Independent components, number

Number of independent components = Number of chemical species —... 

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. 

Binary electrolyte solutions contain just one solute in addition to the solvent (i.e., two independent components in all). Multicomponent solutions contain several original solutes and the corresponding number of ions. Sometimes in multicomponent solutions the behavior of just one of the components is of interest in this case the term base electrolyte is used for the set of remaining solution components. Often, a base electrolyte is acmaUy added to the solutions to raise their conductivity. 

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

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 ... 

Consider the general material balance problem where there are Ns streams each containing Nc independent components. Then the number of variables, N, is given by ... 

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. 

However, care must be taken to avoid the singularity that occurs when C is not full rank. In general, the rank of C will be equal to the number of random variables needed to define the joint PDF. Likewise, its rank deficiency will be equal to the number of random variables that can be expressed as linear functions of other random variables. Thus, the covariance matrix can be used to decompose the composition vector into its linearly independent and linearly dependent components. The joint PDF of the linearly independent components can then be approximated by (5.332). 

Theorem 1 It is necessary and sufficient for qualitative completeness of x that X contain zr independent components transforming according to each IR /W of 6, where zr is the number of times TW is contained in /W(S) . It is also necessary and sufficient for qualitative completeness that the representation of S induced by % is just [/ < >( )]s, i.e., that the induction is regular. 

Note that is a negative number, so that v+ z+ + v = 0.) The electroneutrality principle is equivalent to stating that it is impossible to produce a solution that contains, for example, only cations or an excess of positive charge (i.e., that ions cannot be considered as independent components of solutions). Only entire electrolytes are components that can be added to a solution. 

Molecular multipole components are best described in the molecular (i.e., body-fixed) frame x, y, z. For example, a dipole when aligned with the z-axis is characterized by a single number, the strength p of the dipole. The other two independent components can then simply be expressed in terms of Euler angles or, in this case, of azimuthal and polar angle, q> and 3, between molecular (x, y, z) and laboratory-fixed frame (X, Y, Z). 

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... 

For tensors of higher rank we must ensure that the bases are properly normalized and remain so under the unitary transformations that correspond to proper or improper rotations. For a symmetric T(2) the six independent components transform like binary products. There is only one way of writing xx xx, but since xx x2 = x2 xx the factors xx and x2 may be combined in two equivalent ways. For the bases to remain normalized under unitary transformations the square of the normalization factor N for each tensor component is the number of combinations of the suffices in that particular product. F or binary products of two unlike factors this number is two (namely ij and ji) and so N2 = 2 and x, x appears as /2x,- Xj. The properly normalized orthogonal basis transforming like... 

The mole numbers in Equation (5.63) could represent the species present in the system, rather than the components. However, in such a case they are not all independent. For each independent chemical reaction taking place within the system, a relation given by Equation (5.46) must be satisfied. There would thus be (S — R) independent mole numbers for the system if S is the number of species and R is the number of independent chemical reactions. In fact, this relation may be used to define the number of components in the system [6]. If the species are ions, an equation expressing the electroneutrality of the system is another condition equation relating the mole numbers of the species. The total number of components in this case is C = S — R — 1. 

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. 

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