Components, number of independent

A balance equation can be written for each independent component. Not all the components in a material balance will be independent. 

If there is no chemical reaction the number of independent components is equal to the number of distinct chemical species present. 

Consider the production of a nitration acid by mixing 70 per cent nitric and 98 per cent sulphuric acid. The number of distinct chemical species is 3 water, sulphuric acid, nitric acid. 

If the process involves chemical reaction the number of independent components will not necessarily be equal to the number of chemical species, as some may be related by the chemical equation. In this situation the number of independent components can be calculated by the following relationship  

Number of independent components = Number of chemical species — 

Time Choose the time basis in which the results are to be presented for example kg/h, metric tons/y, unless this leads to very large or very small numbers, where rounding errors can become problematic. 

Choose as the mass basis the stream flow for which most information is given. 

It is often easier to work in moles, rather than weight, even when no reaction is involved. 

For gases, if the compositions are given by volume, use a molar basis, remembering that volume fractions are equivalent to mole fractions up to moderate pressures. 

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Having phases together in equilibrium restricts the number of thermodynamic variables that can be varied independently and still maintain equilibrium. An expression known as the Gibbs phase rule relates the number of independent components C and number of phases P to the number of variables that can be changed independently. This number, known as the degrees of freedom f is equal to the number of independent variables present in the system minus the number of equations of constraint between the variables. 

For three-component (C = 3) or ternary systems the Gibbs phase rule reads Ph + F = C + 2 = 5. In the simplest case the components of the system are three elements, but a ternary system may for example also have three oxides or fluorides as components. As a rule of thumb the number of independent components in a system can be determined by the number of elements in the system. If the oxidation state of all elements are equal in all phases, the number of components is reduced by 1. The Gibbs phase rule implies that five phases will coexist in invariant phase equilibria, four in univariant and three in divariant phase equilibria. With only a single phase present F = 4, and the equilibrium state of a ternary system can only be represented graphically by reducing the number of intensive variables. 

The number of independent components, c, in a given system of interest can generally be evaluated as the total number of chemical species minus the number of relationships between concentrations. The latter may consist of initial conditions (defined by conditions of preparation of the system) or by chemical equilibrium conditions (for chemical reactions that are active in the actual system). Sidebar 7.1 provides illustrative examples of how c is determined in representative cases. 

It is possible to uncouple the expressions for the fluxes by diagonalizing the diffu-sivity matrix through a coordinate transformation. The transformed interdiffusivity matrix will have eigenvalues t as its diagonal entries. For the ternary system, the eigenvalues are the A from Eq. 6.16. There will be N positive eigenvalues Aw in the general case, where N is the number of independent components. 

In terms of the number of independent components, C, Equation (1) may also be written... 

Let the number of basic overall equations be Q it is evident that Q< P. Let us denote the number of substances participating in the reaction as M and the number of independent components, in the sense this notion is used in the Gibbs phase rule, as C then... 

One of the great virtues of the FREZCHEM model is its ability to examine complex chemistries. The number of independent components for the systems examined in this chapter range from four to eight. Earth seawater consisting of Na+, K+, Mg2+, Ca2+, Cl-, SO4-, and alkalinity has seven independent components (six salts and water). The most complex system evaluated is the snowball Earth seawater (eight independent components), which in addition to the above seven components also includes Fe2+. This ability to cope with complexity makes models like FREZCHEM more realistic in describing natural systems than simpler binary and ternary diagrams we demonstrate this point with data from Don Juan Pond, the most saline body of water on Earth. 

Table A.2 is model output for seawater freezing at 253.15 K. Beneath the title, the output includes temperature, ionic strength, density of the solution (p), osmotic coefficient amount of unfrozen water, amount of ice, and pressure on the system. Beneath this line are the solution and gaseous species in the system. The seven columns include species identification, initial concentration, final (equilibrium) concentration, activity coefficient, activity, moles in the solution phase, and mass balance. The mass balance column only contains those components for which a mass balance is maintained. The number of these components minus 1 is generally the number of independent components in the system (in this case, 8 — 1 = 7). The mass balances (col. 7) should equal the initial concentrations (col. 2). This mass balance comparison is a good check on the computational accuracy.

At 263.15 K, the calculated pH is 0.02 this is just above the point where the pH dips into the negative range (Fig. 5.24). Because acidity (H+) is specified as input (0.515 m), the system maintains a mass balance for H+ (Table A.3) this is in contrast to the other examples (Tables A.2, A.4, and A.5) where the mass of H+ is unconstrained. Note that a significant amount of solution H is present as H+ and HSOJ. The number of independent components in this case (4) is again equal to the components under Mass balance (5) - 1. 

