Fixed-Point Iteration Direct Substitution

Excel has very good plotting capabilities. Unfortunately, it is not possible in Excel to simply give a command such as plot(/(x)). It is necessary to produce a list or table of x and/(x) values and to graph the resulting data. This is best illustrated by an example. 

From the graph, it is easy to see that there is one real root between 0 and 5 (the other roots are complex conjugates). Remember that the van der Walls equation of state is an attempt to model the nonideality of the gas. Since the given temperature and pressure are not severe, it is expected that the calculated molar volume from the equation of state should not be greatly different from that predicted by the ideal gas law. From the ideal gas law, the molar volume is 4.29 L/gmol, which is very close to the root shown in the graph. 

To apply this method, the equation must be cast into the fonn 

Note that this can always be accomplished by adding x to each side of f(x) = 0, if necessary. The van der Walls equation can be cast into the following form  

The solution is 4.22998 L/gmol. If more significant digits are required, then more iterations can be carried out. 

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