Chemical equations fractional numbers

Although normally the coefficients in a balanced chemical equation are the smallest possible whole numbers, a chemical equation can be multiplied through by a factor and still be a valid equation. At times it is convenient to use fractional coefficients for example, we could write... 

This is how we know how to balance chemical equations and work out the masses reacting together, because we know that equal molar masses contain the same numbers of particles. In a chemical equation like C + 02 —> C02, we always know that there will be one mole of carbon atoms reacting with one mole of oxygen molecules (02) to give one mole of carbon dioxide. Providing the chemical equation balances, any fraction of moles will also be true. One-tenth of a mole of carbon will require one-tenth of a mole of oxygen gas molecules (02), to make one-tenth of a mole of carbon dioxide gas (C02). 

This equilibrium can be easily studied in the laboratory through a measurement of the vapor density of the equilibrium mixture. In the following formulation the various quantities are listed in columns under the formulas of the compounds in the balanced chemical equation. Let rf be the initial number of moles of N2O4, the equilibrium advancement, and the fraction dissociated at equilibrium = Jn°. 

Mole fraction the ratio of the number of moles of a given component ina mixture to the total number of moles in the mixture. (5.5 11.1) Mole ratio (stoichiometry) the ratio of moles of one substance to moles of another substance in a balanced chemical equation. (3.9) Molecular formula the exact formula of a molecule, giving the types of atoms and the number of each type. (3.6)... 

This property of q is best seen in the equation Parr and I wrote for the equilibration of two systems, C and D. Electrons will flow from the one of lower electronegativity to that of higher electronegativity until a single chemical potential exists. The (fractional) number of electrons transferred is given by ... 

Because the combustion of 1 mol of CH4 with 2 mol of O2 releases 890 kJ of heat, the combustion of 2 mol of CH4 with 4 mol of O2 releases twice as much heat, 1780 kJ. Although chemical equations are usually written with whole-number coefficients, thermochemical equations sometimes utilize fractions, as in the preceding Give It Some Thought Question. 

Because all of the resulting coefficient values are integers, we have found the smallest whole number coefficients that balance the equation, and our result is identical to that found in Example 3.1. If any fractional coefficients were obtained, we would simply multiply through by a constant to get whole numbers. This approach is quite general, and although it is not particularly popular with chemists, it has been used to write computer algorithms for balancing chemical equations. 

The molar interpretation of a chemical equation involves reading the coefficients as the number of moles of the reactants and products. This is still a particulate-level explanation, but we are grouping the particles into counting units that make it easier to translate into a macroscopic-level interpretation. On the molar level, fractional coefficients are acceptable. H2(g) + V2 02(g) H20(g) can be read as one mole of... 

Now you have the same number ( 1 ) of Cl atoms on both sides. Therefore, this is a correct chemical reaction equation. However, the fractional number (in front of a chemical) should be avoided. So you can multiply the whole quotation by2, and will get ... 

About 15 years ago, Deal, Derr and Wilson developed the ASOG group-contribution method based on Wilson s equation where the important composition variables are not the mole fractions of the components but the mole fractions of the functional groups (16, 17). In chemical technology the number of different functional groups is much smaller than the number of molecular species therefore, the group-contribution method provides a very powerful scale-up tool. 

Let s suppose we were producing one molecule of water. In this case, there would be one oxygen atom on the right-hand side of the equation, and to have a balanced chemical equation would require that we have one oxygen atom on the left side. But diatomic oxygen is shown on the left side, and it is not possible to have fractional numbers of molecules. So the smallest number of oxygen molecules on the left side would be one. Based on this analysis, let s tentatively state that m = 1. 

Equation (A2.1.23) can be mtegrated by the following trick One keeps T, p, and all the chemical potentials p. constant and increases the number of moles n. of each species by an amount n. d where d is the same fractional increment for each. Obviously one is increasing the size of the system by a factor (1 + dQ, increasing all the extensive properties U, S, V, nl) by this factor and leaving the relative compositions (as measured by the mole fractions) and all other intensive properties unchanged. Therefore, d.S =. S d, V=V d, dn. = n. d, etc, and... 

