Robert W. Zwanzig

High‐Temperature Equation of State by a Perturbation Method. I. Nonpolar Gases

The perturbation theory described in a previous article is applied to the calculation of the equation of state at high temperature of a polar gas. The electrostatic dipolar potential is considered to be a perturbation on the nonpolar Lennard‐Jones potential. The dipole contribution to the equation of state is expressed in terms of properties of the unperturbed system. These properties are obtained from experimental data on argon and xenon. Numerical tables of the reduced equation of state of the polar gas, over a wide temperature and density range, are presented. The theory is tested on water vapor and on methyl fluoride. Theoretical predictions of the pressure agree with experimental data over a wide density range (up to 0.25 g/cc for water) to within several percent.

Statistical Mechanical Theory of Transport Processes. VII. The Coefficient of Thermal Conductivity of Monatomic Liquids

A molecular theory of the coefficient of thermal conductivity is developed from the general theory of transport processes presented in the first article of this series. The thermal conductivity of liquid argon at its normal boiling point is evaluated using the Lennard‐Jones intermolecular potential and a theoretically determined radial distribution function. The theory leads to an explicit expression for the product of the thermal conductivity and the friction constant of the theory of Brownian motion. With a reasonable estimate of the friction constant, the results of the theory agree satisfactorily with experiment.

Molecular Distribution Functions in a One‐Dimensional Fluid

Exact expressions are derived for the molecular distribution functions in a one‐dimensional fluid whose particles interact with a nearest neighbor pair potential. The pair distribution function for rigid spheres is found to be identical with Frenkels result. The one‐dimensional form of a new set of integral equations for the molecular distribution functions is examined. The superposition principle is found to be exact in a one‐dimensional fluid with nearest neighbor interactions.

Quantum‐Mechanical Calculation of Harmonic Oscillator Transition Probabilities. II. Three‐Dimensional Impulsive Collisions

The previous one‐dimensional calculations of Shuler and Zwanzig for the vibrational transition probabilities for harmonic oscillators undergoing impulsive hard‐sphere collisions with an incident atom have been extended to three dimensions. By the use of the modified wave number approximation, the three‐dimensional scattering problem is related to the one‐dimensional solution and the three‐dimensional transition probabilities are obtained through numerical integration over the previously calculated one‐dimensional probabilities. It is shown that vibrational transitions with | Δv | >1 are strongly allowed (in addition to the | Δv | = 1 transitions) in good agreement with the one dimensional impulsive collision results. The implication of this result on the mode and rate of the dissociation of diatomic molecules is discussed briefly.

The Statistical Mechanical Theory of Transport Processes. VI. A Calculation of the Coefficients of Shear and Bulk Viscosity of Liquids

The theory of the coefficients of shear and bulk viscosity of liquids developed in the third article of this series is applied to the calculation of the coefficients of viscosity of liquid argon at its normal boiling point. The theory of the bulk viscosity, including a previously omitted term due to the rate of dilatation, is presented. With the use of the Lennard‐Jones potential, a radial distribution function which is a much better approximation than the previously used one, and a new approximation to the friction constant, values are obtained for the coefficients of viscosity.

Statistical Mechanical Theory of Transport Processes. XII. Dense Rigid Sphere Fluids

The theory of transport in a dense fluid of rigid spheres is developed from classical statistical mechanics by the use of phase space transformation functions. A modified Maxwell‐Boltzmann integro‐differential equation for the distribution function in μ space is derived, and the difference between this equation and the Enskog equation is discussed. To obtain a formulation of the stress tensor and heat flux solely in terms of binary collisions, it is necessary to allow the time of coarse graining to be very short. The implications of this are discussed with relation to the general principles of the statistical mechanics of transport. The viscosity and thermal conductivity of the dense rigid sphere fluid are calculated. The viscosity is the same as that first computed by Enskog, but the thermal conductivity differs from his calculations.

Errata: Radial Distribution Functions and the Equation of State of Monatomic Fluids

An approximation is obtained for the radial distribution function in a fluid whose molecules interact with the Lennard‐Jones potential. Calculations of the equation of state, the internal energy, and the compressibility are presented and compared with experiment. At temperatures and volumes higher than the critical, agreement with experiment is satisfactory. A systematic method is developed of modifying the theoretical radial distribution function so that calculated thermodynamic quantities agree with their experimental values. In the liquid phase, slight modifications are needed to obtain agreement with experiment.

Kinetics of a Sequence of First‐Order Reactions

Solutions are obtained for the finite set of coupled rate equations ∂Ci / ∂t = αi,i−1Ci−1 + αi,iCi + αi,i+1Ci+1 (i = 0, ···, N), where αi,j are given in general as αi,i−1 = A, αi,i+1 = B, αi,i = − (A + B), except that α0,0 = − α1,0 = − a, αNN = − αN−1,N = − b, α0,−1 = αN,N+1 = 0. Asymptotic expressions are given for the approach to equilibrium as a function of the various rate parameters and the chain length N. For large N, we find that if A   B, the system exhibits a peculiar eigenvalue spectrum, and the relaxation is characterized by two distinct and well‐separated relaxation times, λ1 and λ2 = − a{1 − B[A(1 − a / A}2]−1).

WITHDRAWN: Molecular Distribution Functions in a One-Dimensional Fluid

Exact expressions are derived for the molecular distribution functions in a one-dimensional fluid whose particles interact with a nearest neighbor pair potential. The pair distribution function for rigid spheres is found to be identical with Frenkels result. The one-dimensional form of a new set of integral equations for the molecular distribution functions is examined. The superposition principle is found to be exact in a one-dimensional fluid with nearest neighbor interactions.