Principle of conservation

Quantitative Calculations In precipitation gravimetry the relationship between the analyte and the precipitate is determined by the stoichiometry of the relevant reactions. As discussed in Section 2C, gravimetric calculations can be simplified by applying the principle of conservation of mass. The following example demonstrates the application of this approach to the direct analysis of a single analyte. 

Applying the principle of conservation of mass to carbon, we write... 

Finally, quantitative problems involving multiple analytes and back titrations also can be solved by applying the principle of conservation of electron pairs. 

Cycloaddition involves the combination of two molecules in such a way that a new ring is formed. The principles of conservation of orbital symmetry also apply to concerted cycloaddition reactions and to the reverse, concerted fragmentation of one molecule into two or more smaller components (cycloreversion). The most important cycloaddition reaction from the point of view of synthesis is the Diels-Alder reaction. This reaction has been the object of extensive theoretical and mechanistic study, as well as synthetic application. The Diels-Alder reaction is the addition of an alkene to a diene to form a cyclohexene. It is called a [47t + 27c]-cycloaddition reaction because four tc electrons from the diene and the two n electrons from the alkene (which is called the dienophile) are directly involved in the bonding change. For most systems, the reactivity pattern, regioselectivity, and stereoselectivity are consistent with describing the reaction as a concerted process. In particular, the reaction is a stereospecific syn (suprafacial) addition with respect to both the alkene and the diene. This stereospecificity has been demonstrated with many substituted dienes and alkenes and also holds for the simplest possible example of the reaction, that of ethylene with butadiene ... 

You are probably au fait with the principle of conservation of energy, which introduces the idea of the potential energy U. The kinetic energy and the potential energy of the body can each vary, but their sum is a constant that I will write e. 

An explanation for the finding that concerted [4 -I- 2] cycloadditions take place thermally, while concerted [2 + 2] cycloadditions occur under photochemical conditions, is given through the principle of conservation of orbital symmetry. According to the Woodw ard-Hofmann rules derived thereof, a concerted, pericyclic [4 -I- 2] cycloaddition reaction from the ground state is symmetry-allowed. 

Big or small, simple or complex, energy converters must all subscribe to the principle of conservation of energy. Each one converts energy into some form regarded as useful, and each one diverts energy that is not immediately useful and may never be useful. Because energy is diverted, the efficiency defined as... 

In collisions between two bodies the contact force and the duration of contact are usually unknown. However, the duration of contact is the same for both bodies, and the force on the first body is the negative of the force on the second body. Thus the net change in momentum is zero. This is called the principle of conservation of momentum. 

In fluid mechanics the principles of conservation of mass, conservation of momentum, the first and second laws of thermodynamics, and empirically developed correlations are used to predict the behavior of gases and liquids at rest or in motion. The field is generally divided into fluid statics and fluid dynamics and further subdivided on the basis of compressibility. Liquids can usually be considered as incompressible, while gases are usually assumed to be compressible. 

When fluids are in motion, the pressure losses may be determined through the principle of conservation of energy. For slightly compressible fluids this leads to... 

One molecule (or mole) of propane reacts with five molecules (or moles) of oxygen to produce three molecules (or moles) or carbon dioxide and four molecules (or moles) of water. These numbers are called stoichiometric coefficients (v.) of the reaction and are shown below each reactant and product in the equation. In a stoichiometrically balanced equation, the total number of atoms of each constituent element in the reactants must be the same as that in the products. Thus, there are three atoms of C, eight atoms of H, and ten atoms of O on either side of the equation. This indicates that the compositions expressed in gram-atoms of elements remain unaltered during a chemical reaction. This is a consequence of the principle of conservation of mass applied to an isolated reactive system. It is also true that the combined mass of reactants is always equal to the combined mass of products in a chemical reaction, but the same is not generally valid for the total number of moles. To achieve equality on a molar basis, the sum of the stoichiometric coefficients for the reactants must equal the sum of v. for the products. Definitions of certain terms bearing relevance to reactive systems will follow next. 

Since the continuity equation (equation 9.3) is essentially a restatement of the principle of conservation of mass, wc should not be surprised to learn that it is easily recovered from Boltzmaii s equation by setting k — rn in equation 9.52 ... 

It will be observed that the definition of intrinsic energy by means of the equation (c) implies in itself no physical law, since the value of (U2—Ui) can always be chosen so as to make the values of 2Q and 2A satisfy the equation. We shall now show that the value of (U2 — Ui) is uniquely so defined, and is quite independent of the way in which the process is executed. This is a physical law, which we shall call the Principle of Conservation of Energy. 