For a PVnr system of uniform T and P containing N species and 7T phases at thermodynamic equiUbrium, the intensive state of the system is fully deterrnined by the values of T, P, and the (N — 1) independent mole fractions for each of the equiUbrium phases. The total number of these variables is then 2 + 7t N — 1). The independent equations defining or constraining the equiUbrium state are of three types equations 218 or 219 of phase-equiUbrium, N 7t — 1) in number equation 245 of chemical reaction equiUbrium, r in number and equations of special constraint, s in number. The total number of these equations is A(7t — 1) + r -H 5. The number of equations of reaction equiUbrium r is the number of independent chemical reactions, and may be deterrnined by a systematic procedure (6). Special constraints arise when conditions are imposed, such as forming the system from particular species, which allow one or more additional equations to be written connecting the phase-rule variables (6). 

For each stage J, the following 2C -1- 3 component material-balance (M), phase-equilibrium (E), mole-fraction-summation (S), and energy-balance (H) equations apply, where C is the number of chemical species ... 

For this expression to be valid, in cell A components 1 and 2 must be identical in all respects, so it is a rather special case of an ideal mixture. They are however, allowed to interact differently with the membrane, as described above, xa is the mole fraction of the solute in cell A, while p and p are the number densities of cells A and B respectively. The method was extensively tested against both Monte Carlo and equations of state for LJ particles, and the values of the chemical potential were found to be satisfactory. The method can also be extended to mixtures [29] by making... 

Equation 13 has an important implication a clathrate behaves as an ideally dilute solution insofar as the chemical potential of the solvent is independent of the nature of the solutes and is uniquely determined by the total solute concentrations 2K yK1.. . 2x yKn in the different types of cavities. For a clathrate with one type of cavity the reverse is also true for a given value of fjiq (e.g. given concentration of Q in a liquid solution from which the clathrate is being crystallized) the fraction of cavities occupied 2kVk s uniquely determined by Eq. 13. When there are several types of cavities, however, this is no longer so since the individual occupation numbers 2k2/ki . ..,2k yKn, and hence the total solute concentration... 

In these equations the independent variable x is the distance normal to the disk surface. The dependent variables are the velocities, the temperature T, and the species mass fractions Tit. The axial velocity is u, and the radial and circumferential velocities are scaled by the radius as F = vjr and W = wjr. The viscosity and thermal conductivity are given by /x and A. The chemical production rate cOjt is presumed to result from a system of elementary chemical reactions that proceed according to the law of mass action, and Kg is the number of gas-phase species. Equation (10) is not solved for the carrier gas mass fraction, which is determined by ensuring that the mass fractions sum to one. An Arrhenius rate expression is presumed for each of the elementary reaction steps. 

Bimolecular processes are the primary vehicle by which chemical change occurs. The frequency with which these encounters occur is given by equations 4.3.1 and 4.3.4. However, only an extremely small number of the collisions actually lead to reaction for predictive purposes one needs to know what fraction of the collisions are effective in that they lead to reaction. 

The example reactions considered in this section all have the property that the number of reactions is less than or equal to the number of chemical species. Thus, they are examples of so-called simple chemistry (Fox, 2003) for which it is always possible to rewrite the transport equations in terms of the mixture fraction and a set of reaction-progress variables where each reaction-progress variablereaction-progress variable —> depends on only one reaction. For chemical mechanisms where the number of reactions is larger than the number of species, it is still possible to decompose the concentration vector into three subspaces (i) conserved-constant scalars (whose values are null everywhere), (ii) a mixture-fraction vector, and (iii) a reaction-progress vector. Nevertheless, most commercial CFD codes do not use such decompositions and, instead, solve directly for the mass fractions of the chemical species. We will thus look next at methods for treating detailed chemistry expressed in terms of a set of elementary reaction steps, a thermodynamic database for the species, and chemical rate expressions for each reaction step (Fox, 2003). 

