Solving Equation 4​.2​.4

To solve equation 4.2.4 we use a Newton approximation iterative method.

Eq. 5.1

Eq 5.2

The iterative method consists of picking a first guess for Z (1.0 is a good start) then we calculate Y using equation 5.1 and dY/dZ with equation 5.2. We then make a correction to Z such as:

Eq. 5.3

And the new Z is calculated as previous Z + Cor.

The iterations continue until Cor becomes small enough in absolute value.

Note that a 3rd degree equation can have three real roots. Perry’s states that for T >TC only one root exists but for T < TC at certain pressures, up to 3 positive roots can exist and for the vapour state the largest should be used.

My experience is that starting from Z = 1.0, for all reasonably realistic pressures/temperatures the convergence is to the gas phase Z.