Solving equations with Excel goal seek

6-8 Solving Equilibrium Problems with a Concentration Table and a Spreadsheet 

Br0nsted-Lowry acid enthalpy change neutralization stepwise formation constant 

For the reaction aA + bB cC + t/D, the equilibrium constant is K = [C]l[D], /[A] (B), Solute concentrations should be expressed in moles per liter gas concentrations should be in bars and the concentrations of pure solids, liquids, and solvents are omitted. If the direction of a reaction is changed. K = UK. If two reactions are added. A , = K, K-,. The equilibrium constant can be calculated from the free-energy change for a chemical reaction K = e AcrlRT. The equation AG = AH — TAS summarizes the observations that a reaction is favored if it liberates heat (exothermic, negative AH) or increases disorder (positive AS). Le Chatelier s principle predicts the effect on a chemical reaction when reactants or products are added or temperature is changed. The reaction quotient, Q, tells how a system must change to reach equilibrium. 

The solubility product is the equilibrium constant for the dissolution of a solid salt into its constituent ions in aqueous solution. The common ion effect is the observation that, if one of the ions of that salt is already present in the solution, the solubility of a salt is decreased. Sometimes, we can selectively precipitate one ion from a solution containing other ions by adding a suitable counterion. At high concentration of ligand, a precipitated metal ion may redissolve by forming soluble complex ions. In a metal-ion complex, the metal is a Lewis acid (electron pair acceptor) and the ligand is a Lewis base (electron pair donor). 

Brpnsted-Lowry acids are proton donors, and Br0nsted-Lowry bases are proton acceptors. An acid increases the concentration of H,0+ in aqueous solution, and a base increases the concentration of OH. An acid-base pair related through the gain or loss of a single proton is described as conjugate. When a proton is transferred 

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Solving equations with Excel Goal Seek. Suppose we saturate a solution of 0.001 0 M Nal with Pbl2. From K p for Pbl2, we find [Pb ] as follows ... 

A reasonable way to solve this equation is by trial and error, as shown in the following table. You can create this table by hand or, even more easily, with a spreadsheet. In cell Al, enter a guess for x. In cell A2, enter the formula =A1 (2 A1 H-0.030) 2 . When you guess X correctly in cell Al, cell A2 will have the value 7.9 X 10 . Problem 6-24 gives an even better way to solve this problem with Excel Goal Seek. 

When examining the command fzero( / , xO) in MATLAB, the f defines which problem to solve, the xO is your best guess of the solution, and fzero tells MATLAB to vary x, starting from xO until the/is zero. In Excel s Goal Seek, the analogous steps were to make a cell zero by varying the value of another cell. Goal Seek becomes fzero, a cell with an equation becomes / and another cell becomes xO. 

With 0, Xjjj, a, a, Pp, Pp and M known, this is a quadratic equation in Xp. The equation can be solved by the quadratic formula, although this requires a fair amount of algebra. The equation can also be solved with an Excel spreadsheet using Goal Seek or Solver. 

Prepare an Excel spreadsheet that calculates for a mixture the bubble point at pressures of 15,16,..., 25 atm and produces a plot of the bubble point versus pressure with suitable annotations and title. Use Goal Seek to solve the single nonlinear equation at each pressure. Data for the system are given in Figure 1.4. 

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