Logarithmic Diagram Research Articles

On the theory of stepwise electrode processes

The equations for d.c. polarographic waves of two-step electrode reaction are derived with the stationary plane, expanding plane model of diffusion, as well as with a steady-state approach. The equations for total current and currents due to individual steps are derived and the corresponding logarithmic diagrams are discussed. The procedures for characterization of individual steps are proposed and illustrated by the analysis of data for Cu2+ ion reduction in perchlorate media. The results are compared with those of Jacq et al. who used the same data and the same model. The meaning of overall parameters of the stepwise electrode reaction is discussed and, using the Marcus theory of charge transfer, the relation between the overall standard heterogeneous rate constant and the corresponding rate constants of homogeneous exchange reactions is presented. The meaning of Vetter's “exchange currents of individual steps” is also discussed.

A note on the form of current-potential curves

The forms of the current-potential curves obtained for different models of diffusion by rigorous and approximate methods are compared. It is found that for instantaneous current the irreversible part of the logarithmic diagram (log(i/id−i) vs. potential) obtained by rigorous methods at sufficiently positive potentials agree very well with approximate results (with corresponding diffusion layer thickness (πDt)1/2 for stationary and (3πDt/7)1/2 for expanding plane model). However, at more negative potentials the slope of the logarithmic diagram is no longer equal to αnF/RT in the case of a stationary plane model of diffusion. In the case of an expanding plane at these potentials the diffusion layer thickness is no longer (3πDt/7)1/2 but equal to that of the stationary plane (πDt)1/2. Therefore the analysis of quasi-reversible polarographic waves and related curves should be performed without the factor (3/7)1/2 (erroneously) used earlier, because the irreversible part of the logarithmic diagram at potentials near the reversible potential has no effect on the observed current (which is controlled by diffusion and not by heterogeneous charge transfer) and it is only at these low potentials that the expanding plane diffusion layer solution should be used. The analysis of mean current is also discussed.

A note on the form of current-potential curves

Summary The forms of the current-potential curves obtained for different models of diffusion by rigorous and approximate methods are compared. It is found that for instantaneous current the irreversible part of the logarithmic diagram (log( i/i d −i) vs. potential) obtained by rigorous methods at sufficiently positive potentials agree very well with approximate results (with corresponding diffusion layer thickness (π Dt ) 1/2 for stationary and (3π Dt/7 ) 1/2 for expanding plane model). However, at more negative potentials the slope of the logarithmic diagram is no longer equal to α nF/RT in the case of a stationary plane model of diffusion. In the case of an expanding plane at these potentials the diffusion layer thickness is no longer (3π Dt/7 ) 1/2 but equal to that of the stationary plane (π Dt ) 1/2 . Therefore the analysis of quasi-reversible polarographic waves and related curves should be performed without the factor (3/7) 1/2 (erroneously) used earlier, because the irreversible part of the logarithmic diagram at potentials near the reversible potential has no effect on the observed current (which is controlled by diffusion and not by heterogeneous charge transfer) and it is only at these low potentials that the expanding plane diffusion layer solution should be used. The analysis of mean current is also discussed.

Logarithmic diagrams and gran-curves as an aid in potentiometric titrations in complexometry

A detailed exposition is given of the use of logarithmic diagrams and Gran plots to construct titration curves and to determine accurately the equivalence point in potentiometric complexometric titrations.

On the choice of form factors in non-local quantum electrodynamics

The photon Green function in the approximation of principal logarithmic diagrams is investigated in local quantum electrodynamics. It is supposed that the result of the summation of principal diagrams leads to a non-local theory. Conditions under which the obtained Green function is unambiguous are stated. It turns out that any diagrams of the perturbation theory series, calculated by means of the partially summed photon Green function, are convergent. Within the frame of the procedure proposed, the value for elementary length is obtained:

