Volume 33, Issue 5, 01 November 1960

Theory of the Diffuse Double Layer
View Description Hide DescriptionThe charge moment expansion for potentials of mean force acting between the ions of an electrolyte is reviewed in a form applicable to surface phases. An integral equation is in this manner derived for approximate determination of the average charge distribution near a planar electrode. The solution of the linearized equation is constructed for an electrolyte consisting of charged hard spheres suspended in a dielectric continuum. For very dilute solutions, the predictions of the linearized Poisson‐Boltzmann equation are verified; at higher concentrations, the average space charge in the neighborhood of the electrode tends to alternate in sign as a result of local latticelike ion arrangement imposed effectively by short range ion repulsions. Predicted values of the ζ potential relative to those of the linear Poisson‐Boltzmann theory are reported.

Diffusion of Weak Electrolytes
View Description Hide DescriptionAn expression is obtained by standard irreversible thermodynamics for the diffusion coefficient of a weak electrolyte in rapid equilibrium with an arbitrary number of species. The general result reduces to a simple form if all mobilities are independent of concentration.

New Type of Freezing‐Point Apparatus. Freezing Points of Dilute Lanthanum Chloride Solutions
View Description Hide DescriptionA new type of freezing‐point apparatus is described that reduces the errors of previous types. The freezing‐point depressions of aqueous lanthanum chloride solutions up to 0.04 m are adequately expressed by the Debye‐Hückel approximation for ions with a=6.15 A plus a very small linear term. The ``higher terms'' are computed by combining the equations of Mayer and Kirkwood, using the tables of Poirier, and are found to be very small in this case.
The parameters for the calculation of the activity of water from freezing point depressions, for the Debye‐Hückel theory, and for the Kirkwood parameter B_{ik} are redetermined, and the more general use of B_{ik} is discussed.

Contribution of Lattice Vibrations to the Order‐Disorder Transformation in Alloys
View Description Hide DescriptionThe lattice vibrational partition function for a binary crystal is constructed from the normal mode frequencies in the classical nearest‐neighbor harmonic approximation. The set of frequencies is computed as a function of the short‐range order parameter by the use of the Born‐von Karman analysis and second‐order perturbation theory. The resulting form of the lattice vibrational partition function closely resembles that of the static configurational partition function so that a reexamination of the statistical thermodynamics, including lattice vibrational effects, proves to be straightforward. The derived thermodynamic quantities are compared with experiment for the case of β‐brass. The theoreticalheat capacity discontinuity as calculated by the method of Bethe and Kirkwood is increased from 1.7R to 6.1R by the inclusion of the lattice vibrational contribution. The agreement with the experimental value of about 5R is now satisfactory.

Dielectric Constants of Imperfect Gases. I. Helium, Argon, Nitrogen, and Methane
View Description Hide DescriptionMeasurements from 1 to 100 atm have been made at one or more temperatures between —30° and 70°C by a special transformer ratio arm bridge operating at 10 kc; derived values of the Clausius‐Mossotti function are of sufficient precision (0.1%) to justify comparison with theoretical pair fluctuation, polarizability and multipole moment interaction effects at higher densities. Helium shows a decrease of 0.3% and argon an increase of 0.1% in C‐M at 100 atm, while nitrogen and methane show somewhat larger positive effects, especially at lower temperatures. A decreased effective polarizability of atom pairs at small distances is discussed as a possible explanation of the data for helium and argon, while the effect in nitrogen is consistent with quadrupole‐induced dipole moments of pairs. A theory of octupole‐induced dipole interactions is shown to be capable of explaining the methane results with plausible estimated values of the octupole moment.

Statistical Mechanics of Transport Processes. XIV. Linear Relations in Multicomponent Systems
View Description Hide DescriptionA classical treatment of the time dependence of a phase‐space distribution function for a system near equilibrium is presented. The nonequilibrium distribution function is expressed as a stationary zero‐order function plus a perturbation term and is used to obtain the diffusion and heat fluxes by averaging the appropriate dynamical variables. When only terms linear in the gradients of the local temperature, the chemical potentials, and the velocity of the local centers of mass are retained, the usual linear relations result and explicit expressions for the phenomenological coefficients are obtained. These expressions agree with the results of Mori and of Green and are shown to obey the Onsager reciprocal relations.