The number of components under Mass balance is nine, which means eight independent components for this system. Had we specified a closed carbon system for both CO2 and CH4, then the total number of independent components would have been ten instead of eight because the masses associated with CO2 and CH4 would then be fixed. 

Table A.5 is the output file for salts in the 4.5- to 5.0-km layer, where the system pressure is 484.5 bars (102 bars km-1 x 4.75 km). The temperature of 268.28 K is the freezing point depression for this particular composition and pressure at 268.27 K, ice forms. The pH of this system is 8.02. The number of independent components is seven. This example deals with lithostatic pressures on solutions dispersed in a regolith, which is fundamentally different from the previous examples (Tables A.2-A.4) that dealt with seawaters.

This relation is called the Gibbs phase rule. However, as indicated next, c should be limited to the number of independent components. 

Usually, the combination c — r — a is called the number of independent components, cind, and the phase rule is written as... 

Solution There are three components in a single phase. One chemical reaction, 2NO + Br2 2NOBr, reduces the number of independent components to two. The number of degrees of freedom is 2 — 1 + 2 = 3, one of which is the fixed temperature of 298 K. The other two can be taken as the ratio of the initial pressure of NO to that of Br2 and the total final pressure. First, we calculate the equilibrium constant ... 

Solution There are four components in the system related by one chemical equation. There is also a stoichiometric equation resulting from the fact that all of the hydrogen and oxygen come from water and, therefore, Pco/Pcu4 = 2- The number of independent components is therefore 4 — 2 = 2. With two phases, there are two degrees of freedom. These can be taken as the temperature and pressure of the system. A table is formed as in the two previous examples. (Graphite, a solid, is assumed to have unit activity and does not enter into the equilibrium constant.) The initial amount of water is arbitrarily chosen as 1.0 mol. (An extensive variable characterizing the amount of gas phase can be specified.)... 

In this formulation, N is the number of independent components, Nr the number of independent reactions, z and t are dimensionless space and time according to... 

The concept of composition is an important one. There are many alternative ways of expressing the same idea, that of rank being popular also, which derives from matrices ideally the rank of a matrix equals the number of independent components or nonzero... 

The number of independent components in a phase is the least number of substances whose mole numbers must be specified... 

As discussed in Section 6.1, the number of independent components needing to be specified is reduced by the crystal symmetry. If, for example, the x axis is taken as the [1 0 0] direction, the y axis as the [0 1 0], and the z axis as the [1 0 0], a cubic crystal or a polycrystalline sample with a random crystallite orientation gives ... 

The Cartesian indices refer to an arbitrarily chosen laboratory frame. For certain NLO processes intrinsic permutation symmetry can be used to reduce further the number of independent components. In the case of the Kerr susceptibility, (-w w,0,0), intrinsic permutation symmetry in the last two indices holds, xltJ zx X xx- The most general Kerr susceptibility of an isotropic medium therefore has only two independent components, x9 zz and x9 xx Likewise, the EFISHG susceptibility (-2w w, w,0), important for the evaluation of second-order molecular polarizabilities in solution (see pp. 158 and 162), has only two independent components, x zz and x9]txz, because of intrinsic permutation symmetry in the second and third indices. 

Other important systems are uniaxial isotropic systems, because the widely studied poled polymers belong to this symmetry class (o°w or Co ). of such systems has seven non-vanishing components of which four are independent, = X k, x9k = xSk = X lx and x9k- For the SHG susceptibility X -2u> o),u)) the number of independent components reduces to three because of intrinsic permutation symmetry in the second and third index. If the uniaxial system is created by poling of an isotropic system by an external electric field, e.g. a poled polymer or liquid, then to first order in the applied field, z, the number of independent components of x -2a> w, w) is only two (Kielich, 1968). It is thus equal to the number of independent components of x -2a) a),o),0) because of (28). 

The number of independent components may be further reduced by intrinsic or Kleinman symmetry (cf. p. 131). Intrinsic permutation symmetry, /3 , = holds for the second-order polarizability /3 ,(-2w Kleinman symmetry, i.e. permutation symmetry in all Cartesian indices (cf. p. 131), generally holds only in the limit w—>0. 

The Cartesian indices refer to an arbitrarily chosen laboratory frame. For certain NLO processes intrinsic permutation symmetry can be used to reduce further the number of independent components. In the case of the Kerr susceptibility, w,0,0), intrinsic permutation symmetry in the last two... 

The number of independent components may be further reduced by intrinsic or Kleinman symmetry (cf. p. 131). Intrinsic permutation symmetry, = rsi, holds for the second-order polarizability in the second and... 

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