Equilibrium, chemical conditions

Sz(i) being the total angular momentum of the i spins. The linear combination 5 (l)/a1 + Sz(2)jx2 is now coupled to the dipole-dipole heat bath, whereas the difference Sz(l)/x1 — Sz(2)jx2 remains constant. It may be shown that the chemical equilibrium condition (31) is then equivalent to an equalization of the temperatures of the collective coordinate Sz(l)jx1 + Sz(2)/x2 and of the dipole-dipole heat bath. 

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. 

The systematic procedure is to write as many independent algebraic equations as there are unknowns (species) in the problem. The equations are generated by writing all the chemical equilibrium conditions plus two more the balances of charge and of mass. There is only one charge balance in a given system, but there could be several mass balances. 

It will be assumed that the rates of formation and dissociation of the different complexes (k, k, t2, and k"2) are sufficiently fast in comparison to the diffusion rate so that the complex formations and dissociations are at equilibrium even when current is flowing, i.e., that chemical equilibrium conditions are maintained at any position and time of the experiment [90] ... 

As mentioned in Chapter 2, it is sometimes convenient to use the RGIBBS reactor model instead of the RCSTR model illustrated above. If the reactions are known to be reversible and the forward and reverse reaction rates are fast, a reasonable residence time in the reactor will produce an exit stream with chemical equilibrium conditions. In... 

Section 4.3 elucidates the role of vapor-liquid mass transfer resistances on the feasible products of nonreactive or reactive separation processes. The latter are considered under chemical equilibrium conditions (i.e., they are very fast reactions). The feasible products are denoted as arheotropes. 

In the following, component A is numbered as 1, component B as 2, and component C as 3. At chemical equilibrium conditions, the following relationship holds ... 

In the reactive case, r is not equal to zero. Then, Eq. (3) represents a nonhmoge-neous system of first-order quasilinear partial differential equations and the theory is becoming more involved. However, the chemical reactions are often rather fast, so that chemical equilibrium in addition to phase equilibrium can be assumed. The chemical equilibrium conditions represent Nr algebraic constraints which reduce the dynamic degrees of freedom of the system in Eq. (3) to N - Nr. In the limit of reaction equilibrium the kinetic rate expressions for the reaction rates become indeterminate and must be eliminated from the balance equations (Eq. (3)). Since the model Eqs. (3) are linear in the reaction rates, this is always possible. Following the ideas in Ref. [41], this is achieved by choosing the first Nr equations of Eq. (3) as reference. The reference equations are solved for the unknown reaction rates and afterwards substituted into the remaining N - Nr equations. 

From Eqn. (14) it follows that with an exothermic reaction - and this is the case for most reactions in reactive absorption processes - decreases with increasing temperature. The electrolyte solution chemistry involves a variety of chemical reactions in the liquid phase, for example, complete dissociation of strong electrolytes, partial dissociation of weak electrolytes, reactions among ionic species, and complex ion formation. These reactions occur very rapidly, and hence, chemical equilibrium conditions are often assumed. Therefore, for electrolyte systems, chemical equilibrium calculations are of special importance. Concentration or activity-based reaction equilibrium constants as functions of temperature can be found in the literature [50]. 

Hence the thermal, mechanical, and chemical equilibrium conditions in terms of the intensive properties are... 

The RISM integral equations in the KH approximation lead to closed analytical expressions for the free energy and its derivatives [29-31]. Likewise, the KHM approximation (7) possesses an exact differential of the free energy. Note that the solvation chemical potential for the MSA or PY closures is not available in a closed form and depends on a path of the thermodynamic integration. With the analytical expressions for the chemical potential and the pressure, the phase coexistence envelope of molecular fluid can be localized directly by solving the mechanical and chemical equilibrium conditions. 

In order to linearize equations (87)-(90), for any dependent variable /appearing in these equations, we shall set / = /o + /, where /o is a constant (independent of x and t) that corresponds to quiescent chemical equilibrium conditions and / is small. Since the quiescent value Vq of the velocity v is 1 0 = 0, we have v = v and we shall therefore omit the prime on the small quantity v. Neglecting terms of order higher than the first in / and in each of its derivatives, we then find that equations (87), (88) for i = 2, (89), and (90) become, respectively,... 

