Quantitative Laws

The principle of independent electrochemical reactions applies when several reactions occur simultaneously. It says that each reaction follows its own quantitative laws, irrespective of other reactions. At a given potential, the rates of the different reactions are not at all interrelated, and at a given CD they are merely tied together by relation (13.53). This does not mean that the reactions have no influence on each other at all. One of the reactions may produce changes in the external conditions for other reactions (e.g., in the temperature or solution pH, the amount of impurities adsorbed on the electrode). However, the form of the kinetic equation of each reaction is not affected by these changes. The principle of independent electrochemical reactions is quite general, and rarely violated (we discuss an instance of such a departure in Section 22.2). 

The quantitative laws of electrochemistry were discovered by Michael Faraday of England. His 1834 paper on electrolysis introduced many of the terms that you have seen throughout this book, including ion, cation, anion, electrode, cathode, anode, and electrolyte. He found that the mass of a substance produced by a redox reaction at an electrode is proportional to the quantity of electrical charge that has passed through the electrochemical cell. For elements with different oxidation numbers, the same quantity of electricity produces fewer moles of the element with higher oxidation number. 

What is not so well known about Einstein is that he made contributions to the development of modern chemistry, particularly to the area of quantum mechanics. The Nobel Prize Committee awarded Einstein the Nobel Prize in physics in 1921 for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect. His explanation of the photoelectric effect helped to validate Planck s view of quantized energy, and has become the basis of the quantitative laws of photochemistry. 

The problem of physics is how the actual phenomena, as observed with the help of our sense organs aided by instruments, can be reduced to simple notions which are suited for precise measurement and used for the formulation of quantitative laws. 

Dalton s posmlates can be used to explain three quantitative laws that had been developed proposed his theory. 

A15. Arrhenius, S., Quantitative Laws in Biological Chemistry. Harcourt, New York, 1915. 

The inspiration to this fundamental discovery came to Mayer on a voyage in the Tropics. While in Java he had occasion to bleed some patients, and was struck by the intense red colour of the venous blood. It occurred to him that the reduced combustion in the body must correspond to the reduction in the radiation of heat to the surroundings. As the mechanical work done by a man in the Tropics is about the same as in a, colder climate, it appeared to him that this constant amount of work required a constant amount of heat. Mayer generalised this qualitative result intuitively, and stated his conclusion explicitly in the form of the quantitative law which we have enunciated above, and showed, at the same time, what was then the only... 

Kinetics is concerned with the rates of chemical reactions and the factors which influence these rates. The first kinetic measurements were made before 1820, but interpretation in terms of quantitative laws began with the studies on the inversion of sucrose by Wilhelmy/ the esterification of ethanol with acetic acid by Bethelot and St. Gilles, and the reaction between oxalic acid and potassium permanganate by Harcourt and Esson. These investigations established the relations between rate and concentration of reactants. The important contribution of Arrhenius for the effect of temperature was also made in the nineteenth century. 

The most important stage in the study of specific mass and heat transfer problems is to find general quantitative laws valid for a class of qualitatively similar problems. In many cases, general results of this type can be obtained by the... 

Determination of the quantitative laws governing the rates of the processes in terms of the state variables. These quantitative laws can be obtained from the literature and/or throu an experimental research program coupled with a mathematical modelling program. 

The photophysical properties of a molecule are influenced by its immediate surroundings. Often the effects follow a qualitative empirical rule or, in favorable situations, a well-established quantitative law. Thus, it becomes possible to use photophysical parameters in order to get insight into the microenvironment (e.g. polarity, viscosity) of the emitting molecule. Fluorescence dyes have been used as molecular reporters, not only in solution, but also in membranes, in the solid state, in mixtures of low molecular compounds or... 

The effect of pressure on the reaction rate is not subject to any definite quantitative laws. The results of one reaction may not occur in another. The point is best settled by e q)erimental data. In general, however, increased pressure will result in an increased reaction rate. Thus, Brochet observed that phenol is h3 rogenated very slowly at 150°C at atmospheric pressure using a nickel catalyst but that at 15 atm at the same temperature the reaction was complete and rapid. Maxted and Armstrong and... 

Von Bertalanffy, L., Quantitative laws in metabolism and growth, Q. Rev. Biol.,... 

The basis of chemistry as it grew up in the nineteenth century is Dalton s atomic theory. In this the intuitive idea of ultimate particles is applied to explain definite quantitative laws of chemical composition, those, namely, of constant, multiple, and reciprocal proportions. These rules could only have emerged after a long empirical study... 

In chemical reactions, however spectacular, mass is found to be conserved within the limits of the sensitivity of chemical balances, and this is as it should be if the transformations are mere regroupings of the units of a material substratum. The atoms of ancient speculation fill the roles reqxiired. The quantitative laws of chemical combination then follow and permit the development of chemistry in the form in which it is known today. 

