Compound interest law

In the beginnings of classical physical chemistry, starting with the publication of the Zeitschrift fUr Physikalische Chemie in 1887, we find the problem of chemical kinetics being attacked in earnest. Ostwald found that the speed of inversion of cane sugar (catalyzed by acids) could be represented by a simple mathematical equation, the so-called compound interest law. Nernst and others measured accurately the rates of several reactions and expressed them mathematically as first order or second order reactions. Arrhenius made a very important contribution to our knowledge of the influence of temperature on chemical reactions. His empirical equation forms the foundation of much of the theory of chemical kinetics which will be discussed in the following chapter. 

Money lent at compound interest increases in this way, and hence the above property has been happily styled by Lord Kelvin the compound interest law (Encyc. Brit., art. Elasticity, 1877). A great many natural phenomena possess this property. The following will repay study —... 

Illustration 3.—The variation of atmospheric pressure with altitude above sea-level can be shown to follow the compound interest law. Let pQ be the pressure in centimetres of mercury at the so-called datum line, or sea-level, p the pressure at a height h above this level. Let p0 be the density of air at sea-level (Hg = 1). Now the pressure at the sea-level is produced by the weight of... 

The negative sign indicates that the pressure decreases vertically upwards. This equation is the compound interest law in another guise. The variation in the pressure, as we ascend or descend, is proportional to the pressure itself. Since pJpQ is constant, we have on applying the compound interest law to (13),... 

If any dielectric (condenser) be subject to a difference of potential, the density p of the charge constantly diminishes according to the relation p=be at, where b is an empirical constant and a is a constant equal to the product 47r into the coefficient of conductivity, c, of the dielectric, and the time, t, divided by the specific inductive capacity, /x, i.e., a = Airct/p.. Hence show that the gradual discharge of a condenser follows the compound interest law. Ansr. Show dpjdt = - ap. 

One form of Dalton s empirical law for the pressure of saturated. vapour, p, between certain limits of temperature, 0, istp = aeQ. Show that this is an example of the compound interest law. 

The relation between the velocity V of a certain chemical reactioh and temperature, 0°, is log V = a + b6, where a and b are constants. Show that we are dealing with the compound interest law. What is the logical consequence of this law with reference to reactions which (like hydrogen and oxygen) take place at high temperatures (say 500°), but, so far as we can tell, not at ordinary temperatures ... 

The rate of change of a radioactive element is represented by dNjdt = - rN where N denotes the number of atoms present at the time t, and r is a constant. Show that the law of radioactive change follows the compound interest law . 

Many students, and even some of the text-books, appear to have hazy notions on this question. According to Guy Lussac s law the increase in the volume of a gas at any temperature for a rise of temperature of 1°, is a constant fraction of its initial volume at 0°C. J. Dalton s law (Manchester Memoirs, 3, 595, 1802), on the other hand, supposes the increase in the volume of a gas at any temperature for a rise of 1°, is a constant fraction of its volume at that temperature (the Compound Interest Law, in fact). The former appears to approximate closer to the truth than the latter. (See page 285.) J. B. Gay Lussac (Annales de Chimie, 43,137 1802) says that Charles had noticed this same property of gases fifteen years earlier and hence it is sometimes called Charles law, or the law of Charles and Gay Lussac. After inspecting Charles apparatus, Gay Lussac expressed the opinion that it was not delicate enough to establish the truth of the law in question. But then J. Priestley in his Experiments and Observations on Different Kinds of Air (2, 448, 1790) says that from a very coarse experiment which I made very early I concluded that fixed and common air expanded... 

Wilhelmy s equation, dy/dt — - ay, already discussed in connection with the compound interest law, page 68, may be written... 

Numerous examples of both methods will occur in the course of this work. Some have already been given in the discussion on the Compound Interest Law in Nature, page 56. 

A semi-logarithmic paper (Fig. 133) may be made with distances between say the horizontal columns in fractions of a metre, while the distances between the vertical columns are proportional to the logarithms of the numbers. Functions obeying the compound interest law will plot, on such paper, as a... 

Sj = Ae-0 1 sin btx - s2 = Ae 013 sin bt2 . .. where the negative sign indicates that the displacement is on the negative side. Hence the amplitude of the oscillations diminishes according to the compound interest law,... 

