Low Concentration Limit Research Articles

Generalized van der Waals theory. XI. Gas solubility in simple fluids

The generalized van der Waals theory of mixtures is applied in the low solute concentration limit to the calculation of the Henry's law constant KH, the partial molar enthalpy ΔH5, and the partial molar volume V2. Several versions of the theory are used. Between-species interactions are taken to obey Lorentz-Berthelot rules. Comparison with experiment and earlier theory includes Ar, C6H6 and CCl4 solvents and a number of solute species. At the lowest level of theory simple analytical expressions for KH, ΔH5 and V2 are obtained. The accuracy is semiquantitative with ΔH5 most in error. The accuracy is improved by allowing for the temperature and density dependence of the effective hard-sphere diameter and the density dependence of the excluded volume. The linear and bilinear mixing rules for the excluded volume are compared and the former found preferable. No experimental parameters (beyond accepted LJ(12-6) parameters) are used.

Exchange of labeled nuclei in the CO2--HCO3--solvent system catalyzed by carbonic anhydrase

Silverman et al. (1979. J. Am. Chem. Soc. 101:6734-6740) have reported measurements of the loss of 18O to solvent from the isotopically labeled CO2--HCO3-system and of the mixing of 18O and 13C labels within the system, as catalyzed by human carbonic anhydrase C in the pH range 6-8. This work is an extension of earlier work (Silverman and Tu. 1976. J. Am. Chem. Soc. 98:978-984) on the very similar bovine enzyme. The more recent work is analyzed by its authors in terms of the "hydroxide" model for the apparent pH-dependence of enzymatic activity, a model in which the pH-dependence is associated with the presumed ionization of an H2O ligand of the active-site metal ion to OH-. From a comparison of their data with a solution of the coupled differential equations that describe the kinetics of isotope exchange in terms of the model, Silverman et al. derived a pH-dependent rate of exchange for the water molecule which is formed at the active site of the enzyme during dehydration. By contrast, using the same data and a model in which active enzyme has a water molecule on the metal ion at the active site, and similar differential equations, we derive a value for the rate of exchange of water that is pH-independent. This model has the attraction that it explains the magnetic relaxation rate of solvent water protons in the Co2+-substituted enzyme, whereas the hydroxide mechanism cannot explain these data without the introduction of unfounded ad hoc assumptions; further, the presence of an OH- ligand of the metal has never been demonstrated. We also include an analysis of analogous data for the bovine enzyme. One result of our analysis is that the pKa for activity of the enzyme samples used is near 6.0, implying that the bulk of the data were taken when the enzyme was essentially all active. It is straightforward to account for the pH-dependence of the data near and below the pKa by using an empirically-derived value for the pKa. However, we have recently developed a model for the low pH (inactive) enzyme that has been successful in interpreting a wide range of data, and we show that this new view can explain the few points at low pH quite adequately. Additionally, we consider the recent kinetic results for the human C enzyme, obtained at chemical equilibrium by studies of the linewidths of nuclear magnetic resonances of 13C in labeled substrate (Simonsson et al. 1979. Eur. J. Biochem. 93:409-417) and show that these experiments and those of Silverman et al. are all consistent with kinetic data from nonequilibrium stopped-flow experiments, viewed in terms of our model, in the limit of low substrate concentration. Results at higher concentrations indicate that the Michaelis constants and equilibrium constants differ somewhat.

Spin fluctuations in mixed magnetic chains

The dynamic response of ferro- and antiferromagnetically coupled Heisenberg chains containing impurities is calculated in the low temperature and small concentration limit. Particular attention is given to impurity induced effects in the inelastic neutron cross-section, and explicit expressions are given for spin-wave linewidths.

