Manipulating Equilibrium Expressions

I See the Saunders Interactive General Chemistry CD-ROM, Screen 16.7, Manipulating Equilibrium Expressions. 

The Equilibrium (Mass Action) Expression Gas Phase Equilibria Kp vs. Kp Homogeneous and Heterogeneous Equilibria Numerical Importance of the Equilibrium Expression Mathematical Manipulation of Equilibrium Constants Reversing the Chemical Equation Adjusting the Stoichiometry of the Chemical Reaction Equilibrium Constants for a Series of Reactions Units and the Equilibrium Constant... 

Write the equilibrium expression for the reaction as written, and manipulate the expression as needed to match the other conditions. If we are not confident with these manipulations, we can always write out the specified versions of the equilibrium equations and use them to obtain the needed equilibrium expressions. 

Equilibrium Expressions Heterogeneous Equilibria Manipulating Equilibrium Expressions... 

Using hypothetical equilibria, Table 15.2 summarizes the various ways that chemical equations can be manipulated and the corresponding changes that must be made to the equilibrium expression and equilibrium constant. 

Manipulating equilibrium expressions [ W Section 15.3] Chapter in Review... 

Hint You can use the general method introduced on page 761. First, identify all the species that could be present and the equilibria involving these species. Then identify the two equilibrium expressions that will predominate and eliminate all the species whose concentrations are likely to be negligible. At that point, only a few algebraic manipulations are required.]... 

Example provides practice in manipulating equilibrium constant expressions. 

Instead of using the steady-state approximation in the manipulation of the individual rate expressions, the same result may be reached by assuming that a pseudo equilibrium condition is established with respect to reaction B and that reaction C continues to be the rate limiting... 

Since the anesthesiologist has control over the partial pressure of anesthetic delivered to the lung, it can be manipulated to control the anesthetic gas concentration in the brain, hence the level of unconsciousness. For this reason, anesthetic dose is usually expressed in terms of the alveolar tension required at equilibrium to produce a defined depth of anesthesia. The dose is determined experimentally as the partial pressure needed... 

The idea of equilibrium hinges on the concept of reaction rates. In chemistry rate refers to how much something changes in a unit of time. The something that changes is the concentration of a reactant or a product, usually expressed as molarity. The unit of time is generally the second, although any unit of time can be used. Sometimes it is desirable to manipulate the rate of a reaction in order to speed it up or slow it down. The factors that affect the rate of a reaction are temperature, concentration, surface area, and the use of a catalyst. 

The solution of equilibrium problems is based on algebraic manipulation of the equilibrium constant expression. In typical equilibrium problems the unknown is one part of the K equation shown above and the other parts are given. 

Still another important application of the concept of K equilibrium constants is the coprecipitation of EeP04 and Fe(OH)3 in the removal of phosphorus from water. As in the case of coagulation using alum, it is desired to have a final equation that is expressed only in terms of the constants and the hydrogen ion. Once this is done, the equation can then also be manipulated to obtain an optimum pH for the removal of phosphorus. 

The SELECTOR module is responsible for transforming the internal representation of the reaction system into a form which can readily be solved by the SOLVER module. The equations that are represented by the original FLUX matrix, generated in the INPUT module, may be in an unsolvable form. For example, unknown constant parameters may appear in the same equation with as yet unsolved variable parameters. Also, if there are equilibrium assumptions made about certain reactions, the associated rate constants must be eliminated. Finally, if there are unsolvable parameters, they must be identified, and the associated equations must be eliminated. This process involves a rearrangement of the equations that represent the reaction system, using the FLUX matrix. Other rearrangements may be possible by examining the rate expressions, but the symbol manipulative capability that is needed to accomplish tliis is not yet available in CRAMS. 

The NONLIN module is responsible for intializing the concentration vector, C(t), for l i NRCT. Here NRCT is the number of reactants. If there are no equilibrium reactions, then C i) is set to IC i), the initial concentration vector, for 1 < f < NRCT. If equilibrium reactions do exist, then the type (2) equations (with derivatives set to zero) and the Type (1) and Type (3) equations are all solved simultaneously for the equilibrium concentrations of all reactants. Because the equilibrium equations are generally nonlinear, the Newton-Raphson iteration method is used to solve these equations. Also, since there is no symbol manipulation capability in the current version of CRAMS, numerical differentiation is used to calculate the required partial derivatives. That is, the rate expressions cannot at this time be automatically differentiated by analytical methods. A three point differentiation formula is used 27) ... 

With some manipulation we can express the concentration difference as a pressure difference and see that the diffusion from hot (h) to cold (c) is driven by the concentration gradient and the direction of the reaction is just due to the enthalpy of the reaction. At equilibrium we can write an expression for AG. 

In Eq. (90), the bilinear term is truly a product of linear terms and may be manipulated by standard techniques. The resulting expression for Ak(0 may then be averaged over a nonequilibrium ensemble to obtain >4k(t), or the expression may be multiplied on the left by >4-k(0) and averaged over an equilibrium ensemble to obtain (y4k(t)A-k) In the following, we shall concentrate on the calculation of the time correlation function, which is, of course, identical to the calculation of A it) in the limit of small deviations from equilibrium. 

Quantities in brackets refer to equilibrium solution concentrations of the indicated species. Stability constants are related to standard thermodynamic parameters such as entropy, enthalpy, Gibbs free energy, and temperature, through well-known relationships. The expression in Equation 9.3 can be manipulated using a series of mass balance expressions to relate the equilibrium concentrations of species to the initial concentrations (co) of species used to set up the experiment ... 

Figure 5-1 shows that the disequilibrium ratio (i.e., v- l/v +1) for the reaction catalyzed by SuSy can be influenced by a relative small shift in free energy away from equilibrium. Under normal growing conditions, manipulation of SuSy substrate levels to influence the magnitude of the net flux may therefore be a more effective way of controlling the quantity of cellulose deposited in the cell walls rather than altering its expression level. 

A further generalization of our considerations is concerned with the introduction of the concept of heat baths or thermo-states. If the temperature of system (2) is kept fixed by some appropriate external manipulation, perhaps by making system (2) infinitely large, system (2) is called a heat bath or a thermo-state for system (1). As before, system (1) is in equilibrium with its heat bath if its temperature equals that of the heat bath. This equilibrium condition can now equivalently be expressed as an extremum principle in terms of the macro-variables of only system (1). For simplicity of writing let us denote the variables of system (1) without superscripts in the following. The macro-variable which becomes extremal in the equilibrium is the so-called free energy of system (1) defined as... 

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