Chemical Non-equilibrium Conditions Research Articles

Two-temperature ablative material response model with application to Stardust and MSL atmospheric entries

Ablative material response codes currently in use consider local thermal equilibrium between the solid phases and the pyrolysis gases. For typical entry conditions, this hypothesis may be justified by the fact that the thermal Peclet number within the pores is small, which is a necessary condition for thermal equilibrium in non-reactive materials. However, the validity of this analysis may fall under some circumstances. The thermal Peclet number may become large due to high pyrolysis gas velocities. Additional physical phenomena not accounted for in the Peclet analysis may become non-negligible, such as the change of enthalpy due to chemical reactions. The objective of this study is two-fold. First, a detailed two temperature material response model for porous reactive materials is presented. This model has been implemented and made available in the Porous material Analysis Toolbox based on OpenFOAM (PATO). Second, the model is applied to the Theoretical Ablative Composite for Open Testing (TACOT) in a wide range of conditions to assess the true range of validity of the thermal equilibrium hypothesis. Simulations are carried out on the Stardust and Mars Science Laboratory (MSL) atmospheric entries. The main design variables have been monitored and compared between the two models: temperature evolution and species concentration within the material, pyrolysis gas blowing rate, extension of the pyrolysis zone, and wall recession due to ablation. Results show that under chemical equilibrium conditions, no significant deviation in the monitored quantities are observed, while under chemical non-equilibrium conditions there is a large impact on the species concentration.

Prediction of soil solution concentrations and leaching losses of cadmium in agricultural soils.

Cadmium (Cd) leaching is often estimated in Cd balance models using the product of drainage water (precipitation excess) and soil solution Cd concentration. However, Cd concentrations are seldom available but rather predicted using empirical models. Despite the availability of empirical models that estimate soil solution Cd concentrations, they have limitations for use in New Zealand where total Cd in agricultural soils is low and organic matter (OM) high. This study derived a Freundlich-type model from desorption data and a soil-liquid partitioning coefficient (KD ) model based on sorption data to predict soil solution Cd concentrations from commonly measured soil parameters that could be used to calculate Cd leaching fluxes. Independent soil solution Cd concentrations and Cd fluxes measured in drainage water from field trials were used to validate the predictive capacity of the models. It was found that soil pH and OM content were the most important factors controlling soil solution Cd, along with total Cd. Both models explained 83% of the variation in measured soil solution Cd concentrations in an independent dataset. Comparisons between Cd fluxes predicted using the Freundlich-type model and measured fluxes were within 25% of each other at 6 of 19 field sites studied. However, physical and chemical nonequilibrium conditions in soils and uncertainty in measured values likely contributed to differences between predicted and measured Cd fluxes at other sites. To unravel the impact of nonequilibrium and soil physical conditions on Cd concentrations in drainage water, more data are required on Cd concentrations collected under field conditions. This will allow better validation of the approach used in Cd balance models to calculate Cd leaching from soils.

Formation and growth behavior of intermetallic compound phases in the interfacial reaction of solid Fe / liquid Zn at 450 °C

In the present study, the formation and growth behaviors of intermetallic compound (IMC) layers in the interfacial reaction of solid Fe and molten Zn during hot dipping at 450 °C were investigated. In the early stage of the interfacial reaction, Γ-Fe4Zn9, δ1k-FeZn7, and ζ-FeZn13 phases were formed, among which the growth of the ζ layer was dominant. Columns of the δ1p-Fe13Zn126 phase were formed on the δ1k/ζ interface at approximately 90 s, and the four IMC layers grew simultaneously until 600 s. Subsequently, a layer of the Γ1-Fe21.2Zn80.8 phase was formed along the Γ/δ1k interface and started growing, after which all the equilibrium IMC phases of Γ/Γ1/δ1k/δ1p/ζ grew competitively. The total thickness of the five IMC layers, dtotal, increased proportionally with the square root of the reaction time, t, i.e., dtotal = d0t0.5, which suggested that the growth of the entire IMC layers was controlled by the volume diffusion. However, the values of the time exponent, n, for the individual IMC layers differed depending on their microstructures and chemical concentrations, under equilibrium or nonequilibrium conditions. The formation of the multilayered columnar ζ phase can be attributed to the heterogeneous supersaturation of Fe in the liquid Zn near the columnar ζ / liquid Zn interface during the early stage of the interfacial reaction. The interdiffusion fluxes, J̃ϕ, of Fe and Zn in the ϕ phase (ϕ = Γ, δ1k, and ζ) were estimated from the measured concentration profiles of each phase. This suggested that J̃ϕ caused the growth of each phase as well as the supersaturation of Fe near the interfacial boundaries, thereby resulting in considerable deviations from the equilibrium concentrations. In the later stage of the interfacial reaction, the interfacial concentrations gradually approached the equilibrium concentrations with decreasing J̃ϕ.