Impurity transport calculations for the limiter shadow region of a tokamak

Impurity transport calculations are presented for the scrape-off layer of a tokamak with a poloidal ring limiter. The theory is based on the drift-kinetic equations for the impurity ions in their different ionization states. It is developed in the limit of low impurity concentrations under due consideration of electron impact ionization, Coulomb collisions with hydrogen ions streaming onto a neutralizing surface, a convection along the magnetic field, and a radial drift. The background plasma and the impurity sources at the walls enter the theory as input parameters. Numerical results are given for the radial profiles of density, temperature, particle flux, and energy flux of wall-released impurity ions as well as for the screening efficiency of the scrapeoff layer neglecting impurity re-emission from the limiter

Mixtures of hard ions and dipoles against a charged wall: The Ornstein–Zernike equation, some exact results, and the mean spherical approximation

The Ornstein–Zernike equation for a mixture of ions and dipoles near a hard charged wall is obtained. It is shown that the same exact contact and monotonicity theorems, previously derived for the primitive (continuum dielectric) case, also are valid for this model. Rather simple expressions for the contact density, potential difference, capacitance, and distribution functions are obtained in the mean spherical approximation (MSA). These expressions reduce to previously known results in the limits of low and high concentrations of ions. It is found that cooperative alignment of the dipoles near the wall results in an increased potential difference and reduced capacitance of the double layer compared to that calculated when the solvent is represented by a continuum dielectric.

Track effects in radiation chemistry. Concentration dependence for the scavenging of hydroxyl by ferrocyanide in nitrous oxide-saturated aqueous solutions

The radiation chemical yield for oxidation of ferrocyanide by OH produced in the fast electron radiolysis of N/sub 2/O-saturated aqueous solutions is found to increase from a low concentration limiting yield of 5.2 to values in excess of 6.7 for ferrocyanide concentrations above 0.1 M. This observed dependence reflects competition between the scavenging reaction and track recombination processes involving OH radicals which occurs on a time scale greater than 10/sup -9/s. By extrapolation an additional yield of -- 1.5 is estimated for OH radicals which recombine at still shorter times. These scavenging studies show that significant amounts of recombination occur within the track even at times as long as 10/sup -6/s. Secondary radicals produced within the track are also subject to recombination processes as is illustrated by the low concentration limit which shows that most OH radicals produced from N/sub 2/O within the track do not escape from the track. In general, for concentrations of reactive solutes greater than -- 10/sup -4/ M, i.e., for k(S) > 10/sup 6/ s/sup -1/, scavenging of OH from the track will contribute significantly to the observed yields and this effect must be taken into account in comparing the results from different radiolytic systems.

Thermochemistry and lower combustion limit of ammonium perchlorate in presence of methylammonium perchlorates

The heats of combustion of mono-, di-, tri- and tetramethylammonium perchlorates have been determined by bomb calorimetry. The data have been used to explain why the thermal behavior of ammonium perchlorate (AP) is considerably modified in presence of these compounds as shown by differential thermal analysis. Above a particular concentration of methylammonium perchlorate (MAP), AP ignites in a single step around 290°C. The minimum concentration of a MAP (mono-, di-, tri- or tetra-) needed to cause ignition of AP in a single step depends on intramolecular “elemental stoichiometric coefficient” of the mixtures that has the same value regardless of the MAP. Furthermore, the calorimetric values of these mixtures are the same. The heat evolved on ignition of such a composition appears to determine the lower concentration limit of combustion of its mixture with AP.

A polarization approach to the scattering of elastic waves—II. Multiple scattering from inclusions

