Solving van der waals equations for molar volume?

Question

I need to solve the van der waals equation for the molar volume. I have reduced to equation to this form:

x^3 + x^2 + x = 0 where x is the molar volume.

How do you solve for x?

Answer 1

Like many things in math, there are many ways of doing this.

The first and easiest is plug it into a calculator (graphing, like a TI-89).  Either directly solve it, or graph it and look for the intercepts (use the table to confirm).

A second, very common way is to factor.  For instance, you can factor that equation into

x * (x^2 + x + 1) = 0.

From here, recall the fundamental theorem of algebra that states that if X * Y = 0, either X = 0 or Y = 0:

X = 0 is a solution, and X^2 + x + 1 = 0 can be further solved to find more solutions.

From here, the easiest way is to use the quadratic formula:

(-b +- SQRT( b^2 - 4ac))/2a, though if you plug that in you'll find that there are no real solutions.  I don't remember what van der waals equation requires, but if it doesn't accept imaginary answers, then stop.

Finally, a good way to do this type of problem from the beginning is use synthetic division:

http://www.okc.cc.ok.us/maustin/Syntheti...