Analysis of the shape of deformed pebbles

The shape of an ellipsoidal pebble may be defined by the lengths of its semidiameters x, y and z, and by r = √ xyz. The shape may be plotted as a point on a triangular diagram of sides log( x r ), log( y r ), log( z r ). The shapes of a group of related ellipsoids may be plotted to produce a scatter diagram which may then be contoured to give a modal shape and shape-dispersion of the group. The symmetrical nature of the triangular logarithmic plot is convenient for representing all three axes equivalently. The graphical method of statistical analysis permits an accurate measurement of modal shape, and provides a method of determining whether a population of pebbles is homogeneous in regard to shape. The dispersion patterns are of three kinds which are: 1. (1) inherited from the original conglomerate. 2. (2) representing random perturbations in the strain. 3. (3) tracing paths of progressive strain. The strain inside pebbles (not necessarily that in a metaconglomerate as a whole) may be measured in terms of the axial extensions using the logarithmic parameters in ( x r ), In ( y r ), In ( z r ), or ϵ x , ϵ y , ϵ z . The magnitude of strain may be expressed in terms of a parameter E, the logarithmic strain interval, which is a summation of the natural or logarithmic strains and is applicable to any type of strain. The strain interval E is simply related to the plot interval S which is the separation of any two points on the triangular diagram. Triangular logarithmic shape diagram, for the sparagmite, Bygdin, -Fetlar, Barberton and Goat Island Conglomerates and the South Mountain oolites are compared. The diagrams imply an initial scatter of shapes and an increase in the range of shapes with increasing strain. In most cases the shape distribution implies a linear deformation path, the most common being y r equals a constant near unity. This is a strain of the type ϵ y = 0, ϵ x = − ϵ z which is a plane strain in xz. An exception is the Bygdin Conglomerate for which the shape distribution suggests a two-stage strain. Two components of strain cannot be determined by this method. The first is dilitation which is detectable only by examination of the passage into unstrained material, as done by Cloos (1947, p.884). The second is a component of internal rotation which is detectable only from the fabric symmetry (inside the pebbles). The strain which is found by this method is the pure component of strain in the interior of the pebbles.

Graphical representation of homogeneous chemical equilibria in volcanic gas systems

The logarithms of the partial pressures of the components of the gaseous phase are plotted against the logarithm of the partial pressureof the oxygen under conditions of constant temperature, constant total pressure, and constant ratios of all elements except oxygen. The result is a quantitative survey of the effect of oxidation on the gases present. The equilibrium composition of an individual volcanic gas sample or of any mixture of gases may be obtained from the diagram when the atomic ratio of the oxygen is also fixed. In constructing logarithmic partial pressure diagrams, it is essential to know which molecular species may become dominant (contain virtually all of a certain element) under the conditions of the diagram.

芝草の播種量と発芽率との関係

Germination tests of gtasses under several conditions of the sown densty and the water supply were carried out to make sure of the relation between the sown density and the germinating rate.The results were obtained as follows.The germinating rate in high density plots with insufficient water went down significantly and moreover, the germination was delayed for 3 days, in comparison with sufficient water plots.(Table 1, 2 and Fig. 1). This was certified also by the water content per oneseed.(Table 3, 4 and Fig. 3).Those phenomena were considered as the results of the scramble for water.Here, the authors introduce the following formulae on the relation between the sown density and the germinating rate._??_ (1)where, r: germinating rateρ: sown densityh: quantity of waterThis formula explains that the augmentaion of the germinating rate is inversely proportionate to the increase of the sown density at a certain water supply.The following formulae are obtained by working out this differential epuation (1).log r=-a·logδ+log R (2)or, r=R·ρ-a (3)moreover, with avalable space per one seed (δ) instead of density (ρ), Formula (3) is shown as follows;r=R·δa (4)where, the constant (a) is determined by water condition, and the constant of integration (R) is the maximum germinating rate under sufficient water condition.The r-ρ or r-δ curves are shown in Fig 4 and 5 in logarithmic diagrams. The character of these curves is explained in Fig 6, 7 and 8.In conclusion, these results are noteworthy for the determination of the sown density of grasses.

Selection of Partition Chromatographic Systems from Distribution Diagrams: Determination of Dextromethorphan in Cough Sirups

Abstract Ionization and extraction constants for dextromethorphan and 14 other alkoids are given. By constructing the logarithmic distribution diagrams for mixtures of these compounds, Celite partition chromatographic systems can easily be selected for their separation; both free base and ionpair extraction provide useful systems. A general procedure for the isolation of dextromethorphan from any of those alkoids with which it is currently formulated is presented; the results of analyses of standard samples and of commercial sirups are given. Some other separations do not appear to be feasible. However, the distribution diagrams clearly show the reasons for this fact

Evaluation of titration analyses with logarithmic concentration diagrams

Examines the use of logarithmic concentration diagrams in the evaluation of strong and weak acid-base titrations.