The integral equation formulation given in the accompanying paper (Part I) is applied to the study of plane waves propagating in a composite material comprising a possibly anisotropic elastic matrix containing aligned elastic inclusions, positioned randomly. An asymptotic solution is given for waves of low frequency. To lowest order, the equations are related closely to those which define the static overall moduli and it is shown that low frequency waves propagate as though the material were homogeneous, with elastic moduli equal to the static overall moduli and density equal to the mean density of the composite. Estimates of the overall moduli that follow from the formulation coincide, for a certain class of composites, with estimates obtainable from the Hashin-Shtrikman variational principle; explicit bounds for the overall moduli of bodies containing aligned penny-shaped cracks, aligned rigid discs and aligned rigid needles follow as by-products of the analysis. Continuing the perturbation solution generates an imaginary term in the series denning the wave number of any one of the plane waves; this corresponds to attenuation of the mean wave by scattering. The lowest order approximation to the attenuation coefficient is found explicitly. It involves the product of the concentration of the inclusions, a factor containing an integral of the pair distribution function and a concentration-dependent term that reduces exactly to one of the scattering cross-sections discussed in Part I in the low concentration limit. Examples show that, at any given frequency, attenuation increases initially with concentration, attains a maximum and then declines as concentration increases further.

Density of states of a disordered Hubbard model using the modified moments method

We apply the modified-moments method to compute the density of states of the impurity band of a doped semi-conductor in the intermediate region of impurity concentrations. This method is used to correct the density of states obtained by interpolating between the high and low concentration limit asymptotic expressions. The calculation is based on the Hubbard model in the atomic limit without spin ordering. The overlap integral is assumed to be a Gaussian function of the impurity separation. Use is made of the first seven moments of the exact distribution and of the low and high concentration limit approximations previously calculated. The first six moments are employed to determine the orthogonal polynomial expansion of the density of states while the seventh moment is used as a check on the accuracy of the distribution obtained. The results are similar to the previous ones using a truncated Edgeworth series for the correction term but the present method has the advantage of being a more systematic approach.

Dynamic response of a disordered ferromagnetic chain: Alloy transfer matrix approximation

The alloy transfer matrix approximation is used to study the uniform dynamic susceptibility of a disordered ferromagnetic chain. This approximation allows for a consistent treatment of diagonal and off-diagonal disorder. The results, in the limit of low concentrations, are in agreement with the exact single impurity ones. Intensities and lineshapes for infrared absorption are calculated for finite impurity concentrations and different values of the relative anisotropy parameter of a model alloy.

Calculation of the concentration limits of flammability of individual compounds and their mixtures V. Lower flammability limits of aliphatic saturated halogen-containing compounds

The existing methods for calculating the lower concentration limit of flammability (C/) [1, 2], based on the constancy of the adiabatic flame temperature, give considerable deviations (of the order of magnitude of 30-40 rel.%) from the value measured for chlorine-contain ing compounds. On transition to bromine-containing compounds, the error is still larger. The explanation for this is that these calculation methods do not take into account the known [3-5] fact of the inhibiting influence of the decomposition products of the halogencontaining compounds on the processes occurring in flame. This influence can be taken into account if the method for calculating the lower concentration limit of flammability is used. This method was developed on the principle of equality of branching (6br) and cleavage (flcl) probabilities in the chain reactions proceeding in flame, and the thermophysical properties of the system studied [6]. The method consists in the search for a kinetic equation expressing the critical condition:

Specific heat enhancement and electron-phonon interaction in random binary alloys

A simple model for treating the electron-phonon interaction in the low-temperature region in random binary alloys is developed within the coherent potential approximation. This approach exhibits the correct behaviour in the low-concentration limits and it is easy to use in numerical calculations. As an example it has been applied to Nb-Mo and Fe-Cr alloys.