U = constant. The Principle of Conservation of Energy is usually expressed in the form that the intrinsic energy of an absolutely isolated system of bodies is constant and independent of all changes of state which may occur subject to the condition that the system remains isolated. Since in this case we have absolutely no means of examining the energy content of the system, the statement appears somewhat indefinite. 

The names for these mechanisms vary throughout the literature. For example, the Sgi mechanism has also been called the Sp2, the Se2 (closed), and the Se2 (cyclic) mechanism. The original designations, Se 1, Se2, and so on, were devised by the Hughes-Ingold school. It has been contended that the SeI mechanism violates the principle of conservation of orbital symmetry (p. 1068), and that the Se2 (back) mechanism partially violates it Slack, D.A. Baird, M.C. J. Am. Chem. Soc., 1976, 98, 5539. 

It must be emphasized once again that the rules apply only to cycloaddition reactions that take place by cyclic mechanisms, that is, where two s bonds are formed (or broken) at about the same time. The rule does not apply to cases where one bond is clearly formed (or broken) before the other. It must further be emphasized that the fact that the thermal Diels-Alder reaction (mechanism a) is allowed by the principle of conservation of orbital symmetry does not constitute proof that any given Diels-Alder reaction proceeds by this mechanism. The principle merely says the mechanism is allowed, not that it must go by this pathway. However, the principle does say that thermal 2 + 2 cycloadditions in which the molecules assume a face-to-face geometry cannot take place by a cyclic mechanism because their activation energies would be too high (however, see below). As we shall see (15-49), such reactions largely occur by two-step mechanisms. Similarly. 2 + 4 photochemical cycloadditions are also known, but the fact that they are not stereospecific indicates that they also take place by the two-step diradical mechanism (mechanism... 

In 15-58, we used the principle of conservation of orbital symmetry to explain why certain reactions take place readily and others do not. The orbital symmetry principle can also explain why certain molecules are stable though highly strained. For example, quadricyclane and hexamethylprismane are thermodynamically much less stable (because much more strained) than their corresponding isomeric dienes, norbomadiene and hexamethylbicyclo[2.2.0]hexadiene (108). Yet the... 

The fraction of free sites in Eq. (155) is found from the principle of conservation of sites ... 

The principles of conservation of mass and momentum must be applied to each phase to determine the pressure drop and holdup in two phase systems. The differential equations used to model these principles have been solved only for laminar flows of incompressible, Newtonian fluids, with constant holdups. For this case, the momentum equations become... 

For batch, isothermal polymerizations, the principle of conservation of population yields for alcohol adducts... 

Equation (37) is the quantum statistical analogue of Liouville s equation. To find the quantum analogue of the classical principle of conservation of phase density the solution to (37) is written in the form... 

As can be seen, the process does not violate the principle of conservation of charge. 

It should be noted that equation 7.25 represents the interaction of forces acting on bubbles in a swarm while equation 7.24 represents the principle of conservation of mass (continuity). Both equations must be satisfied simultaneously. 

The first law of thermodynamics puts forward the principle of conservation of ener. Written for a general open system (where flow of material in and out of the system can occur) it is... 

Once the "synthesis tree" has been elaborated, we must proceed to the evaluation of the alternative pathways and compare them with possible synthetic schemes in order to optimise the chosen route and make it as self-consistent as possible. However, all synthetic plans must be flexible enough to allow new alternative solutions when things do not happen as anticipated. In this sense. Woodward referred very often to opportunism and of taking advantage of the "surprises" which may occur during the execution of a synthesis. Through the different stages of a synthesis new aspects may evolve and even important discoveries may be made. Such was the case, for instance, in the vitamin B12 synthesis in which the considerations of the stereochemistry of an intermediate, opposite to the one anticipated, led Woodward to the discovery of the principle of conservation of orbital symmetry [29]. 

Eighteenth-century and early nineteenth-century views of the nature of heat were founded on the principle of conservation of caloric. This principle is an eminently attractive basis for rationalizing simple observations such as temperature changes that occur when a cold object is placed in contact with a hot one. The cold object seems to have extracted something (caloric) from the hot one. Furthermore,... 

One hundred fifty years ago, the two classic laws of thermodynamics were formulated independently by Kelvin and by Clausius, essentially by making the Carnot theorem and the Joule-Mayer-Helmholtz principle of conservation of energy concordant with each other. At first the physicists of the middle 1800s focused primarily on heat engines, in part because of the pressing need for efficient sources of power. At that time, chemists, who are rarely at ease with the calculus, shied away from... 

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