In Section A.l, the general laws of thermodynamics are stated. The results of statistical mechanics of ideal gases are summarized in Section A.2. Chemical equilibrium conditions for phase transitions and for reactions in gases (real and ideal) and in condensed phases (real and ideal) are derived in Section A.3, where methods for computing equilibrium compositions are indicated. In Section A.4 heats of reaction are defined, methods for obtaining heats of reaction are outlined, and adiabatic flame-temperature calculations are discussed. In the final section (Section A.5), which is concerned with condensed phases, the phase rule is derived, dependences of the vapor pressure and of the boiling point on composition in binary mixtures are analyzed, and properties related to osmotic pressure are discussed. 

For nonideal gases the above simplifications are not applicable and it would appear to be best to use equation (21) directly as the chemical equilibrium condition. However, it is conventional when dealing with nonideal gases to replace p,- by the fugacity defined below. By solving equation (15) for Ni and using equation (25) to express Ni in terms of p, it is found that... 

In systems where heterogeneous chemical equilibria prevail, both chemical and phase equilibrium conditions must be simultaneously satisfied. In practice, this means that the chemical equilibrium condition—Eq. (51) in the discrete description, and Eq. (60) in the continuous one—must be satisfied in one phase, and the phase equilibrium condition [/( or fi(x) to be the same in all phases] must be satisfied this clearly guarantees that the chemical equilibrium condition is automatically satisfied in all phases. 

Once this interpretation has been established, MODEL.LA. (a) generates all the requisite modeling elements and (b) constructs the modeling relationships, such as material balances, energy balance, heat transfer between jacket and reactive mixture, mass transport between the two liquid phases, equilibrium relationships between the two phases, estimation of chemical reaction rate, estimation of chemical equilibrium conditions, estimation of heat generated (or consumed) by the reaction, and estimation of enthalpies of material convective flows. In order to automate the above tasks, MODEL.LA. must possess the following capabilities ... 

If the reaction kinetics are expressed in concentrations only, an ideal solution is assumed. Furthermore, under chemical equilibrium conditions the two reaction rates (to products and from products) should be the same. Because the chemical equilibrium constant K must be expressed in activity coefficients, an inconsistency might occur. 

Chemical equilibrium conditions are assumed for all reactions. The chemical model consists of eight components UOj"1", VO4-, CO2-, K+, Ca2+, H+, and HFO as the sorbent. Four minerals, carnotite (K2(U02)2(V04)2), tyuyamunite (Ca(U02)2(V04)2), calcite (CaC03), and gypsum (CaS04 2H2O), are allowed to participate in precipitation-dissolution reactions. The detailed chemical model (reactions and parameters) is presented in Morrison et al. (1995a). No ionic strength correction was made for activity coefficients. 

For high Da the column is dose to chemical equilibrium and behaves very similar to a non-RD column with n -n -l components. This is due to the fact that the chemical equilibrium conditions reduce the dynamic degrees of freedom by tip the number of reversible reactions in chemical equilibrium. In fact, a rigorous analysis [52] for a column model assuming an ideal mixture, chemical equilibrium and kinetically controlled mass transfer with a diagonal matrix of transport coefficients shows that there are n -rip- 1 constant pattern fronts connecting two pinches in the space of transformed coordinates [108]. The propagation velocity is computed as in the case of non-reactive systems if the physical concentrations are replaced by the transformed concentrations. In contrast to non-RD, the wave type will depend on the properties of the vapor-liquid and the reaction equilibrium as well as of the mass transfer law. 

With the chemical potential and pressure obtained in the form of the closed expressions (4.A.9) and (4.A.11) in Chapter 4, the phase coexistence envelope can be localized directly by solving the mechanical and chemical equilibrium conditions (1.134) and (1.135) for the vapor and liquid phase densities, Pvap and puq, whether or not the solution exists for all intermediate densities. Provided the isotherm is continuous across all the region of vapor-liquid phase coexistence, Eqs.(1.134) and (1.135) are exactly equivalent to the Maxwell construction on either pressure or chemical potential isotherm. This stems from the fact that the RISM/KH theory yields an exact differential for the free energy function (4.A. 10) in Chapter 4, which thus does not depend on a path of thermodynamic integration. 

Chemical equilibrium Condition in which a chemical reaction and its reverse are occurring at equal rates... 

The possible existence of complexes of impurity atoms in heavily doped germanium and gallium arsenide is discussed. An attempt is made to establish the extent to which such complexes can explain the experimental discrepancy between the electron density and the dopant concentration, account for the negative magnetoresistance, and explain the effect of heat treatment on the investigated crystals under chemical equilibrium conditions. 

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