Particle-Size Distribution—The size-frequency distribution of ground materials differs from the usual type of frequency-distribution in chance sampling. As a general rule the number of particles increases with decreasing particle-diameter. Martin (1923) has shown that frequency-distribution follows the law of compound interest, namely... 

The law of compound interest on investment is the same as the law of bacterial growth. If Pq is the initial value of the principal amount, the interval at which compounding occurs, and r the interest rate for the interval expressed as a fraction, the principal at time t will... 

Henry s law constants for most of the compounds of interest can be found in the literature.54 Figure 18.11 shows Henry s law constants for TCE, EDC and several gasoline compounds.19 These data are derived from water solubility data and the equilibrium vapor pressure of pure liquids at certain temperatures, and may be extrapolated correctly to field design work. Temperature has a major effect on Henry s constant and on stripper performance. Each rise of 10°C in temperature... 

Although not all facets of the reactions in which complexes function as catalysts are fully understood, some of the processes are formulated in terms of a sequence of steps that represent well-known reactions. The actual process may not be identical with the collection of proposed steps, but the steps represent chemistry that is well understood. It is interesting to note that developing kinetic models for reactions of substances that are adsorbed on the surface of a solid catalyst leads to rate laws that have exactly the same form as those that describe reactions of substrates bound to enzymes. In a very general way, some of the catalytic processes involving coordination compounds require the reactant(s) to be bound to the metal by coordinate bonds, so there is some similarity in kinetic behavior of all of these processes. Before the catalytic processes are considered, we will describe some of the types of reactions that constitute the individual steps of the reaction sequences. 

In order to plot the absorption spectrum of a compound or complex ion, we must be able to carefully control the wavelengths from the broad spectrum of wavelengths emitted by the source so that we can measure the absorbance at each wavelength. Additionally, in order to perform quantitative analysis by Beer s law, we need to be able to carefully select the wavelength of maximum absorption, also from this broad spectrum of wavelengths, in order to plot the proper absorbance at each concentration. These facts dictate that we must be able to filter out the unwanted wavelengths and allow only the wavelength of interest to pass. 

The use of infra-red or ultraviolet spectroscopy to examine the molecular groups present in a chemical compound is familiar to any chemist. One of the main uses of this technique is to apply a range of electromagnetic frequencies to a sample and thus identify the frequency at which a process occurs. This can be characteristic of, say, the stretch of a carbonyl group or an electronic transition in a metal complex. The frequency, wavelength or wavenumber at which an absorption occurs is of most interest to an analytical chemist. In order to use this information quantitatively, for example to establish the concentration of a molecule present in a sample, the Beer-Lambert law is used ... 

Despite the broad definition of chemometrics, the most important part of it is the application of multivariate data analysis to chemistry-relevant data. Chemistry deals with compounds, their properties, and their transformations into other compounds. Major tasks of chemists are the analysis of complex mixtures, the synthesis of compounds with desired properties, and the construction and operation of chemical technological plants. However, chemical/physical systems of practical interest are often very complicated and cannot be described sufficiently by theory. Actually, a typical chemometrics approach is not based on first principles—that means scientific laws and mles of nature—but is data driven. Multivariate statistical data analysis is a powerful tool for analyzing and structuring data sets that have been obtained from such systems, and for making empirical mathematical models that are for instance capable to predict the values of important properties not directly measurable (Figure 1.1). 

It consists of three parameters, which are C (i. e., the equilibrium concentration of the chemical compound of interest in solution), Q (i.e., the maximum number of moles of a pollutant adsorbed per mass adsorbent), and q (i.e., the number of moles of adsorbate per mass adsorbent at equilibrium). The Toth model (Eq. 17) reduces to Henry s law at very low concentrations and exhibits saturation at high concentrations. 

Although both dimer and trimer are known to exist, the trimer is the more easily produced. Typically, the dimer is only seen as a by-product of the trimer synthesis. Unfortunately, over the past decade law enforcement agencies have encountered TATP more frequently both in the United States and abroad. Numerous questions regarding its synthesis and properties have surfaced in Internet news groups as well, and there is no sign of abatement in interest about the compound. 

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