Quantum Corrections for Transport Coefficients
View Description Hide DescriptionThe equation of motion of a quantum‐mechanical two‐particle system is solved without approximation in an expansion in Planck's constant. First‐order quantum corrections comprising all contributions to the coefficient of ℏ ^{2} in the expansion are derived for the differential scattering cross section and the kinetic‐theory transport coefficients. Numerical results are obtained for a simple model of molecular interactions, and lack of agreement is found with work based on the WKB solution of the Schrödinger equation.

Optical and Other Electronic Properties of Polymers
View Description Hide DescriptionThe properties of a polymer are related to the properties of its constituent groups and to its geometry. That is, relations between polymerwave functions and group wave functions are obtained. The equations generalize a result of Kirkwood to include the effect of time‐average fields. The main assumption in the derivation is that no exchange of electrons between groups occurs. The usefulness of these wave functions is illustrated by expressions for the absorption coefficient,polarizability,optical rotation,dipole moment, and stability of a polymer.

Ensemble Method in the Theory of Irreversibility
View Description Hide DescriptionWe describe a new formulation of methods introduced in the theory of irreversibility by Van Hove and Prigogine, with the purpose of making their ideas easier to understand and to apply. The main tool in this reformulation is the use of projection operators in the Hilbert space of Gibbsian ensemble densities. Projection operators are used to separate an ensemble density into a ``relevant'' part, needed for the calculation of mean values of specified observables, and the remaining ``irrelevant'' part. The relevant part is shown to satisfy a kinetic equation which is a generalization of Van Hove's ``master equation to general order.'' Diagram summation methods are not used. The formalism is illustrated by a new derivation of the Prigogine‐Brout master equation for a classical weakly interacting system.

On Molecular Chaos and the Kirkwood Superposition Hypothesis
View Description Hide DescriptionWe consider properties which are possessed by typical functions of very many variables such as the n‐particle molecular distribution function of a dynamical system. We show that such a function ``selected at random'' will in all likelihood approximately satisfy the molecular chaos condition and, to a better approximation, will very likely satisfy the Kirkwood superposition relation.

Theory of ``Melting'' of the Helical Form in Double Chains of the DNA Type
View Description Hide DescriptionPrevious theoretical treatments of the transition between the helical and random forms of the desoxyribose nucleic acid(DNA) molecule are extended to include formally the explicit consideration of the dissociation into two separate chains and the consideration of the effects of the ends of the chains. An approximate form for the fraction of base pairs that are bonded is obtained in terms of two parameters, a stability constant for base pairing and a constant representing the interaction of adjacent base pairs. The matrix method of statistical mechanics proves to be adaptable to this problem. Some numerical examples are worked out for very long molecules, for which case it is found that the effect of concentration is small.

Density Correlations, Critical Opalescence, and the Free Energy of Nonuniform Fluids
View Description Hide DescriptionWe show the equivalence of three prior approaches to the theory of critical opalescence: (a) the direct study of the long range behavior of the radial distribution function, due to Ornstein and Zernike, (b), the indirect approach by Rocard, who obtained the local pressure in a system of nonuniform density, and (c), the indirect approach by Debye, who obtained the free‐energy density in a system of nonuniform density. The molecular aspects of the theory are relieved to some extent from the confines of regular solution theory. The consequences of the Kirkwood superposition approximation, and its necessary revision, are examined.

Ultrasonic Attenuation in the Critical Region
View Description Hide DescriptionThe interaction of forced sound waves with critical density fluctuations is discussed quantitatively. The nonlinear equations of motion are analyzed to distinguish three contributions to the instantaneous density and temperature variation: (1) the spontaneous fluctuations that are defined as the solution to an initial value problem, (2) the macroscopically observed sound wave which oscillates with harmonic time dependence, and (3) the result of an interaction between (1) and (2). The three contributions satisfy three equations, which are discussed separately. A completely general combination of these equations to compute ultrasonicabsorption and attenuation is avoided, as the requisite thermodynamic and transport coefficients are inadequately known. Because, however, a thermal relaxation mechanism has previously been treated on the basis of an oversimplified multiphase picture of spontaneous fluctuations, the present calculations are carried to completion on the assumption that only the local heat capacity is affected by the density fluctuations. The sound wave together with the fluctuating heat capacity produce temperature fluctuations, whose relaxation is the source of the absorption.

