Algebraic equation

An equation algebraically equivalent to Eq. XI-4 results if instead of site adsorption the surface region is regarded as an interfacial solution phase, much as in the treatment in Section III-7C. The condition is now that the (constant) volume of the interfacial solution is i = V + JV2V2, where V and Vi are the molar volumes of the solvent and solute, respectively. If the activities of the two components in the interfacial phase are replaced by the volume fractions, the result is... 

To solve these equations algebraically, the inequality signs must first be removed by introducing slack variables Sj and Si such that ... 

Solve the following system of equations algebraically by substitution ... 

We derived Equation (4.30) from first principles, using pure mathematics. An alternative approach is to prepare a similar equation algebraically. The result of the algebraic derivation is the Gibbs-Duhem equation ... 

A sound decomposition strategy should be applicable to any type of mathematical model of a physical process. Therefore, the set of system equations might include linear or nonlinear equations algebraic, differential, difference, or integral equations continuous or discrete variables with the following restrictions ... 

In this section, you learned how to calculate the enthalpy change of a chemical reaction using Hess s law of heat summation. Enthalpies of reaction can be calculated by combining chemical equations algebraically or by using enthalpies of formation. Hess s law allows chemists to determine enthalpies of reaction without having to take calorimetric measurements. In the next section, you will see how the use of energy affects your lifestyle and your environment. 

Solution of these three equations for the three unknowns Tc, Pc, Vc leads, as before, to (2.46)-(2.48). (Note, however, that this trick of equating algebraic coefficients does not work for more general equations of state, so the calculus-based derivation is clearly preferred.)... 

A kinematic wave may be called linear if the relationship between the flow and the concentration can be expressed by one or more linear equations, algebraic or differential. The term linear may also be applied when a diffusion term is included in the continuity equation as is done in 3 of Lighthill ... 

By a reactor model, we mean a system of equations (algebraic, ordinary, or partial differential, functional or integral) which purports to represent a chemical reactor in whole or in part. (The adequacy of such a representation is not at issue here.) It will be called linear if all its equations are linear and simple if its input and output can be characterized by single, concentration-like variables, Uo and u. The relation of input and output will also depend on a set of parameters (such as Damkohler number. Thiele modulus, etc.) which may be denoted by p. Let A(p) be the value of u when w0 = 1. Then, if the input is a continuous mixture with distribution g(x) over an index variable x on which some or all of the parameters may depend, the output is distributed as y(x) = g(x)A(p(jc)) and the lumped output is... 

The theoretical solution to the equations for electrode processes nearly always has to involve approximations, not only for numerical but also for analytical solutions—such as, for example, the assumption that there is no convection within the diffusion layer of hydrodynamic electrodes. In other cases, of complex mechanism, it is not even possible to resolve the equations algebraically. There is another possibility for theoretical analysis, which is to simulate the electrode process digitally. 

The key concept is that of independent equations. Algebraic equations are independent if you cannot obtain any one of them by adding and subtracting multiples of any of the others. For example, the equations... 

State-space models relate the variation in state variables over time to their values in the immediate past and to inputs with differential or difference equations. Algebraic equations are then used to relate output variables to state variables and inputs at the same time instant. Consider a system of first-order differential equations (Eq. 4.44) describing the change in state variables and a system of output equations (Eq. 4.45) relating the outputs to state variables ... 

Steps 3-6 We do not need to rearrange the equation algebraically, but when we plug in our values with their units, we see that we need to convert 85 mL to liters to get our units to cancel correctly. 

Instead of approximating an equation and then solving the approximate equation algebraically, we can apply the graphical method to obtain a numerical approximation to the correct root. This method is sometimes very useful because you can see what you are doing and you can usually he sure that you do not obtain a different root than the one you want to find. The equation to be solved is written in the form... 

You can solve the equation algebraically for Finitiai- Then substitute in the given quantities to solve for the volume of 2.00 MHCl needed to prepare 1.00 L of a 0.646 MHCl solution. 

Solve the ideal gas equation algebraically forn j Then, calculate the moles of H2 by substituting the known quantities into the equation. 

If Steady states (i.e., dose-concentration pairs) are reached by two subsequent doses and desired target concentration has not been reached, it is possible to solve the Michaelis-Menten equation algebraically by using simultaneous equations or by use of a graphical method to determine the patient s and V. ... 

This gives readers the invaluable opportunity to use and implement their code in a numerical library that involves some of the most appealing algorithms in the solution of differential equations, algebraic systems, optimal problems, data... 

A process model usually consists of three types of equations algebraic equations (AE), ordinary differential equations (ODE) and/or partial differential equations (PDE). AEs mainly arise in steady-state models, ODEs occur in steady-state models with spatial variation and also in dynamic models, and PDEs appear in models with variations in two or more dimensions including time. These equations have to be solved for model validation (Figure 4.3) and then in the subsequent applications of the model. 

C. Numerical solutions (e.g., differential equations, algebraic equations)... 

Consider the optimal design of a unit operation or a chemical plant or consider a problem of optimal process control. In these situations as in many other similar situations, the constraint equations (algebraic, differential, or algebraic-differential) constitute the most significant part of the overall problem. 

Solve this equation algebraically for molality (m), then substitute A7f and Kf irrto the equation to calculate the molality. 

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