For problems involving gradients in chemical species, the convection-diffusion equations for the species are also solved, usually for N— 1 species with the Nth species obtained by forcing the mass fractions to sum to unity. Turbulence can be described by a turbulent diffusivity and a turbulent Schmidt number, Sct, analogous to the heat transfer case. 

This form of the partition coefficient, analogous to that used for Fe-Mg fractionation between olivine and melt (see Chapter 1), is necessary only for the rare cases where trace substitution affects Cj and Cp substantially. A number of reviews (O Nions and Powell, 1977 Michard, 1989) describe the various sorts of partition coefficients expressed either in mass-fractions, atom fractions, or normalized to a major element and their respective merits. If the discussion is restricted to a narrow range of chemical compositions (e.g., basaltic systems, Irving, 1978, Irving and Frey, 1984), enough experimental information exists on trace-element partitioning to resort to the wonderfully simple equation (9.1.1). 

In the case of classic chemical kinetics equations, one can get in a few cases analytical solution for the set of differential equations in the form of explicit expressions for the number or weight fractions of i-mcrs (cf. also treatment of distribution of an ideal hyperbranched polymer). Alternatively, the distribution is stored in the form of generating functions from which the moments of the distribution can be extracted. In the latter case, when the rate constant is not directly proportional to number of unreacted functional groups, or the mass action law are not obeyed, Monte-Carlo simulation techniques can be used (cf. e.g. [2,3,47-52]). This technique was also used for simulation of distribution of hyperbranched polymers [21, 51, 52],... 

STANJAN The Element Potential Method for Chemical Equilibrium Analysis Implementation in the Interactive Program STANJAN, W.C. Reynolds, Thermosciences Division, Department of Mechanical Engineering, Stanford University, Stanford, CA, 1986. A computer program for IBM PC and compatibles for making chemical equilibrium calculations in an interactive environment. The equilibrium calculations use a version of the method of element potentials in which exact equations for the gas-phase mole fractions are derived in terms of Lagrange multipliers associated with the atomic constraints. The Lagrange multipliers (the element potentials ) and the total number of moles are adjusted to meet the constraints and to render the sum of mole fractions unity. If condensed phases are present, their populations also are adjusted to achieve phase equilibrium. However, the condensed-phase species need not be present in the gas-phase, and this enables the method to deal with problems in which the gas-phase mole fraction of a condensed-phase species is extremely low, as with the formation of carbon particulates. 

Mossbauer spectroscopy involves the measurement of minute frequency shifts in the resonant gamma-ray absorption cross-section of a target nucleus (most commonly Fe occasionally Sn, Au, and a few others) embedded in a solid material. Because Mossbauer spectroscopy directly probes the chemical properties of the target nucleus, it is ideally suited to studies of complex materials and Fe-poor solid solutions. Mossbauer studies are commonly used to infer properties like oxidation states and coordination number at the site occupied by the target atom (Flawthome 1988). Mossbauer-based fractionation models are based on an extension of Equations (4) and (5) (Bigeleisen and Mayer 1947), which relate a to either sums of squares of vibrational frequencies or a sum of force constants. In the Polyakov (1997)... 

The chemical shift and line width observed for each water content were taken as weighted averages of the values in the free and bound states, and from two equations expressing these averages, Fb and Ff, the mole fractions of bound and free Na+ ions, respectively. were extracted. Fb significantly increases as the approximate hydration number that might be expected for a SOs Na pair is approached from the direction of considerable hydration. 

Figure 9.38, for example, shows the application of the chemical mass balance approach to the fine particle fraction of particles collected at a location in Philadelphia (Dzubay et at., 1988 Olmez et al., 1988 Gordon, 1988). If the set of equations (II) fitted the data perfectly, the sum of the contributions of the various sources would be 100% for each element. Clearly, from the top frame, this is not the case for a number of elements, and both positive and negative deviations from 100% can be seen. However, the contributions of several sources are clear Si and Fe from soil, Ni, V, and Ca from oil-fired power plants, Ti from a paint pigment plant, La, Ce, and Sm from a catalytic cracker, K, Zn, and Sn from an incinerator, Sb from an antimony roaster, and Pb and Br from motor vehicles. 

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