Resistance to Steady Laminar Flow of Pseudoplastic Fluids through Pipes and Valves

The well-known logarithmic friction factor diagram for laminar pipe flow can be extended in the following two respects: For application to non-Newtonian fluids, by incorporating in the plot a modified form of the consistency curve for a given material. Methods are discussed of obtaining these curves, and of transforming them into the required shape. For prediction of flow resistance through valves and fittings, by use of an auxiliary diagram based on results of appropriate tests. The general procedure is outlined, and it is stressed that results cannot be relied on quantitatively, until test data are obtained for wider ranges of sizes and types than hitherto available.

Stress-Strain Diagram of the Steel Constituted of Hard Patches and Free Ferrite

The influence of microstructural parameters on the true stress-true strain diagrams between 0.02 strain and the necking strain was investigated in the steels whose microstructure was constituted of tempered martensite patches and free ferrite.The results obtained are as follows:(1) Under some conditions, the logarithmic true stress-logarithmic true strain diagram does not make a straight line but it consists of two straight lines having different slopes.(2) The appearance of the knee, which is the intersecting point of two straight lines, is controlled by the value of the volume fraction of tempered martensite f, the mean free path between tempered martensite patches α2, the tempered martensite patch size β and tempering temperature.(3) The stress at the knee is closely related to the yield strength of tempered martensite structure. It is supposed from this that, at the knee, the macroscopic yieldings of tempered martensite patches start.(4) The lowering of α2, β and tempering temperature and the rising of f introduce the increasing flow stress at a constant strain.(5) To explain the above experimental results, a model representing the microstructure of steel used in the present work was proposed. In the model, all the tempered martensite and some portion of free ferrite were connected in parallel and furthermore this unit and remaining portion of free ferrite were connected in series as shown in Fig. 6(c).

On the giant sea-salt particles in the atmosphere: II. Theory of the Vertical Distribution in the 10-m Layer Over the Ocean

The vertical distribution of giant sea-salt particles in the 10-m layer over the sea surface, where humidity, eddy diffusivity, etc., sharply change with height, is analytically studied. A steady state with no horizontal gradient and with a logarithmic distribution of wind speed is assumed, but the vertical distributions of temperature and vapor pressure are taken into account in such a way that each droplet is assumed to instantaneously attain its equilibrium salinity for the environmental air at each level, and to remain as a droplet of supersaturated salt solution when the relative humidity is lower than 75%. It is deduced that the vertical distribution of the number concentration of each class interval of mass of salt contained in a droplet should be approximately expressed by a straight line on a logarithmic diagram. Also, the logarithm of the ratio of the concentration of the particles at the sea surface and at the 10-m level is proportional to the two thirds power of the mass of salt, and inversely proportional to the wind speed at the 10-m level. The proportionality factor is approximately a function of the relative humidity at the 10-m level and a friction factor. This proportionality factor is calculated for various values of the variables and the effect of wind speed and relative humidity on the vertical distribution of particles is predicted. When this result is applied to data of the concentration and the effective production rate of sea-salt particles at the 10-m level compiled in the previous paper, complicated features disappear to give very simple patterns of the production rate or the surface concentration of sea water droplets as a function of wind force and of mass of salt. This represents predicted processes of production of sea-salt particles at the ocean surface, and gives some agreement with the author's 1961 wind flume experiment. DOI: 10.1111/j.2153-3490.1965.tb01429.x

The asymptotic behavior of the scattering amplitudes and the renormalisation group method

Differential equations of the renormalization group with respect to the momentum transfer are used to find the asymptotic behavior of the scattering amplitude at high energies (S). Application of the method is illustrated by two models of the quantum field theory, establishing an interesting relation between the invariant charge and the Regge exponent. The class of diagrams taken into account also gives an essentially larger contribution to the asymptotic behavior in S'' than those corresponding to the terms previously obtained by summing the principle'' logarithmic diagrams. (H.D.R.)