Soil properties of sugarcane fields controlling triazine leaching potential

Atrazine and ametryn are pre- and post-emergence herbicides widely used in sugarcane fields. Both are characterised by moderate to high mobility in soil and are regarded as a threat to hydrologic systems. Physical and chemical soil properties are highlighted as one of the factors governing the fate of pesticides in hydrologic systems. The aim of our study was to estimate the leaching potential of atrazine and ametryn in soils of the saline alluvial plain of Tucumán cultivated with sugarcane. Miscible displacement experiments were carried out in triplicate using undisturbed topsoil cores from four farm fields. Field 1 was in plant, fields 2 and 3 were in second ratoon, and field 4 was in third ratoon. Experimental breakthrough curves were fitted to the convection–dispersion equation; results showed that hydraulic transport was in equilibrium conditions, meaning that all soil water was involved in solute convective transport. Atrazine had asymmetric breakthrough curves, revealing chemical non-equilibrium conditions, with at least two sorption–desorption sites participating in the process. In contrast, ametryn concentrations in leached samples were low and inconsistent; therefore, no breakthrough curves could be described. Of the total applied, recoveries reached 10% and 2% for leached atrazine and ametryn respectively. For an estimated annual recharge of 100 mm, these leached amounts would exceed international reference values for drinking water.

Balancing Non-Equilibrium Driving with Nucleotide Selectivity at Kinetic Checkpoints in Polymerase Fidelity Control.

High fidelity gene transcription and replication require kinetic discrimination of nucleotide substrate species by RNA and DNA polymerases under chemical non-equilibrium conditions. It is known that sufficiently large free energy driving force is needed for each polymerization or elongation cycle to maintain far-from-equilibrium to achieve low error rates. Considering that each cycle consists of multiple kinetic steps with different transition rates, one expects that the kinetic modulations by polymerases are not evenly conducted at each step. We show that accelerations at different kinetic steps impact quite differently to the overall elongation characteristics. In particular, for forward transitions that discriminate cognate and non-cognate nucleotide species to serve as kinetic selection checkpoints, the transition cannot be accelerated too quickly nor retained too slowly to obtain low error rates, as balancing is needed between the nucleotide selectivity and the non-equilibrium driving. Such a balance is not the same as the speed-accuracy tradeoff in which high accuracy is always obtained at sacrifice of speed. For illustration purposes, we used three-state and five-state models of nucleotide addition in the polymerase elongation and show how the non-equilibrium steady state characteristics change upon variations on stepwise forward or backward kinetics. Notably, by using the multi-step elongation schemes and parameters from T7 RNA polymerase transcription elongation, we demonstrate that individual transitions serving as selection checkpoints need to proceed at moderate rates in order to sustain the necessary non-equilibrium drives as well as to allow nucleotide selections for an optimal error control. We also illustrate why rate-limiting conformational transitions of the enzyme likely play a significant role in the error reduction.

Numerical analysis and characterization of the dissociated flowfield inside a 1.6 MW arc-heated wind tunnel facility

The present work focuses on a Computational Fluid Dynamics (CFD) investigation of the high-enthalpy flow produced by the 1.6 MW arc-heated wind tunnel facility of the University of Texas at Arlington. The numerical analysis is dedicated to the aerothermodynamic characterization of the internal flowfield of a nominal M=1.8 conical nozzle. Due to dramatic changes in the thermomechanical response of TPS materials when specific conditions of temperature, species composition, and pressure are reached, the knowledge of the flow conditions at the nozzleʼs exit is an essential element for the identification of appropriate test conditions during the design of the experiment first, and the interpretation of the results later. Thermal and chemical nonequilibrium conditions must be considered in the high-temperature nozzle flow when the characteristic times for the chemical reactions and vibrational energy are comparable to the characteristic flow time. The Navier–Stokes equations are solved with the implicit finite volume density-based solver of the CFD code Fluent. From the numerical characterization of the nozzle flow, new facility performance envelopes have been generated providing analytical correlations between the relevant aerothermodynamic parameters at the nozzle exit plane and the input facility operation conditions. The impact of the choice of thermochemical models on the calculated aerothermodynamic quantities is also investigated by a comparative study of the results obtained when both the Park and Gupta models are implemented to simulate the nonequilibrium flow.

