximation for estimation of the third coefficient. The third coefficient is much larger than was indicated by earlier theories now known to be inapplicable to dilute solutions, and plots of π/c vs. c must be appreciably curved if the slope is great. Similar expr...

The Theory of Ionic Solutions

Abstract: The virial development for the osmotic pressure of a solution may be used, if the potentials of average force of the solute molecules at infinite dilution are known, to compute the deviations from perfect solution behavior. The expression for the logarithm of the activity coefficient can thus be obtained as a sum of coefficients multiplied by powers of the concentration. For an ionic solution, with 1/R2 forces, the series is only conditionally convergent. By summing certain additive terms occurring in the coefficients over all powers of concentration convergence can be attained.The integrations necessary to obtain terms correct up to and including c32 are performed. The results are given in terms of certain functions which can readily be computed.

The Effect of the Electrostatic Polarization of the Solvent on Electronic Absorption Spectra in Solution

Abstract: Regarding the solvent as a continuous dielectric medium, it is shown that its effect on the Franck‐Condon absorption of light by solute molecules must be expressed in terms of the electronic polarization part of its dielectric constant, K = n2. Using methods based both on quantum theory and on classical dispersion theory, it is shown that the red shift of absorption in solution depends directly on f, the oscillator strength, and inversely either on a3 (a is the radius of the spherical solute molecule) or the polarizability α. The expression Δν(cm−1)=const. (f/νa3)[(n2−1)/(2n2+1)] with two possible values of the constant, and alternatively with the substitution of α for a3, is tested on experimental data for isoprene, benzene, bromine, and iodine. Good quantitative agreement is obtained for the (V, N) transitions of isoprene and benzene. If the strong ultraviolet absorption of bromine and iodine solutions is regarded as the displaced (V, N) transition, the quantitative agreement is poor, although qualitati...

Light Scattering in Multi‐Component Systems

Abstract: A general equation is developed for the fluctuation of refractive index in multi‐component systems, thus permitting the interpretation of turbidities for such systems in the absence of angular dissymmetry. Applications to several special cases of interest are presented.

The Vibrational Spectra of Molecules and Complex Ions in Crystals III. Ammonium Chloride and Deutero‐Ammonium Chloride

Abstract: The infra‐red spectra of thin non‐scattering films of NH4Cl and ND4Cl were obtained at 28°, −78° and −190°C. A convenient low temperature transmission type cell usable for such films is described. No indication of fine structure due to free rotation of the NH4+ ions was found. Instead, evidence is presented for the existence, both above and below the λ‐point, of a torsional lattice mode involving the NH4+ ions. The limiting frequencies of the torsional oscillations were observed at about 390 and 280 cm−1 for NH4Cl and ND4Cl, respectively. These values agree quite well with the frequencies calculated on the basis of a purely electrostatic potential function. The spectra of the low temperature modifications indicate strongly that the structures belong to the space group Td1 in which the NH4+ ion symmetry is Td. Of the eight observed bands, two are assigned to the triply degenerate fundamentals ν3 and ν4, one to the overtone 2ν4, one to the combination ν2+ν4 which resonates strongly with ν3, one to the combi...

Molecular Motion in Certain Solid Hydrocarbons

Abstract: An investigation has been made of the variation of the nuclear resonance absorption line width with temperature for four long chain aliphatic compounds and eight aromatic hydrocarbons. The aliphatic compounds are dimorphous; in the lower temperature modification it is concluded that the molecules are rigid at the lowest temperatures, but that an increasing number of molecules rotate about their length as the temperature increases; in the upper temperature modification all the molecules rotate. The naphthalene crystal lattice is found to be rigid up to the melting point. Benzene and anthracene, however, show sharp line‐width transitions at about 110° and 190°K, respectively. It is suggested that in benzene this is due to tunnelling or rotation of the molecules about their hexagonal axes. The explanation for anthracene is less clear, but it is suggested that each molecule rotates about its long diad axis. The xylenes, mesitylene, and hexamethylbenzene show internal rotation of the CH3 groups at all temperat...

