Chemical equations fractional coefficients

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... 

Figure 9.16 Kinetic fractionation during crystal growth. Steady-state distribution of melt concentrations in the vicinity of a solid growing at the rate v for trace elements with different solid-liquid fractionation coefficients [equation (9.6.5), Tiller et al. (1953)]. The stippled area indicates the steady-state chemical boundary-layer with thickness <5 = <5>/v.

This reaction quotient is a fraction. The numerator is the product of the chemical species on the right hand side of the equilibrium arrow, each one raised to the power of that species coefficient in the balanced chemical equation. The denominator is the product of the chemical species on the left hand side of the equilibrium arrow, each one raised to the power of that species coefficient in the balanced chemical equation. It is called Qc, in this case, since molar concentrations are being used. If this was a gas phase reaction, gas pressures could be used and it would become a Qp. 

Generally, it is best to let xrepresent the substance with the smallest coefficient in the chemical equation. This helps to avoid fractional values of x in the equilibrium expression. Fractional values make solving the expression more difficult. 

Because the coefficients of a balanced chemical equation can represent moles, it is acceptable to use fractions in an equation. For example, you can write the equation 2H2(g) + 02(g) - 2H20w as... 

Explain why it is acceptable to use fractional coefficients in a balanced chemical equation. 

This fact can be demonstrated as follows. Let us determine the value of the well-known Flory parameter x, which corresponds to the 6 point (i.e. to the point of inversion of the second virial coefficient of the solution of rods) in the Flory theory of Ref.9). This can be done by expanding the chemical potential of the solvent in the isotropic phase (Eq. (16) of Ref.9 ) into powers of the polymer volume fraction in the solution, and by equating the coefficient at the quadratic term of this expansion to zero this procedure gives Xe = 1/2 independently of p. On the other hand, it is well known26,27) that the value of x decreases with increasing p and that X < 1 at p > 1. The contradiction obtained shows that the expressions for the thermodynamic functions used in Ref.9) are not always correct... 

The definition of A/f of a substance refers to a reaction in which me mole of the substance is formed. We put one mole of C2H50H(f) on the right side of the chemical equation and put the appropriate elements in their standard states on the left. We balance the equation without chan ng the coefficient of the product, even if we must use fractional coefficients on the left. 

Can the subscripts in a chemical formula be fractions Explain. Can the coefficients in a balanced chemical equation be fractions Explain. Changing the subscripts of chemicals can balance the equations mathematically. Why is this unacceptable ... 

In the case of chemical reactions this coefficient describes the same ratio of the element isotopes before reaction (in the reagent) and after (in products). For instance, at moisture evaporation oxygen fractionating coefficient is determined by the equation... 

In this equation, which is known as the Darken equation, X indicates the mole fraction of Co or of Ni. The equation assumes local equilibrium everywhere and that D is a chemical or interdiffusion coefficient in a chemical potential gradient. The matrix is Co Ni O. D is plotted as a function of concentration for both Ni and Co diffusing in the mixed oxide at 1300°C and 1445°C in Figure 25.15a. 

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. 

Although this is considered a properly balanced equation, many chemists do not ordinarily use fractional coefficients in chemical equations. To convert the fractional coefficient into an integer value, one simply multiplies the entire balanced equation by the proper integer value to convert the fractional value into the smallest integer value. In this case, one would multiply the entire equation by a factor of 2 to produce the final balanced equation with all integer coefficients ... 

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... 

In the mid-nineteenth century, Cato Guldberg and Peter Waage studied the equilibrium mixtures of a wide variety of chemical reactions. They observed that at a constant temperature in an equilibrium mixture of reactants and products regardless of the initial concentrations, the reaction quotient has a constant value. The reaction quotient (QJ is a fraction with product concentrations in the numerator and reactant concentrations in the denominator—with each concentration raised to a power equal to the corresponding stoichiometric coefficient in the balanced chemical equation. For the general reaction. 

The effective MTC is the group of terms within the braces in Equation 13.11. The concentration differences across the mixed surface layer of thickness h (m), is the interface vapor-phase concentration, Cai(mg/m ), minus the vapor-phase concentration at z = h, Cah (mg/m ). In addition to the kinetic transport parameter Dbs/h, the MTC contains the thermodynamic parameter ratio Kd/H, which imparts the sorbed-phase chemical loading fraction characteristic of the solid particles. When mobilized by the macrofauna this fraction significantly enhances the magnitude of the MTC. For certain strongly sorbed chemicals the coefficient can be very large so that this mechanism dominates the rate of chemical movement from within the soil layers to the soil-air interface. See Example 13.6.1 below for numerical verification of this behavior. 

Once all reactants and products have been added, the correct chemical equation should be displayed at the bottom of the window, and the equation status should read Balanced. If not, select a reactant or product and choose Edit to adjust the stoichiometric coefficient. Fractions maybe used for stoichiometric coefficients. Finally, enter a temperature, and the corresponding equilibrium constant will be displayed. 

If we had chosen to describe composition in terms of elements, we would need to carry the elemental compositions of all species, minerals, and gases, as well as the coefficients of the independent chemical reactions. Our choice of components, however, allows us to store only one array of reaction coefficients, thereby reducing memory use on the computer and simplifying the forms of the governing equations and their solution. In fact, it is possible to build a complete chemical model (excluding isotope fractionation) without acknowledging the existence of elements in the first place ... 

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). 

No heat is evolved when pure components that form an ideal solution are mixed. The validity of this statement can be shown from consideration of the temperature coefficient of the chemical potential. Again, from Equation (14.6) at fixed mole fraction. 

In these equations is the partial molal free energy (chemical potential) and Vj the partial molal volume. The Mj are the molecular weights, c is the concentration in moles per liter, p is the mass density, and z, is the mole fraction of species i. The D are the multicomponent diffusion coefficients, and the are the multicomponent thermal diffusion coefficients. The first contribution to the mass flux—that due to the concentration gradients—is seen to depend in a complicated way on the chemical potentials of all the components present. It is shown in the next section how this expression reduces to the usual expressions for the mass flux in two-component systems. The pressure diffusion contribution to the mass flux is quite small and has thus far been studied only slightly it is considered in Sec. IV,A,6. The forced diffusion term is important in ionic systems (C3, Chapter 18 K4) if gravity is the only external force, then this term vanishes identically. The thermal diffusion term is impor-... 

---