Aeration / Stripping of n-Butyl Mercaptan in aqueous environment

Μany applications in water quality management have a common key water quality parameter, dissolved oxygen, resulting to the critical role of aeration. On the other hand, in municipal and industrial wastewater, especially where aeration is applied, the presence of volatile organic compounds (VOCs) causes several concerns including a direct threat to humans, partly due to their emission from treatment tanks. pH, temperature and Henry’s Law govern VOCs’ speciation and consequently their emission characteristics. Limited data and simplifications of available mass-transfer models pose obstacles to a realistic approach, especially in the presence of a chemical equilibrium, for example in the case of mercaptans. In the present study the importance of oxygen transfer and stripping of a VOC (n-butyl mercaptan) on aeration’s overall effectiveness are examined separately. Clean water oxygenation and stripping of mercaptan to an inert gas (nitrogen) were studied aiming to consider mass transfer aspects and to investigate the influence of chemical equilibrium between ionic and neutral form of the target compound in neutral and alkaline solutions. Using appropriate mass transfer relationships (dynamic method), experimental data were analyzed for the determination of overall mass transfer coefficient ( kOL,O2α ) of oxygen. Correlating kOL,O2 α with the corresponding mass transfer coefficient of n-butyl mercaptan in neutral solutions (calculated according the model proposed by Matter-Muller et al. [1]), a value of ratio βy of 0.566 is found, close to the reported values of other VOCs with similar values of Henry’s constant. At alkaline pH however the conventional simplified model fails to predict realistic values of mass-transfer coefficients. A coupled differential algebraic equation system, based on mass balances, taking into account dissociation of the compound to be stripped and assuming chemical non-equilibrium conditions during stripping, was developed. Reaction parameter k2 was calculated with non-linear least-squares analysis. The model predicts satisfactorily the experimental data and it provides a useful tool for the semibatch stripper design in situations where a reversible reaction is involved. At pH values below 8.5 mercaptan concentration falls exponentially whereas above 10.5 it tends to linearity. The bubble equilibrates and mercaptan transferred depends upon solubility and not diffusivity. Especially after depletion of initial neutral compound, transport depends upon neutral/ionic form speciation. The effectiveness of stripping n-butyl mercaptan, at a given pH, is mainly determined by a proportionality constant considered as “fugacity capacity” (removal effect on the process) and by a reversible reaction rate constant k2 (kinetic effect on the process). The ‘’fugacity capacity” is determined by hydrophobicity (i.e. low solubility and high limiting activity coefficient) rather than pure-component volatility (i.e. vapor pressure or boiling point). High limiting activity coefficient promotes mercaptan emission due to established vapor-liquid equilibrium, while the low reaction parameter k2, controls neutral compound quantity. At high pH, where ionic form predominates, experimental data showed that stripping was almost independent of the gas flow rate applied. A strong sensitivity of the model to uncertainty of γ∞ was found: γ∞ controls emission rate and through this the dynamic variations of neutral/ionic concentration profiles whereas reaction rate law parameter k2 controls the neutral/ionic transformation and it is the crucial quantity which governs the process at high pH values.

Exploring Chemical and Thermal Non-equilibrium in Nitrogen Arcs

Plasma torches operating with nitrogen are of special importance as they can operate with usual tungsten based refractory electrodes and offer radical rich non-oxidizing high temperature environment for plasma chemistry. Strong gradients in temperature as well as species densities and huge convective fluxes lead to varying degrees of chemical non-equilibrium in associated regions. An axi-symmetric two-temperature chemical non-equilibrium model of a nitrogen plasma torch has been developed to understand the effects of thermal and chemical non-equilibrium in arcs. A 2-D finite volume CFD code in association with a non-equilibrium property routine enabled extraction of steady state self-consistent distributions of various plasma quantities inside the torch under various thermal and chemical non-equilibrium conditions. Chemical non-equilibrium has been incorporated through computation of diffusive and convective fluxes in each finite volume cell in every iteration and associating corresponding thermodynamic and transport properties through the scheme of ‘chemical non-equilibrium parameter’ introduced by Ghorui et. al. Recombination coefficient data from Nahar et. al. and radiation data from Krey and Morris have been used in the simulation. Results are presented for distributions of temperature, pressure, velocity, current density, electric potential, species densities and chemical non-equilibrium effects. Obtained results are compared with similar results under LTE.

