Page 210 - Physical Chemistry

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Use of a Spreadsheet to Solve Equations Section 6.5
A spreadsheet can be used to solve the cubic equation (6.43). The directions will Simultaneous Equilibria
be given for Excel 2003. Begin by entering a guess for x in cell A1. Since we know x
is between 0 and 1, we guess x as 0.5. Then enter the formula =A1^
3-3.995*A1^2+5.269*A1-2.250 into cell B1.
Excel contains a program called the Solver that will adjust the values in user-
specified cells so as to make the values in other cells satisfy conditions set by the user.
(The corresponding programs in Gnumeric and Quattro Pro are called the Solver and
the Optimizer, respectively.) To invoke the Solver, choose Solver on the Tools menu.
If you don’t see Solver on the Tools menu, choose Add-Ins on the Tools menu, click
the box for Solver Add-In, and click OK. If you don’t see Solver Add-In in the Add
Ins box, you need to click Browse and find the Solver.xla file. [To start the Solver in
Excel 2007, click the Data tab and then click Solver. If you don’t see Solver on the
Data tab, click the Office Button at the upper left, click Excel Options at the bottom,
click Add-ins, and click Go next to the Manage box (which should show Excel Add-
ins); in the Add-ins box, click the Solver Add-in check box and click OK.]
In the Solver Parameters dialog box that opens, enter B1 in the Set Target Cell
box, click Value of after Equal To and enter 0 after Value of. In the By Changing Cells
box, enter A1 to tell Excel that the number in A1 (the value of x) is to be varied. To
have Excel solve the problem, just click on Solve in the Solver Parameters box. After
a moment, Excel displays the Solver Results box telling you that it has found a solu-
tion. Click OK. Cell B1 now has a value very close to zero and cell A1 has the desired
solution 0.9317. . . .
If you again choose Solver from the Tools menu and click Options in the Solver
Parameters box, you will see the default value 0.000001 in the Precision box. Excel
stops and declares that it has found a solution when all the required conditions are sat-
isfied within the specified precision. With the default precision, the value in B1 will
7
be something like 3 10 . To verify the accuracy of the solution, it’s a good idea to
change the Precision from 10 6 to 10 10 , rerun the Solver, and verify that this does not
significantly change the answer in A1. Then restore the default precision.

6.5 SIMULTANEOUS EQUILIBRIA
This section shows how to solve a system with several simultaneous ideal-gas reactions
Figure 6.8
that have species in common. Suppose the following two ideal-gas reactions occur:
Variation of G, H, and TS with
112 CH H O ∆ CO 3H 2 extent of reaction j in the
2
4
(6.47)
122 CH 2H O ∆ CO 4H 2 synthesis of NH (g) at 500 K and
3
2
4
2
4 bar for an initial composition of
Let the initial (0 subscript) numbers of moles be 1 mol of N and 3 mol of H . The
2
2
H-versus-j curve is linear. Since
n 1 mol, n 0,H 2 O 1 mol, n 1 mol, n 1 mol, n 0,CO 2 mol n is negative for the reaction, S
0,CH 4 0,CO 2 0,H 2
decreases as j increases. (Of
[The reactions (6.47) are industrially important in the production of hydrogen from course, S reaches a maximum
univ
natural gas. The reverse of reaction (1) is one of the reactions in the Fischer–Tropsch when G reaches a minimum.)
process that produces hydrocarbons and water from CO and H (formed from the re-
2
action of coal with air and steam, which is an example of coal gasification). During
World War II, Germany was cut off from oil supplies and used the Fischer–Tropsch
process to produce gasoline.]
Using P /P° x P/P° n P/n P°, we get as the equilibrium conditions for these
tot
i
i
i
reactions:
n 1n 2 3 P 2 n 1n 2 4 P 2
CO
K° H 2 a b , K° CO 2 H 2 a b (6.48)
P,1
P,2
n
n CH 4 H 2 O P°n tot n CH 4 1n H 2 O 2 2 P°n tot