Diffusion modulated donor–acceptor energy transfer in a disordered system

This mainly theoretical paper treats, in a disordered system, the time-dependence of irreversible donor–acceptor energy transfer in the limit of low acceptor concentrations. The transfer is modulated by diffusion of energetic donor molecules relative to acceptors. Though donor dynamics will be touched upon, the main emphasis of the paper is on the risetime dynamics of the concentration of energetic acceptors in the nanosecond and subnanosecond regimes following excitation of donors with a brief pulse of light. It has been found that the fastest acceptor risetimes occur in the limit of slowest donor diffusion. The reason for this apparent anomaly, which reverses intuition, is that diffusion aids attainment of a steady state, and the steady state donor distribution gives rise to slower acceptor risetimes than transient contributions. The result has been verified in some preliminary experiments where quasidonor diffusion, caused by donor–donor energy transfer, has been retarded by dilution with inert molecules. Faster risetimes also result from shortening the range of the donor–acceptor transfer interaction. A very striking effect on acceptor risetimes can come about from the presence of local order of donors surrounding an acceptor. Providing the peak of the donor radial distribution function is sufficiently close to the acceptor compared with a characteristic transfer distance, acceptor risetimes can be substantially speeded up. Because of the sensitivity of acceptor dynamics to local order and transfer behavior, the possibility is raised of using the acceptor as a ’’probe’’ for these effects in future picosecond and subpicosecond experiments.

Intermolecular interaction dynamics and optical dephasing: Picosecond photon echo measurements in mixed molecular crystals

Picosecond optical coherence techniques are employed to study dynamical intermolecular interactions in low concentration mixed molecular crystals, i.e., pentacene in naphthalene at 1.4 K. A concentration-dependent broadening of the optical homogeneous line is observed. Stimulated photon echo measurements rule out excited state transport as the broadening mechanism. The photon echo results exhibit a linear concentration dependence and a theory is proposed that accounts for the broadening in terms of phonon-induced fluctuations in guest–guest dipole–dipole interactions. Detailed considerations indicate that the system is best described in terms of localized excitations rather than delocalized impurity band states. In the low concentration limit, there is additional broadening of the homogeneous line beyond that produced by the lifetime. A model is presented describing this broadening in terms of phonon-induced modulation of the guest–host van der Waals interactions, i.e., in terms of a fluctuating crystal shift.

The Current-voltage Behavior of Ion Channels: Important Features of the Energy Profile of the Gramicidin Channel Deduced from the Conductance-voltage Characteristic in the Limit of Low Ion Concentration

The conductance-voltage (G-V) characteristic of a single-filing, multi-barrier, multi-occupancy channel depends in the limit of low ion concentration upon only two parameters: the voltage dependence of the entry step and the ratio of the rate constant for leaving the channel to that for crossing its middle (14,17,20). We show that the G-V shape in this low concentration limit can be measured accurately using a triangular wave, many-channel technique and demonstrate that the observed shape is incompatible with that expected if the only important rate limiting barrier at low concentration were at the channel mouth. Instead the central barrier turns out, surprisingly, in view of the markedly sublinear I-V shape at low concentration, to be even slightly larger than the exit barrier. Additionally, we find that it is not possible to fit both the G-V shape and the concentration dependence of the zero-current conductance simultaneously with a 3-barrier 2-site model. However, by adding additional sites to yield a 3-barrier 4-site model either of the type 3B4S" where the extra site in each channel half is external to the mouth of the channel or of the type 3B4S' where the extra site is internal to the mouth of the channel, we obtain good agreement. Additionally, using the flux ratio data of Procopio and Andersen (19) to discriminate between 3B4S and 3B4S" models, we find the 3B4S" model to be the only satisfactory one.

Frequency crossing signals in thermal conductivity at low concentrations of phonon-scattering centres

Analytical expressions are derived for the size and width of the frequency crossing signals that are observed in the thermal conductivity of a dielectric when the frequencies of two resonant scattering processes are made equal. The expressions apply in the low concentration limit and are extended to higher concentrations by computer analysis.

Quasielastic light-scattering studies of micellar sodium dodecyl sulfate solutions at the low concentration limit

Correlation functions of scattered light intensity of carefully purified sodium dodecyl sulfate (SDS) solutions were measured as a function of tenside concentration and NaCl concentration of the aqueous phase. The correlation functions were analyzed by taking into account the influence of the Coulomb interaction between the micelle (macroion) and small electrolyte ions on the diffusion coefficient. Values of the hydrodynamic radius, the aggregation number, and the effective surface charges were obtained. The aggregation number increases from N = 27 to N = 95 upon increasing the NaCl concentration from 0 to 0.05 mole per liter, while it remains constant when the salt concentration increases further up to 0.2 mole per liter. The effective charge of the micelles decreases with increasing NaCl content in the whole concentration region studied. These results could be interpreted qualitatively in terms of a model which relates the existence of an equilibrium size of the micelles to the balance between hydrophobic and Coulomb interactions. Our results lead to the conclusion that at least up to an NaCl concentration of 0.2 mole per liter the SDS-micelles exhibit an oblate spherical shape rather than a cylindrical form.