Measurement of the vibrational temperature of oxygen behind a shock wave front under thermal and chemical nonequilibrium conditions

In shock tube experiments the profiles of light absorption in oxygen are obtained for the wavelength interval 200–260 nm over the temperature range 4000–10800 K. Using these data, the vibrational temperature profiles are measured for oxygen molecules behind the shock front. The method of determination of the vibrational temperature of oxygen is based on comparing absorption measurements and detailed absorption spectrum calculations for oxygen in the Schumann-Runge system.

Application of continuous time random walk theory to nonequilibrium transport in soil

Continuous time random walk (CTRW) formulations have been demonstrated to provide a general and effective approach that quantifies the behavior of solute transport in heterogeneous media in field, laboratory, and numerical experiments. In this paper we first apply the CTRW approach to describe the sorbing solute transport in soils under chemical (or) and physical nonequilibrium conditions by curve-fitting. Results show that the theoretical solutions are in a good agreement with the experimental measurements. In case that CTRW parameters cannot be determined directly or easily, an alternative method is then proposed for estimating such parameters independently of the breakthrough curve data to be simulated. We conduct numerical experiments with artificial data sets generated by the HYDRUS-1D model for a wide range of pore water velocities ( υ) and retardation factors ( R) to investigate the relationship between CTRW parameters for a sorbing solute and these two quantities ( υ, R) that can be directly measured in independent experiments. A series of best-fitting regression equations are then developed from the artificial data sets, which can be easily used as an estimation or prediction model to assess the transport of sorbing solutes under steady flow conditions through soil. Several literature data sets of pesticides are used to validate these relationships. The results show reasonable performance in most cases, thus indicating that our method could provide an alternative way to effectively predict sorbing solute transport in soils. While the regression relationships presented are obtained under certain flow and sorption conditions, the methodology of our study is general and may be extended to predict solute transport in soils under different flow and sorption conditions.

Modelling of an argon plasma jet generated by a dc arc

The modelling of a dc plasma source operating at low power and at low mass flow rate is presented. The physical description of the electric arc in volume with chemical and thermal non-equilibrium conditions is given: two-temperature model, ionization–neutralization process and arc–gas interactions are discussed. The modelling assumptions are justified by the local non-dimensional numbers characterizing the system (Damkhöler, Reynolds and Knudsen numbers). Special attention is focused on the influence of the arc power on the thermal non-equilibrium between electrons and heavy particles and the resulting ionization rate. The kinetic thermal non-equilibrium is shown as a decreasing function of the arc discharge current and it is established that the electron density never reaches equilibrium conditions, even at the throat exit. Calculations have been performed for a stationary arc confined to the throat. The gas is argon and the flow is axisymmetric and stationary.

Strangeness enhancement at LHC

We study production of strangeness in the hot QGP fireball under conditions achieved at LHC, and use these results to obtain soft (strange) hadron multiplicities. We compare the chemical equilibrium and non-equilibrium conditions and identify characteristic experimental observables.

Radiative transfer in gases under thermal and chemical nonequilibrium conditions: Application to earth atmospheric re-entry

It is well known that the Kirchhoff's law, connecting emission to absorption through the Planck's function, is no more valid when the medium is not at local thermal and chemical equilibrium. The aim of the present paper is to analyze the effects of this nonequilibrium on radiative transfer, and in particular, for atmospheric air re-entry applications. A general formulation is proposed in the framework of multi-temperature description of the thermodynamic state of the gas mixture. Application of this model to simulate an earth re-entry experiment first leads to a reasonable agreement and then outlines important effects of chemical nonequilibrium, especially through bound–free transitions.

