Van Der Waals' Equation Using Newton Raphson

Solving the volume using van der Waals’ equation

I.

INTRODUCTION

a. ENGINEERING PROBLEM STATEMENT Van der Waals equation is

        = 

, where a and b are constants that

can be found in tables, R is the gas constant,  p is the pressure,  V  is  is the volume , n is the number of moles and T  is  is the temperature. When solving the pressure and temperature, we simply substitute the given parameters and will directly yield to the answer. However, trouble comes when solving the volume V because it occurs at two different places (based on the equation above) and thus the equation is not linear in V, and by further simplifying it, the equation of volume will result to a cubic expression. Solving cubic is not as simple as solving a quadratic.

b. THEORY OF SOLUTION We want to find V such that  f(V)= 0 by rewriting the van der Waals equation to

        = 0

. Newton-Raphson approximation is given as

 +1 =     ′

)−  (+ − −.  and thus the formula for van der Waals equation is +1  =      −+ 

                3   = 0     3            = 0      23     0   3  0 = 0

By taking the derivative with respect to V

Therefore

 ′ =     233 c. NUMERICAL ANALYSIS

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The standard way of finding roots of equation is using the Newton-Raphson Method. The program will require 7 inputs that are: the number of mole (n), the pressure (p), the Temperature in Kelvin (T) , the value of gas constant a and b, and number of iterations. In NRM, it must start out with a ‘near enough value to the actual root’ of Vi, so here, in van der Waals equation, the ideal gas law will give the first approximation that is V= nRT/p. With these, solving for better approximate of the volume will be made possible.