The quantitative laws of chemical combination provide clear pointers to the molecular theory of matter, which increases progressively in vividness and realism with the application of Newton s laws to the motions of the particles. The interpretation of phenomena such as the pressure and viscosity of gases and the Brownian motion, and the assignment of definite magnitudes to molecular speeds, masses, and diameters render it clear that a continual interchange of energies must occur between the molecules of a material system, a circumstance which lies at the basis of temperature equilibrium and determines what in ordinary experience is called the flow of heat. It is responsible indeed for far more than this, and a large part of physical chemistry follows from the conception of the chaotic motion of the molecules. This matter must now be examined more deeply. 

The essential character of thermal phenomena becomes clear, the conditions of coexistence of solids, liquids, and gases in systems of any number of chemical components are explained, the dependence of equilibria upon concentrations, upon pressure, and upon temperature is defined. The conceptions of entropy and free energy, of statistical equilibrium and energy distribution, provide quantitative laws which describe the perpetual conflict of order and chaos, and which prescribe in a large measure not only the shapes assumed by the material world but also the pattern of its possible changes. 

Section 4.3.3.1 outlines the quantitative laws linking the limiting current to the concentration in consumed species in simple systems. 

The ejq)erimental MTDSC observations on anhydride-cured and amine-cured epoxies, described in the previous section, will now be modelled to illustrate the benefits of the technique to obtain a quantitative law of cure kinetics for such thermosetting systems. Because cure kinetics are often complicated by diffusion limitations and/or mobility restrictions, the effect of diffusion has to be incorporated into the overall reaction rate law. For this purpose, both heat capacity and non-reversing heat flow signals for quasi-isothermal and non-isothermal cure experiments are used. 

Perhaps the first quantitative law governing the properties of solutions was published by William Henry in 1803. Henry was studying the solubility of gases in liquids and found that this solubility was proportional to the gas partial pressure (Henry 1803). He did not express his results as an equation, but published tables of data from which the proportionality could be extracted. An interesting review of the current status of Henry s law has been given by Rosenberg and Peticolas (Rosenberg and Peticolas 2004). 

The most important question in the further development of this line of research appears, therefore, to be In the present state of our knowledge regarding these substances and the experimental methods available, what prospect exists beyond the qualitative treatment hitherto practised successfully, of arriving at satisfactory quantitative laws which will serve not only to express clearly the particular phenomena involved but also to correlate particularly those data which describe the state of the dispersed high polymeric substances These include the dimensions, or, at least, the average dimensions, of the particles which are free to move in solution,... 

Arrhenius and Madsen, Z. phys. Chem.y 1903, xliv, 7 Arrhenius, Z. phys. Chem.y 1903, xlvi, 415 Madsen, Brit. Med. y.y 1904, II, 567-74 Craw, Z. phys. Chem.y 1905, lii, 569 Arrhenius, Immunochemistryy New York, 1907 id.y Quantitative Laws in Biological Chemistryy 1915 Thorvald Madsen was Director of the Serum Institute, Copenhagen. 

This study can be used to predict the rock properties in the EDZ in the tunnel excavation, slope cutting or foundation excavation. It is shown that the deformation and strength parameters are greatly influenced by the confining pressures and plastic strain. More work would be carried out to find the quantitative law among them to advance a suitable constitutive model and strength criterion. 

The subject of stoichiometry (a name introduced by Richter (p. 680)) comprises the quantitative laws of chemical composition. These are stated in the law of constant composition (Proust), the law of multiple proportions (Higgins and Dalton), and the law of reciprocal proportions or equivalents (Richter). The first and third are dealt with in this chapter, the second in Chapts. XV and XVI. They are all consequences of Dalton s atomic theory, which forms the main topic of Chapt. XVII. 

Dalton s experiments on gaseous diffusion (p. 773) did not seem to have the force which we now attribute to them, even when the quantitative law was established by Graham. Graham objected that cold should be produced by the expansion and that the mixing of different gases by diffusion occurs at different rates Something more, therefore, must be assumed than that gases are vacua to each other, in order to explain the whole phenomena observed in diffusion. T. S. Thomson attempted to reconcile Dalton s theory of mixed... 

Faraday seems to have been reluctant to emphasise this quantitative relationship. For him this was not yet a law of nature, for laws were special, not to be lightly enunciated, and in the case of quantitative laws difficult to establish by experiment because there might always be exceptions. All his earlier discoveries -electro-magnetic rotations, induction and the dynamo etc - had been qualitative effects where it was possible to demonstrate these phenomena and thus make them accessible to an audience in such a way that they would accept the visible experimental reasoning. He had departed from this with the identity of electricities, as he had had to resort in many cases to using the experiments of others to establish this result. In his electro-chemical work Faraday had to make more departures from what had been his normal practice. He had proposed a quantitative relationship, a new theory of electrochemical action and a theory of matter to support this - things which Faraday had never done before as a mature scientist. As I... 

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