Die Hydratation der Stärke

ZusammenfassungZur vergleichenden Untersuchung der Wassersorption von Kartoffel‐, Weizen‐, Mais‐ und Reisstärke wurden die Gleichgewichtswerte bei Wassersättigung sowie die Ad‐ und Desorptionsisothermen nach zwei verschiedenen Verfahren aufgenommen. Weiterhin wurde der Einfluß verschiedener Trocknungsarten auf die Größe der Sorption untersucht. Der Einfluß der Stärkestruktur auf die Wassersorption wurde an verkleisterten und mittels Methanol ausgefällten Kartoffel‐, Weizen‐, Mais‐ und Reisstärken untersucht.Die durchgeführten Messungen führten zu folgenden Ergebnissen: Die Werte für die Wassersorption nehmen in der Reihenfolge Reis‐, Mais‐, Weizen‐, Kartoffelstärke zu; dabei sind die Unterschiede zwischen den Stärkearten bei Wassersättigung am größten, vermindern sich bei abnehmender rel. Feuchte und verschwinden bei den Getreidestärken unterhalb 68% rel. F. und zwischen Kartoffel‐ und Getreidestärken unterhalb 35%. Bei den unverkleisterten Stärken ist die Größe der Wassersorption in allen Feuchtigkeitsbereichen bei gleicher Temperatur von der angewandten Methodik — Messung bei Atmosphärendruck innerhalb von 20 bis 120 Tagen bzw. Messung unter vermindertem Luftdruck in 10–50 Stunden— praktisch unabhängig. Durch thermische Trocknungsverfahren wird bei allen vier Stärkearten die Fähigkeit zur Wasseraufnahme bei Wassersättigung der Atmosphäre herabgesetzt. Dieselben Stärken zeigten dagegen nach Methanolentwässerung und Vakuumtrocknung eine erhöhte Fähigkeit zur Wasseraufnahme. Durch Verkleistern und anschließende Methanolfällung erhaltene Quellstärken zeigten bei Wassersättigung eine auf über 40 % H2O erhöhte Sorptionsfähigkeit. Dabei sind die Unterschiede zwischen den Stärkearten fast völlig verschwunden. Unterhalb 79 bis 68 % rel. F. sind die Sorptionsisothermen mit denen der unverkleisterten Stärken praktisch identisch. Die Isothermen zeigen bei Auswertung nach der Freundlich'schen Isothermengleichung log a = f (lg x) bei doppelt logarithmischer Darstellung zwischen 0 und 68 % rel. F. mit unterschiedlichen Neigungskoeffizienten angenähert linearen Verlauf und gehen bei zunehmender Wassersättigung ohne merklichen Übergang in einen steil ansteigenden Ast über. Aus der Abweichung von der Gültigkeit der Freundlich‘schen Isothermen wird auf eine Wasserbindung der Stärke im Bereich höherer rel. Feuchtigkeiten durch Kapillarkondensation geschlossen, deren Größe von der Oberflächenbeschaffenheit des Sorbens und den Sorptionsbedingungen abhängig ist. Unterhalb dieses kritischen Punktes der Isothermen ist die Wasserbindung der Stärke vorwiegend eine für die Stärkekohlenhydrate spezifische Stoffeigenschaft und erfüllt die für einen reinen Adsorptionsvorgang geltende Freundlich‘sche Gleichung.

Logarithmic diagrams in trace analysis

Logarithmic diagrams in trace analysis

Pitting Strength for Rolling Contact of Curved Surface

Endurance tests were carried out for rolling contacts of roller on roller, ball on saddle type specimen, and ball on ball, lubricating with machine oil. The limit values of their pitting strength indicated by Hertz's maximum pressure increased remarkably with the decrease of cling factors. In the case of rolling contact of ball on ball, the contact surface deformed plastically marking a flat band, and the logarithmic diagram of the width of this contact area versus repetition numbers was in linear relation. From this fact we can practically estimate the variation of maximum pressures acting on the curved surfaces. The fatigue diagram for the constant equivalent shear stresses calculated for these varying maximum pressures coincided with the fatigue diagram in the case of the constant shear stresses for the contact of rollers considered as a problem of plane strain. Applying these results, we have tried to explain the life of ball bearings.

