Question

Numerical method to solve van-der waals equation

Answer 1

Step 1 of 2
The ideal gas law is given by

Ethyl alcohol withandat a temperature ofand pressureand have molal volume as,

An alternative equation of state for gases is according to van der Waals equation:

Now thus the root is,

Put the values of all variable in function is,

Differentiate the above function,

Thus, the value of,

The numerical methods discussed in Chapter 5 and Chapter 6 to perform the computation. The value of the root is determine asis the molal volume and a and b are empirical constants that depend on the particular gas.
Step 2 of 2
The value of molal volume v can be determined by Newton-Raphson iterative method the formula is,

First Iteration:
Put the value of the values of p, a, b, R and T the expression where,

Put the values of all variable in function is,

And the value of,

Second Iteration:
Put the value of the values of p, a, b, R and T the expression where,

Put the values of all variable in function is,

And the value of,

Use the ideal gas law and get the result of the accurate root in few iteration:

After 3 iterations, the molal volume is which nearly same as calculated value.