Ultraviolet photoelectron spectroscopy of zinc-copper systems during atomic diffusion

We have measured the energy-distribution curves (EDC's) ($h\ensuremath{\nu}=21.2$ eV) during atomic diffusion in the following Zn-Cu samples: an evaporated Zn substrate with 120-\AA{} Cu overlayer, an evaporated Zn substrate with 55-\AA{} Cu overlayer, and a Zn(0001) face with 55-\AA{} overlayer. Owing to diffusion, the intensity of the Cu-$3d$ structure decreases with time and that of the Zn-$3d$ structure increases. In the case of an evaporated substrate, a fast decay of the Cu signal is seen owing to Cu diffusion along Zn grain boundaries; a lower extreme for this diffusion coefficient ${D}_{\mathrm{C}\mathrm{u},\mathrm{Z}\mathrm{n}}^{\ensuremath{'}}$ is between 8 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}15}$ and 8 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}14}$ ${\mathrm{cm}}^{2}$/sec. Diffusion of Zn along Cu grain boundaries also takes place with $(3\ensuremath{\lesssim}{D}_{\mathrm{Z}\mathrm{n},\mathrm{C}\mathrm{u}}^{\ensuremath{'}}\ensuremath{\lesssim}20)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}17}$ ${\mathrm{cm}}^{2}$/sec. In all the samples a slower bulk diffusion of Cu in Zn is seen; in this case the decay of the Cu signal can be fitted with a simple theoretical model giving $(3\ensuremath{\le}{D}_{\mathrm{Cu}}\ensuremath{\le}5)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}19}$ ${\mathrm{cm}}^{2}$/sec. This diffusion takes place in conditions of chemical-concentration gradient as shown by the EDC's, where a marked modification of Cu-$3d$ structure is seen during diffusion; the most evident spectroscopic result is a shift (up to 1.6 eV) of the Cu-$3d$ peak towards higher binding energies. The chemical-concentration conditions originate a diffusion coefficient which is about two orders of magnitude greater than that of Cu in Zn in the limit of low concentration. We discuss qualitatively the importance of the energy associated with electron states in determining this increase of ${D}_{\mathrm{Cu}}$, and we show, on a spectroscopic basis, that about an order of magnitude can be accounted for in this way.

Electronic transport in substitutionally disordered systems

The MCPA theory of electronic transport in disordered systems is applied to the calculation of some transport coefficients of one-dimensional and of three-dimensional simple cubic disordered AB-alloys. In the limit of low concentrations, the behaviour of the DC-conductivity of the one-dimensional alloy is studied analytically, and for arbitrary concentrations the DC- and AC-conductivity and the thermopower of one- and three-dimensional alloys are calculated numerically within the MCPA. For all cases investigated essential improvements of the corresponding CPA-results are found, in particular caused by the vertex corrections. Furthermore, Monte-Carlo-results for the DC-conductivity of one-dimensional alloys are presented and compared to the corresponding CPA- and MCPA-results. The question of localization is discussed in this context and it is shown that there may be a nonvanishing conductivity in one-dimensional disordered systems.

On the theory of dipolar fluids and ion–dipole mixtures

The Stillinger–Lovett condition for the second moment of ion–ion distribution functions is obtained from first principles for an ion–dipole mixture. The Debye–Hückel form for the asymptotic behavior of the potential of mean force between ions is shown to emerge for arbitrary density of the dipolar solvent in the limit of low ionic concentration.