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Book reviewed in this article:Statistical Techniques in Market Research. By Robert FerberAmerican Transportation in Prosperity and Depression. By Thor HultgrenManufacturing Business. By P. “W. S. AndrewsChanges in the Cost‐of‐Living and the Distribution of Income since 1938. By Dudley SeersI.L.O.: Coal Mines Committee: Beneral Report, 1949Japan's Textile Industry. By John R. StewartIndustrial Profits in India. By M. C. MunshiPolitical Economy in Australia. Historical Studies by J. A. La NauzeMill's Utilitarianism. Reprinted with a study of the English Utilitarians. By John PlamenatzThe Social Problems of an Industrial Civilization with an Appendix on the Political Problem. By Elton MayoMonetary Management. By E. A. GoldenweiserMonetary Theory and Fiscal Policy. By Alvin H. HansenPost‐War Monetary Plans and Other Essays. By John H. WilliamsInflationary and Deflationary TendenciesShadows of Hyper‐inflation. By S. K. MuranjanPost War Inflation in India–A Survey. By N. V. SovaniOverhead Costs. By W. Arthur LewisThe Cost of Distribution. By S. J. Lengyel and E. M. BeecroftEmployment and Equilibrium. By A. C. PigouStudies in Income and Wealth, Vol. XI. Conference on Research in Income and WealthReports of the Commodity Prices Board. Edited by N. V. SovaniFive Lectures on Economic Problems. By George J. StiglerExpectation in Economics. By G. L. S. ShackleEconomic Calculation in the Socialist Society. By T. J. B. HoffEnterprise in a Free Society. By C. E. GriffinUn Sistema di Logodiagrammi Tipici (A Standardized System of Logarithmic Diagrams). By A. de Pietri‐TonelliProblemes 'Economic Internationale. By Jean WeillerThe Consumer and the Economic Order. By W. C. Waite and R. CassadyUnemployment, Full Employment and India. By Nabagopal DasAn Introduction to the Geography of Newfoundland. By B. V. GutsellThe Economic Survey of Asia and the Far East in 1948. United Nations Economic Commission for Asia and the Far EastMajor Economic Changes in 1948.Economic Survey, 1919–1939. By W. Arthur LewisWorld Economic Report, 1948Urban Land Economics. By Richard U. RatcliffThe Agrarian Problems of Madras Province. By V. V. SayanaNeue Deutsche Agrarpolitik (New German Agricultural Policy). By Willy NeulingInternational Control of Sugar, 1918–1941, By B. C. SwerlingBalance of Payments Yearbook, 1938–1946‐1947Balance of Payments of Nigeria, in 1936. By Penelope A. BowerInsight into Labour Issues. Edited by Richard A. Lester and Joseph ShisterYear Book of Labour Statistics, 1947‐48Indian Industrial Labour. By T. N. RastogiLocal Government Areas, 1834–1945. By D. V. Lipman

About the Practical Use of the Logarithmic Diagram of Bjerrum

About the Practical Use of the Logarithmic Diagram of Bjerrum

High-Pressure Capillary Flow

A further development is given in this paper of the theory of the flow of liquids in capillaries under inlet pressures that are high enough to cause an appreciable change in viscosity, such that the viscosity can no longer be treated as uniform. The results are put in the form of Poiseuille's law, with a correction factor. Starting from any given empirical relation between viscosity and pressure, an expression for the corresponding correction factor can be obtained by integration. Conversely, if the form of the function connecting viscosity with pressure is unknown, it can be determined by differentiation of the observed flow-pressure graph connecting rate of flow with inlet pressure. For the usual case it can be shown that the rate of flow approaches an asymptotic limiting value. If for any reason the logarithmic viscosity-pressure diagram takes an upward bend, the viscosity increasing more rapidly than before, the rate of flow must actually decrease with further increase of inlet pressure, and the flow-pressure graph will pass through a maximum. The foregoing analysis makes possible several different methods for computing the viscosity-pressure characteristics of a lubricating oil, or other liquid, from the simple experimental procedure of observing the efflux out of a long metal capillary into the free atmosphere. The paper includes a review of the calculations by S. Kiesskalt for flat capillaries; and indicates under what conditions the extrusion type of high-pressure gage proposed by C. Barus might be realized. The theory is illustrated by the experiments of C. Barus on marine glue to 30,000 pounds per square inch, and by more recent experiments of the authors on pressure-gun and cup grease, and on castor oil, to approximately 45,000 pounds per square inch. The long capillary method is less sensitive and less precise but much more rapid than the two methods previously used in the work of the A. S. M. E. Special Research Committee on Lubrication. From the data presented on castor oil it appears that all three methods are in reasonable agreement.