Equilibrium equations simultaneous solution

Simultaneous solution of these equilibrium relations (coupled with the conservation equations x+ x-f = 1 and x/ + x/ = 1) gives the coexistence curve for the two-phase system as a function of pressure. 

If he selects the still pressure (which for a binary system will determine the vapour-liquid-equilibrium relationship) and one outlet stream flow-rate, then the outlet compositions can be calculated by simultaneous solution of the mass balance and equilibrium relationships (equations). A graphical method for the simultaneous solution is given in Volume 2, Chapter 11. 

Key features of Pourbaix diagrams are the points of intersection between the coexistence lines. In a simple diagram, three compounds of a dependent component can coexist at these points. Thus, if compounds i, j and k coexist at a point, 3 coexistence lines must radiate from the point, ij, jk and ik. The coordinates of potential triple intersection points can be determined by simultaneous solution of pairs of equations. For example the coordinates of the equilibrium point between sulphur, SOjj- and HS- are determined by solution of the equations... 

Equilibrium The physical process (reaction) of adsorption or ion exchange is considered to be so fast relative to diffusion steps that in and near the solid particles, a local equilibrium exists. Then, the so-called adsorption isotherm of the form q = f(Ce) relates the stationary and mobile-phase concentrations at equilibrium. The surface equilibrium relationship between the solute in solution and on the solid surface can be described by simple analytical equations (see Section 4.1.4). The material balance, rate, and equilibrium equations should be solved simultaneously using the appropriate initial and boundary conditions. This system consists of four equations and four unknown parameters (C, q, q, and Ce). 

This optional chapter provides tools to compute the concentrations of species in systems with many simultaneous equilibria.3 The most important tool is the systematic treatment of equilibrium from Chapter 8. The other tool is a spreadsheet for numerical solution of the equilibrium equations. We will also see how to incorporate activity coefficients into equilibrium calculations. Later chapters in this book do not depend on this chapter. 

For a solution of two weak acids with comparable values of Ka, there is no single principal reaction. The two acid-dissociation equilibrium equations must therefore be solved simultaneously. Calculate the pH in a solution that is 0.10 M in acetic acid (CH3C02H, Ka = 1.8 X 10 5) and 0.10 M in benzoic acid (C6H5C02H, Ka = 6.5 X 10 5). (Hint Letx = [CH3C02H] that dissociates and y = [C HsCCAH] that dissociates then [H30+] =x + y.)... 

In the case of very weak acids, [H3O+] from the dissociation of water is significant compared with [H30+] from dissociation of the weak acid. The sugar substitute saccharin (C7H5NO3S), for example, is a very weak acid having Ka = 2.1 X 10-12 and a solubility in water of 348 mg/100 mL. Calculate [H30+] ina saturated solution of saccharin. (Hint Equilibrium equations for the dissociation of saccharin and water must be solved simultaneously.)... 

Unconstrained nonlinear optimization problems arise in several science and engineering applications ranging from simultaneous solution of nonlinear equations (e.g., chemical phase equilibrium) to parameter estimation and identification problems (e.g., nonlinear least squares). 

The solution to a multi-component, multi-phase, multi-stage separation problem is found in the simultaneous or iterative solution of the material balances, the energy balance and the phase equilibrium equations (see Chapter 1). This implies that a sufficient number of design variables are specified so that the number of remaining unknown variables exactly equals the number of independent equations. When this is done, a separation process is said to be specified. 

Since Pf is a function of temperature only, Raoult s law is a set of N equations in the variables T, P, y,, and, . There are, in fact, N - 1 independent vapor-phase mole fractions (the y,- s), N - 1 independent liquid-phase mole fractions (the x, s), and T and P. This makes a total of 21V independent variables related by N equations. The specification of N of these variables in the formulation of a vapor/liquid equilibrium problem allows the remaining N variables to be determined by the simultaneous solution of the N equilibrium relations given here by Raoult s law. In practice, one usually specifies either T or P and either the liquid-phase or the vapor-phase composition, fixing 1 + N - 1) = N variables. 

The calculation of temperatures and equilibrium compositions of gas mixtures involves simultaneous solution of linear (material balance) and nonlinear (equilibrium) algebraic equations. Therefore, it is necessary to resort to various approximate procedures classified by Carter and Altman (Cl) as (1) trial and error methods (2) iterative methods (3) graphical methods and use of published tables and (4) punched-card or machine methods. Numerical solutions involve a four-step sequence described by Penner (P4). 

For stripping, the entering liquid and gas concentrations are known. The fraction stripped and therefore the exit liquid concentration is also known, but the exit gas concentration is unknown. Therefore point 2 - at the bottom of the column is fixed, but point 1 - at the top of the column - is not fixed. The maximum exit and minimum gas flow rate gas concentration is obtained when the operating line intersects the equilibrium curve, as shown by the dashed line in Figure 6.16. The intersection is given by Equation 6.21.1S, which is also obtained by the simultaneous solution of a component balance and an equilibrium relation. 