Numerical Modeling of an Ar–H2 Radio-Frequency Plasma Reactor under Thermal and Chemical Nonequilibrium Conditions

The species densities and the thermal and chemical nonequilibrium phenomena in an Ar–H2 radio frequency inductively coupled plasma reactor used for hydrogenation of materials have been investigated through numerical simulation. The mathematical model consists of a two-temperature fluid dynamics model and a chemical kinetics model that takes into account the effect of local chemical nonequilibrium. Computations are carried out for the rf plasma running at 11.7 kW and 27 kPa for different Ar–H2 mixtures and for pure argon. Predicted results for the electron and heavy-species temperatures, the species densities, as well as the degree of thermal and chemical nonequilibrium, are presented in detail. It is found that the electron and hydrogen atom densities in the reactor and in the near-wall region of the torch are strongly altered by nonequilibrium effects. The hydrogen atom density remains high in the reactor zone, and peaks in a region that has been found to be attractive for material processing. Deviations from thermal and chemical equilibrium are greatly reduced by the addition of hydrogen to an argon plasma.

Particle yield fluctuations and chemical nonequilibrium in Au-Au collisions atsNN=200GeV

We study charge fluctuations within the statistical hadronization model. Considering both the particle yield ratios and the charge fluctuations we show that it is possible to differentiate between chemical equilibrium and non-equilibrium freeze-out conditions. As an example of the procedure we show quantitatively how the relative yield ratio $\ensuremath{\Lambda}/{K}^{\ensuremath{-}}$ together with the normalized net charge fluctuation $v(Q)=\ensuremath{\langle}(\ensuremath{\Delta}Q){}^{2}\ensuremath{\rangle}/\ensuremath{\langle}{N}_{\mathrm{ch}}\ensuremath{\rangle}$ constrain the chemical conditions at freeze-out. We also discuss the influence of the limited detector acceptance on fluctuation measurements, and show how this can be accounted for within a quantitative analysis.

Degradation and sorption of atrazine, hexazinone and procymidone in coastal sand aquifer media

The degradation, sorption and transport of atrazine, hexazinone and procymidone in saturated coastal sand aquifer media were investigated in batch and column experiments. The pesticides were incubated with sterilised and non-sterilised groundwater or a mixture of groundwater and the aquifer material in the dark at 15 degrees C for 120 days. The estimated half-lives of the pesticides (and their ranges) in the mixture of groundwater and aquifer sand were 36 (31-40), 54 (40-77) and 84 (46-260) days for atrazine, procymidone and hexazinone, respectively. Compared with the relevant results for the groundwater-sand mixture phase, the estimated half-life of pesticides in the groundwater phase alone was shorter for procymidone (21 days) but longer for hexazinone (134 days); atrazine was not degraded in the groundwater phase. Chemical degradation appeared to have played the predominant role in the degradation of hexazinone and procymidone in the aquifer system, while both chemical and biological processes seemed to be important for the degradation of atrazine. Batch isothermal experiments were carried out at pH 4.6-4.7 to obtain sorption coefficients under equilibrium conditions. The isothermal data of the pesticides fitted well with the non-linear Freundlich function with an exponent of sorption coefficient that was greater than one. Contrary to reports in the literature, sorption of atrazine was the greatest, and procymidone was slightly more sorbed than hexazinone. A column experiment was conducted at a typical field-flow velocity of 0.5 m day(-1) over 60 days to study pesticide attenuation and transport in flow dynamic conditions. Retardation factors, R, derived from a two-site sorption/desorption model were 8.22, 1.76 and 1.63 for atrazine, procymidone and hexazinone, respectively. Atrazine displayed the lowest mobility and the mobility of procymidone was only slightly less than that of hexazinone, which is consistent with observations in the batch experiment. A possible explanation for these observations is that ionic atrazine is bound to oppositely charged ionic oxides, and ionic oxides have less effect on the sorption of the non-ionic procymidone. The significant tailing in the pesticide breakthrough curves (BTCs) in comparison with the bromide BTC, together with model-simulated results, suggests that the transport of the pesticides was under chemical non-equilibrium conditions with R values that were less than their equivalent values predicted using the batch equilibrium isothermal data. As a result of non-linear kinetic sorption, retardation factors of the pesticides in groundwater systems would not be constant and will decrease with decreasing pesticide concentrations and increasing flow velocities. Hence, the use of equilibrium isotherm data will probably over-predict the sorption of pesticides in groundwater systems. Rhodamine WT, a commonly used groundwater tracer, was significantly retarded (R = 5.48) and its BTC was much more spread out than the bromide BTC. Therefore, it would not be a good tracer for the indication of groundwater flow velocity and dispersion for the coastal sand aquifer system. In contrast to some aquifer media, the dye tracer was unsuitable as a marker of the appearance of atrazine in a coastal sand aquifer system.