Light Scattering and Spectrophotometry of Colloidal Solutions

Abstract: During the past several years it has been demonstrated that the light scattered from dilute solutions of macromolecules or colloidal particles is intimately related to the weight, size, and interaction of the solute species. Theoretical developments and improved techniques have made possible the determination of the molecular weight, dimension and activity coefficient of a number of polymers and proteins in solution. In these investigations the intensity of light scattered at various angles from a monochromatic beam passing through the solution has been measured by means of specially designed photometers. However, it would appear that the equivalent information could be derived from a different means of observation—that of transmission measurements at various wave‐lengths. If this possibility could be exploited the common techniques of spectrophotometry could replace the more specialized ones now used. The extent to which this is feasible is explored in this paper.As a first step it is necessary to review...

Light Scattering Arising from Composition Fluctuations in Multi‐Component Systems

Abstract: A general theory of Rayleigh scattering due to composition fluctuations in multi‐component systems is developed with the aid of the grand canonical ensemble of Gibbs. It reduces to the usual expression for systems of two components, but contains previously neglected terms arising from thermodynamic interactions between solutes in systems of more than two components. The theory is used to interpret the turbidity measurements of polystyrene in benzene‐methanol mixtures of Ewart, Roe, Debye, and McCartney.

Nuclear Magnetic Resonance Line Shape for a Triangular Configuration of Nuclei

Abstract: The nuclear resonance line shape observed in solids can often be accounted for quantitatively and (in the absence of quadrupole effects) may provide information concerning the orientation and internuclear distances of dipolar nuclei. As an example of this method, the line shapes for a rigid and rotating triangular configuration of such nuclei are calculated and compared with recent experiments.

Critique of the Free Volume Theory of the Liquid State

Abstract: A deduction of the free volume theory of the liquid state from the general principle of statistical mechanics and certain well‐defined approximations is undertaken. The Gibbs configuration integral is expressed as a sum of integrals corresponding to single and multiple occupancy of the cells of a reference lattice. The integral corresponding to single occupancy is evaluated with the approximate probability density, expressed as a product of functions of the coordinates of individual molecules, which leads to minimum free energy under the restraints of constant temperature and volume. The minimization of the free energy gives an integral equation for the probability density within each cell of the lattice. A first approximation of the solution of this equation yields a partition function identical with that of the Lennard‐Jones‐Devonshire free volume theory.

The Intrinsic Viscosity, Translational and Rotatory Diffusion Constants of Rod‐Like Macromolecules in Solution

Abstract: A theory of the intrinsic viscosities, translational and rotatory diffusion constants of rod‐like macromolecules in solution is developed on the basis of a rigid‐chain model made up of monomeric elements separated by rigid bonds. The general methods of previous papers are used to take proper account of the hydrodynamic interactions of the monomeric units of the molecule. The theory developed leads to results showing qualitatively, marked similarity to that obtained for cylindrical and ellipsoidal particles. The intrinsic viscosities, translatory and rotatory diffusion constants are shown, as in the case of flexible macromolecules, to be related through a pair of parameters. A method for obtaining the molecular weight of rod‐like macromolecules is proposed, based on the combination of intrinsic viscosity and rotatory diffusion constant measurements.

Molecular Orbital Calculations of the Lower Excited Electronic Levels of Benzene, Configuration Interaction Included

Abstract: The lower excited π‐electron levels of benzene are calculated by the non‐empirical method of antisymmetrized products of molecular orbitals (in LCAO approximation) including configuration interaction. All configurations arising from excitation of one or two electrons from the most stable configuration are considered, and all many‐center integrals are retained. The results are in better agreement with experiment and valence‐bond calculations than those obtained previously by Craig in a calculation neglecting many‐center integrals. Configuration interaction is found to change the order of the 1B1u and 1E2g states but leave unchanged the order of the 3B1u and 3B2u states, in agreement with the assignments 1A1g—3B1u and 1A1g—1E2g for the experimental bands at 3.8 and 6.2 ev.

