Mathematics quadratic equations

On page 235-241 is the explicit solution used in Excel format to make studies, or mathematical experiments, of any desired and possible nature. The same organization is used here as in previous Excel applications. Column A is the name of the variable, the same as in the FORTRAN program. Column B is the corresponding notation and Column C is the calculation scheme. This holds until line 24. From line 27 the intermediate calculation steps are in coded form. This agrees with the notation used toward the end of the FORTRAN listing. An exception is at the A, B, and C constants for the final quadratic equation. The expression for B was too long that we had to cut it in two. Therefore, after the expression for A, another forD is included that is then included in B. 

The concentration of the acetic acid that has not dissociated is considered to be approximately equal to the initial concentration of the acid because the extent of the dissociation of weak acids is small. However, often it is not negligible and then to solve for the equilibrium concentrations, use of the quadratic equation or other special mathematics is needed. 

This is a quadratic equation in [Fe(SCN)2+[. It has two mathematical solutions but only one is physically possible ... 

Step 3. Substitute the equilibrium concentrations into the equilibrium equation for the reaction and solve for x. If you must solve a quadratic equation, choose the mathematical solution that makes chemical sense. 

Quadratic equations arise frequently in the mathematical descriptions of common physical and chemical processes. For instance, silver chloride is only very slightly soluble in water. It has been determined experimentally that the solubility product Ksp of silver chloride at 25C is 1.56 x 10 12 M2, meaning that in a saturated solution the concentrations of silver ion and chloride ion satisfy the relationship... 

Before we go into the mathematical framework behind wave mechanics, we will review one more mathematical concept normally seen in high school imaginary and complex numbers. As discussed in Section 1.2, for a general quadratic equation ax2 + bx+c =... 

Solving a quadratic equation always yields two roots. One root (the answer) has physical meaning. The other root, while mathematically correct, is extraneous that is, it has no physical meaning. The value of x is defined as the number of moles of A per liter that react and the number of moles of B per liter that react. No more B can be consumed than was initially present (0.100 M), so x = 0.309 is the extraneous root. Thus, x = 0.099 is the root that has physical meaning, and the extraneous root is 0.309. The equilibrium concentrations are... 

This formula gives two roots, both of which are mathematically correct. A foolproof way to determine which root of the equation has physical meaning is to substimte the value of the variable into the expressions for the equilibrium concentrations. For the extraneous root, one or more of these substimtions will lead to a negative concentration, which is physically impossible (there cannot be less than none of a substance present ). The correct root will give all positive concentrations. In Example 17-7, substitution of the extraneous root x = 0.309 would give [A] = (0.300 — 0.309) M = —0.009 M and [B] = (0.100 — 0.309) M = —0.209 M. Either of these concentration values is impossible, so we would know that 0.309 is an extraneous root. bu should apply this check to subsequent calculations that involve solving a quadratic equation. 

Mathematically, the quadratic equation also always has a negative answer for X (which we are not interested in since it makes no chemical sense). That value of X can be obtained by constraining the answer to be less than zero in the Subject to the Constraints box. Click on Add. For Cell Reference, click on Cl. Adjust the arrow to move to <=. In the Constraint dialogue box, type 0. Then click OK. Click on Solve, and you see the answer x = —0.40565 (and formula = —9E-07). 

The intensity decrease is clearly hyperbolic and therefore a mathematical binding analysis can be performed using the following quadratic equation obtained from the definition of the equilibrium constant ... 

This problem could be solved more easily if we could assume that (0.10 - x) 0.10. If the assumption is mathematically valid, then it would not be necessaiy to solve a quadratic equation, as we did above. Re-solve the problem above, making the assumption. Was the assumption valid What is our criterion for deciding ... 

The main interest of the surface response methodology is that we can know the value of each response studied, in every point of the experimental domain. If a mathematical approach is used, each response can be repre.sented by a quadratic equation of the surface response ... 

It can be seen from Figure 10.3 that for a polymer composition of 30 mole% diethyl fumarate, two monomer compositions of DEF are obtained. This is referred to as multiplicity of compositions. It is not desirable to operate the industrial reactors at this composition. Mathematically, a quadratic equation can lead to two roots, real and imaginary. This leads to an interesting criterion for copolymer compositional stability. It is conceivable that for some set of reactivity ratios, the composition may not be stable. Equation (10.26) rearranged to provide/i the composition of the monomers in a CSTR for a desired copolymer composition would be... 

In addition to basic arithmetic and algebra, four mathematical operations are used frequently in general chemistry manipulating logarithms, using exponential notation, solving quadratic equations, and graphing data. 

A quadratic equation has two mathematical solutions. You obtain one solution by taking the upper (positive) sign in and the other by taking the lower (negative) sign. You get... 

Only a few basic mathematical skills are required for the study of general chemistry. But to concentrate your attention on the concepts of chemistry, you will find it necessary to have a firm grasp of these basic mathematical skills. In this appendix, we will review scientific (or exponential) notation, logarithms, simple algebraic operations, the solution of quadratic equations, and the plotting of straight-line graphs. 

We can take the dissociation of the acid into consideration when we perform a calculation with an aqueous solution of a weak acid. However, a quadratic equation results and the slight increase in accuracy rarely justifies the additional mathematical effort required. 

Mathematical toobox 4.1 Quadratic equations A quadratic equation is an equation of the form ax + bx+ c=0... 

Appendix 1 Mathematical Procedures A1 Al.l Exponential Notation A1 A1.2 Logarithms A4 A1.3 Graphing Functions A6 A1.4 Solving Quadratic Equations A7 A1.5 Uncertainties in Measurements AlO Appendix 2 The Quantitative Kinetic Molecular Model A13... 

In cases where the equation defining a particular physical situation is a second-degree equation (or even one of higher order), there arises a problem that is not present when one simply considers the pure mathematics, as we have done above. Since quadratic equations necessarily have two roots, we must decide, in cases where both roots are not the same, which root correctly represents the physical simation, even tiiough both are mathematically correct. For example, consider the equilibrium equation... 

If the state and control variables in equations (9.4) and (9.5) are squared, then the performance index become quadratic. The advantage of a quadratic performance index is that for a linear system it has a mathematical solution that yields a linear control law of the form... 

We now develop a mathematical statement for model predictive control with a quadratic objective function for each sampling instant k and linear process model in Equation 16.1 ... 

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