Transport properties in reacting mixture of polyatomic gases

The mathematical modeling of transport properties in reacting gases on the basis of kinetic equations for the distribution functions is given in this paper. Thermal and chemical nonequilibrium conditions are considered. The influence of rotational and vibrational excitation and chemical reactions on the pressure tensor, heat flux and diffusion velocity is investigated. The generalization of the Chapman-Enskog method is used at the different levels of the strong nonequilibrium mixture description. Three nonequilibrium models are presented: the level approach, the generalized multi-temperature approach and the one-temperature approach. The macroscopic equations for macroparameters are derived from the kinetic theory treatment, and the expressions for the pressure tensor, diffusion velocity and heat flux as well as for the transport coefficients are given. The role of the different rates of the various energy exchanges, the anharmonism of molecular vibrations and chemical reactions in the transport properties of reacting mixtures is discussed.

Wetting under chemical equilibrium and nonequilibrium conditions

The thermodynamics of a solid-liquid-vapor system both under chemical equilibrium and nonequilibrium conditions, based on the model of Gibbs, is discussed. Under chemical equilibrium conditions, the degree of wetting or nonwetting of a flat and nondeformable solid by the liquid is defined by Young's equation in terms of the static interfacial tensions. Under chemical nonequilibrium conditions, mass transfer across an interface results in a transient decrease in the corresponding specific interfacial free energy and the interfacial tension by an amount equal to the free energy of the effective chemical reaction per area at that interface. When the reaction is between the solid and the liquid, this transient lowering of the interfacial tension can cause the liquid drop to spread on the solid substrate if the interfacial energy reduction is large enough and also if the diffusion rates of the reacting components in the solid phase are slow enough relative to the flow rate of the liquid to cause the liquid at the periphery of the drop to be in dynamic contact with unreacted solid.

Thermoelastic Behavior of Natural Rubber

Abstract Stress—temperature measurements of natural rubber were carried out up to the elongation ratio, α, of 2.0. An automatic stress relaxometer was constructed for this purpose which can be completely enclosed in a controlled environment. Experiments were so conducted as to minimnze possible chemical effects and nonequilibrium conditions. Relative internal energy contribution to stress, fe/f, is calculated as a function of α in terms of statistical and thermodynamic theories. Both of these yield similar results. It is shown that in the region of low strains (1.0<α<1.5),fe/f decreases rapidly with increasing α, but appears to remain constant at 1.5<α<2.0. This observation is not in agreement with the prediction of the current statistical theory of rubber elasticity, which stipulates that the energy effects are intramolecular and independent of deformation. Implications of these findings are discussed. It is suggested that perhaps at low strains the intermolecular interactions are large in comparison with intramolecular energies, but become relatively insignificant at higher elongation ratios. The temperature coefficient of unperturbed chain dimensions is also calculated from thermoelastic data. It is constant only in the region 1.5<α<2.0. Finally, a new, more exact derivation of the Elliott—Lippmann anisotropy factor in terms of the statistical theory is given in the Appendix.

Thermoelastic Behavior of Natural Rubber

Stress-temperature measurements of natural rubber were carried out up to the elongation ratio, α, of 2,0. An automatic stress relaxometer was constructed for this purpose which can be completely enclosed in a controlled environment. Experiments were so conducted as to minimize possible chemical effects and nonequilibrium conditions. Relative internal energy contribution to stress, fe/f, is calculated as a function of α in terms of statistical and thermodynamic theories. Both of these yield similar results. It is shown that in the region of low strains (1.0<α<1.5), fe/f decreases rapidly with increasing α, but appears to remain constant at 1.5<α<2.0. This observation is not in agreement with the prediction of the current statistical theory of rubber elasticity, which stipulates that the energy effects are intramolecular and independent of deformation. Implications of these findings are discussed. It is suggested that perhaps at low strains the intermolecular interactions are large in comparison with intramolecular energies, but become relatively insignificant at higher elongation ratios. The temperature coefficient of unperturbed chain dimensions is also calculated from thermoelastic data. It is constant only in the region 1.5<α<2.0. Finally, a new, more exact derivation of the Elliott-Lippmann anisotropy factor in terms of the statistical theory is given in the Appendix.