Determination of Molecular Shape by Osmotic Measurement

Abstract: The second coefficient A2 of osmotic pressure π of high‐polymer solutions is calculated in the case of the rigid ovaloid. A2 is a function of the molecular shape and thus we can determine the shape of high polymers.

Light Scattering Investigation of the Structure of Polystyrene

Abstract: Using light scattering, the mean square molecular radius and length of a number of samples of polystyrene have been measured as a function of solvent and temperature. Molecular weights, intrinsic viscosities, and ``osmotic second virial coefficients'' have also been determined on many of the samples. From the data it may be concluded: (1) that polystyrene molecules are extended and stiffened by steric interactions of the phenyl groups; (2) that the molecules are largely unbranched; (3) that the heat of dilution is negative in some solvents, contrary to the usual situation in hydrocarbon systems; (4) that the osmotic second virial coefficient depends strikingly on molecular weight for a given solvent; (5) that there is a good correlation between this coefficient and the mean square radius at a given molecular weight; (6) that the intrinsic viscosity is strictly proportional to a power of the molecular weight over the molecular weight range from 2,000 to 2,000,000, but that deviations occur at lower molecular weights; (7) that the intrinsic viscosity and mean square radius are functions of solvent at high molecular weights but not at very low molecular weights. Theoretical concepts necessitated by these observations are discussed briefly.

Effects of Dielectric Saturation upon the Diffuse Double Layer and the Free Energy of Hydration of Ions

Abstract: The theory of the diffuse double layer is developed for the case of a dielectric of variable dielectric constant. Using experimental data by Malsch, calculations have been made of the properties of the diffuse double layer. It is found that those properties which are experimentally significant are almost unaffected by the onset of dielectric saturation. The Born equation for the free energy of hydration of ions is elaborated to include the effects of dielectric saturation, and it is found that these effects are negligible in the case of the alkali halide ions.

The Energy of Interaction between Two Hydrogen Atoms

Abstract: The energy of interaction between two hydrogen atoms at intermediate to large separations is determined by the use of the variational principle applied to a wave function in which both ionic and polarization terms are included. The binding energy is known accurately in the neighborhood of the equilibrium separation from Rydberg's analysis of the experimental vibrational spectrum. At very large separations the van der Waals' energy has been calculated accurately by Pauling and Beach. Our calculations extend from the smallest to the largest internuclear distances. For the normal hydrogen molecule we compute a binding energy of 4.25 ev as compared to the experimental value of 4.74 ev. By comparing our calculations at small separations with Rydberg and at large separations with Pauling and Beach we obtain a most likely potential energy curve for all separations. Similarly we obtain a most likely potential energy curve for the 3Σ excited state of hydrogen by comparing with the accurate results of James, Coolid...

Radial Distribution Functions and the Equation of State of a Fluid Composed of Rigid Spherical Molecules

Abstract: The integral equation for the radial distribution function of a fluid of rigid spherical molecules has been integrated numerically in the Kirkwood approximation and in the Born‐Green approximation over a wide range of densities. The distribution functions so obtained have been used to calculate the equation of state and excess entropy of the fluid. The results are compared with those obtained by means of the free volume theory of the liquid state.

Molecular Orbital Calculations on Porphine and Tetrahydroporphine

Abstract: The first three electronic transitions of porphine and the first five electronic transitions of tetrahydroporphine are predicted as simple one‐electron LCAO molecular orbital transitions (with overlap included). The electron densities and bond orders in the ground states are also computed. The observed levels are tentatively assigned to the predicted levels with the help of comparisons with spectra of other aza‐amine derivatives of conjugated hydrocarbons. Probable degeneracies and polarizations are determined. The blue shift and loss of intensity of the first electronic transition, in going from tetrahydroporphine to the larger conjugated system of porphine, is typical of the change from a ``long‐field'' to a ``round‐field'' molecule (see previous paper) and is shown to occur in several other ``long‐field'' systems when additional conjugated atoms are added at the side of the system.