Conditions for chemical equilibrium

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. 

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. 

Rearranging yields the chemical equilibrium condition for a single reaction in its most general sense ... 

For the equihbrium properties and for the kinetics under quasi-equilibrium conditions for the adsorbate, the transfer matrix technique is a convenient and accurate method to obtain not only the chemical potentials, as a function of coverage and temperature, but all other thermodynamic information, e.g., multiparticle correlators. We emphasize the economy of the computational effort required for the application of the technique. In particular, because it is based on an analytic method it does not suffer from the limitations of time and accuracy inherent in statistical methods such as Monte Carlo simulations. The task of variation of Hamiltonian parameters in the process of fitting a set of experimental data (thermodynamic and... 

A phase boundary for a single-component system shows the conditions at which two phases coexist in equilibrium. Recall the equilibrium condition for the phase equilibrium (eq. 2.2). Letp and Tchange infinitesimally but in a way that leaves the two phases a and /3 in equilibrium. The changes in chemical potential must be identical, and hence... 

The equilibrium condition for the distribution of one solute between two liquid phases is conveniently considered in terms of the distribution law. Thus, at equilibrium, the ratio of the concentrations of the solute in the two phases is given by CE/CR = K, where K1 is the distribution constant. This relation will apply accurately only if both solvents are immiscible, and if there is no association or dissociation of the solute. If the solute forms molecules of different molecular weights, then the distribution law holds for each molecular species. Where the concentrations are small, the distribution law usually holds provided no chemical reaction occurs. 

A chemical system is therefore unable to perform work. The equilibrium condition for an electrochemical system is expressed by... 

Equation (2.57) can be compared with Eqn. (2.50). Noting that the standard value of the vacancy chemical potential of a crystal is only slightly dependent on T, Nv is in essence exponentially dependent on 0/T). This Arrhenius-type of temperature dependence is also found for the interstitials, since in view of the site-preserving formation reaction Aa+V) = Af +VA, the equilibrium condition for this defect formation reaction shows that... 

We turn our attention in this chapter to systems in which chemical reactions occur. We are concerned not only with the equilibrium conditions for the reactions themselves, but also the effect of such reactions on phase equilibria and, conversely, the possible determination of chemical equilibria from known thermodynamic properties of solutions. Various expressions for the equilibrium constants are first developed from the basic condition of equilibrium. We then discuss successively the experimental determination of the values of the equilibrium constants, the dependence of the equilibrium constants on the temperature and on the pressure, and the standard changes of the Gibbs energy of formation. Equilibria involving the ionization of weak electrolytes and the determination of equilibrium constants for association and complex formation in solutions are also discussed. 

The postulates of Nernst are those that are required when we wish to determine equilibrium conditions for chemical reactions from thermal data alone. In order to calculate the equilibrium conditions, we need to know the value of AGe for the change of state involved. We take the standard states of the individual substances to be the pure substances at the chosen temperature and pressure. The value of AH° can be determined from measurements of the heat of reaction. We now have... 

Note that the phase transfer is treated like a chemical reaction. When the equilibrium conditions for these three reactions are introduced into equation 8.4-1, we obtain... 

In terms of chemical potential p of the various species, the equilibrium condition for the above reaction is [55]... 

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]. 

To show how the inside loop may be modified to handle chemical equilibrium, consider for simplicity the case of specified pressure, and reaction occurring only in the liquid phase. For the jth reaction the equilibrium condition may be expressed as ... 

Here, p is the chemical potential for the / th component, and X can be any of a number of external forces that causes an external parameter of the system, a, to change by an amount da. Under conditions of constant temperature and pressure, the first two terms on the right side of Eq. 2.2 drop out. If there are no external forces (X) acting on the system, the equilibrium condition for the transport of matter requires that the value of the chemical potential for each component be the same in every phase (at constant T and P). 

Given mathematical expression, these laws lead to a network of equations from which a wide range of practical results and conclusions can be deduced. The universal applicability of this science is shown by the fact that it is employed alike by physicists, chemists, and engineers. The basic principles are always the same, but the applications differ. The chemical engineer must be able to cope with a wide variety of problems. Among the most important are the determination of heat and work requirements for physical and chemical processes, and the determination of equilibrium conditions for chemical reactions and for the transfer of chemical species between phases. 

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. 

Science is fundamentally empirical—it is based on experiment. The development of the equilibrium concept is typical. From observations of many chemical reactions, two Norwegian chemists, Cato Maximilian Guldberg (1836-1902) and Peter Waage (1833-1900), proposed the law of mass action in 1864 as a general description of the equilibrium condition. For a reaction of the type... 

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,... 

If surface equilibrium prevails, then it is relatively straightforward to generalize the interface condition to chemical processes that are more complex than equations (1) and (8). This of interest, since propellant materials often experience processes of this type for example, NH4CIO4 undergoes dissociative sublimation into NH3 and HCIO4 [33]. For a general process in which the condensed material is transformed to 1 the surface equilibrium condition (for an ideal gas mixture and a solid whose thermodynamic properties are independent of pressure) is... 

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... 

Equilibrium conditions for chemical reactions in nonideal solutions are conventionally expressed in terms of activities, defined as... 

We have been restricting our attention to the equilibrium condition for a single reaction, (17). It is also worthwhile to investigate the problem of computing equilibrium compositions for systems in which many reactions may occur simultaneously. Explicitly, the problem is the following given the pressure, the temperature, and the total number of moles of atoms in the system (irrespective of the number of moles of the chemical compounds in which these atoms may appear), find the number of moles of all chemical species (compounds and free atoms) in the system at equilibrium. 

The second scheme, which is more generally used, involves a hybrid procedure patterned after the methodology of Section 2.11. Here one distinguishes between pure condensed phases, indexed by the symbol s, and components forming homogeneous mixtures, indexed by the symbol j. For the pure condensed phases one adopts Eq. (3.6.2) in the specification of the chemical potential for species in solution it is conventional to introduce Eq. (3.5.21). The equilibrium condition for the reaction = 0 is now specified by... 

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