It is possible to predict the concentrations of the various molecular species in a solution of phosphoric acid, but the calculations involved, which require the simultaneous solution of several equilibrium equations, are rather complicated. In a 1 F solution of phosphoric acid, H3PO4, it is found that the concentration of hydrogen ion and of dihydrogen phosphate ion, H2P04, is 0.083 mole/1, the concentration of monohydrogen phosphate ion, HP04, is 6.2 X 10 moles/1, and the concentration of phosphate ion, P04 -, is 1 X 10 moles/l. 

The symbols represent the concentrations of the various ions and molecules in the equilibrium solution. The concentrations of free bromine and hypobromous acid were obtained by solving Jones and Hartmann s equations simultaneously and substituting the experimental values for the total bromine (C), total bromide (B), and hydrogen-ion concentration [H+]. As the concentration of hypobromous acid was extremely small, it was neglected in equation 36. Equation 37 was obtained by substituting in equation 35 the value of [Br3 ] obtained from equation 33 and the value of [Br( ] obtained from equation 34 and then solving for [Br ]. 

As we have seen, a direct numerical approach is often quite difficult because rigorous simultaneous solutions of equilibrium relationships lead to equations of third, fourth, or higher order these equations are obviously not amenable to convenient numerical resolution. 

Calculate the concentration of all species present in a solution containing 1.00 M and 0.010 M Cd(N03)2 at 25°C. Because the solution is strongly acidic, Cd-OH complexing need not be considered. Solving the problem requires the simultaneous solution of equilibrium constant expressions and mass-balance equations involving the aqueous species. We are given stepwise formation (equilibrium) constant expressions for the Cd-Cl complexes that can be reformatted to give... 

In problems where the flux ratios are known (e.g., condensation and heterogeneous reacting systems where the reaction rate is controlled by diffusion) the mole fractions at the interface are not known in advance and it is necessary to solve the mass transfer rate equations simultaneously with additional equations (these may be phase equilibrium and/or reaction rate equations). For these cases it is possible to embed Algorithms 8.1 or 8.2 within another iterative procedure that solves the additional equations (as was done in Example 8.3.2). However, we suggest that a better procedure is to solve the mass transfer rate equations simultaneously with the additional equations using Newton s method. This approach will be developed below for cases where the mole fractions at both ends of the film are known. Later we will extend the method to allow straightforward solution of more complicated problems (see Examples 9.4.1, 11.5.2, 11.5.3, and others). 

The class of simultaneous solution methods in which all of the model equations are solved simultaneously using Newton s method (or a modification thereof) is one class of methods for solving the MESH equations that allow the user to incorporate efficiencies that differ from unity. Simultaneous solution methods have long been used for solving equilibrium stage simulation problems (see, e.g., Whitehouse, 1964 Stainthorp and Whitehouse, 1967 Naphtali, 1965 Goldstein and Stanfield, 1970 Naphtali and Sandholm, 1971). Simultaneous solution methods are discussed at length in the textbook by Henley and Seader (1981) and by Seader (1986). 

The thermodynamic modeling and calculation procedures for S-L-V equilibrium in binary systems for RESS/PGSS involve simultaneous solution of the phase equilibrium relations for the two components, namely, the SCF solvent, 1, and the solid solute, 3, for all three phases, S, L, and V, as given by the following equations ... 

The column section can be solved by simultaneous solution of the component mass balance and energy balance equations and the vapor-liquid equilibrium relations. Additional equations include the temperature, pressure, and composition dependence of the equilibrium coefficients and enthalpies. The equations for stage j are as follows ... 

These are the equilibrium equations of the y-tp model that are to be solved for the unknown variables among T, p, x, y. Hi, or from the known variables simultaneously with the mass balances and/or other applicable constraint equations. A common method of solution of the simultaneous equations is to use K values. From Equation (4.494) the K values are formed. 

A natural response to the limitations of both geochemical equilibrium models and the solute transport models (see 10.3 for a discussion) is to couple the two. Over the last two decades, a number of models that couple advective-dispersive-diffusive transport with fully speciated chemical reactions have been developed (see reviews by Engesgaard and Christensen, 1988 Grove and Stollenwerk, 1987 Mangold and Tsang, 1991). In the coupled models, the solute transport and chemical equilibrium equations are simultaneously evaluated. 

Stagewise calculations require the simultaneous solution of material and energy balances with equilibrium relationships. It was demonstrated in Example 1.1 that the design of a simple extraction system reduces to the solution of linear algebraic equations if (1) no energy balances are needed and (2) the equilibrium relationship is linear. 

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