Dielectric Relaxation of Polar Liquids
View Description Hide DescriptionThe statistical theory of the dielectric relaxation of polar liquids is developed using the fluctuation‐dissipation approach to linear dissipative phenomena, and an expression is derived relating the complex dielectric constant to a time‐dependent microscopic correlation function. It is found that a finite number of microscopic relaxation times leads to an equal number of macroscopic decay times, and, in the case of a single relaxation time τ0, the decay time is given by

Some Further Remarks on the Coefficient of Self‐Diffusion in Simple Dense Fluids
View Description Hide DescriptionThe self‐diffusion coefficient in a simple dense fluid is calculated from the autocorrelation function of the momenta. If the randomization of momentum is rapid, the resultant diffusion coefficient differs from that first suggested by Kirkwood, Buff and Green by only (2/π)^{½}. The physical basis of this result is considered. A very simple method of demonstrating the necessity for molecular correlations in the approach to equilibrium is discussed.

Scaled Particle Theory of Fluids
View Description Hide DescriptionA statistical thermodynamictheory has been developed employing distance scaling as a coupling procedure. This is an extension to real fluids of the technique applied by Reiss, Frisch, and Lebowitz to rigid‐sphere systems. One considers molecules interacting with pair potential u(r), except for one particle which interacts with potential u(r/λ). This single particle, essentially a scaled version of a normal molecule, is termed a λ‐cule. It is convenient to restrict discussion to potentials with rigid cores at r=a and cutoffs at γa. Attention is focused on a function, θ(λ, ρ, T), which reduces to G of footnote reference 1 in the case of rigid spheres. The pressure, chemical potential, and work of expanding a λ‐cule are simply related to θ. One can write θ exactly for λ<1/2γ and simple connection conditions hold at λ=1/2γ. An integral condition and λ=∞ condition on θ also exist. While θ is not completely specified, the foregoing conditions determine much of its behavior.

Kinetic Theory of Dense Gases
View Description Hide DescriptionThe Boltzmann equation is found to be determined to arbitrary order in the density as a consequence of the molecular chaos assumption. A three‐body collision approximation is given explicitly and shown to be identical to that implied by the Bogolyubov assumptions; the relation between the molecular chaos assumption and the Bogolyubov assumptions is discussed. It is shown that the Enskog modification of the Boltzmann equation for dense gases composed of rigid spheres is correct, subject to the molecular chaos assumption. The current disagreement over the correct form for this equation is discussed.

On the Theory of the Critical Point of a Simple Fluid
View Description Hide DescriptionThe consequences of a new system of integral equations for the theory of the critical point are discussed. Reasons are given for believing that the fundamental assumption of the Ornstein‐Zernicke theory about the direct correlation function is incorrect.

Effect of Binding of Ions and Other Small Molecules on Protein Structure. VIII. On Isomerization of Serum Albumin in Acidic Media
View Description Hide DescriptionThe complex electrophoretic patterns of BSA in 0.02M NaCl‐HCl are due to reaction boundaries arising from a set of rapidly established isomerization equilibria and a slow, apparently irreversible reaction. The slow reaction is of only minor importance for electrophoresis in acetate buffer, although a second irreversible process occurs in this solvent at protein concentrations of 0.2% and lower. This latter reaction accounts for the bimodality of the patterns obtained at low field strength with 0.2% BSA in acetate buffer. Resolution of those peaks corresponding to the rapidly established isomerization equilibria is a consequence of changes in either pH or conductance in the reaction boundary during nonideal electrophoresis.Equilibrium constants can, nevertheless, be computed from extrapolated values of the various electrophoretic mobilities without recourse to area measurements. Our observations suggest the following set of consecutive isomerization equilibria:where, as in the reaction scheme of Aoki and Foster, the F form of the protein has a larger electrophoretic mobility than the N form, whereas that of the I form is intermediate between those of the N and F forms.