### We have seen that from a molar material balance applied to a two-phase system in equilibrium, and the definition of K_{i}, we can derive the Rachford and Rice Objective Function:

$$F\left({\alpha}_{g}\right)={\displaystyle \sum}_{i=1}^{n}\frac{{z}_{i}\left({K}_{i}-1\right)}{1+{\alpha}_{g}\left({K}_{i}-1\right)}=0$$ (13.3)

Equation (13.3) is a non-linear equation in one variable, and the Newton Raphson procedure is usually implemented to solve it. In general, Newton Raphson is an iterative procedure with a fast rate of convergence. The method calculates a new estimate, ${\alpha}_{g}{}^{new}$ , which is closer to the real answer than the previous guess, ${\alpha}_{g}{}^{old}$ , as follows:

$${\alpha}_{g}^{new}={\alpha}_{g}^{old}-\frac{F\left({\alpha}_{g}^{old}\right)}{F\text{'}\left({\alpha}_{g}^{old}\right)}$$ (13.6)

Substituting (13.3) and (13.4) into (13.6),

$${\alpha}_{g}^{new}={\alpha}_{g}^{old}+\frac{{\displaystyle \sum}_{i=1}^{n}\frac{{z}_{i}\left({K}_{i}-1\right)}{1+{\alpha}_{g}^{old}\left({K}_{i}-1\right)}}{{\displaystyle \sum}_{i=1}^{n}\frac{{z}_{i}{\left({K}_{i}-1\right)}^{2}}{{\left[1+{\alpha}_{g}^{old}\left({K}_{i}-1\right)\right]}^{2}}}$$ (13.7)

In this iterative scheme, convergence is achieved when

$$\left|{\alpha}_{g}^{new}-{\alpha}_{g}^{old}\right|<\epsilon $$ (13.8)

where $\epsilon $ is a small number ($\epsilon \text{}=\text{}1.0\text{}x\text{}{10}^{\u2013\text{}9}$ is usually adequate). After solving for ${\alpha}_{g}$ , the liquid molar fraction and composition of each of the phases can be calculated as follows:

$$\text{LiquidMolarFraction:}{\alpha}_{l}=1-{\alpha}_{g}$$ (13.9a)

$$\text{PercentageofLiquid:}\%L=100*{\alpha}_{l}$$ (13.9b)

$$\text{PercentageofVapor:}\%V=100*{\alpha}_{g}$$ (13.9c)

$$\text{VaporPhaseComposition:}{y}_{i}=\frac{{z}_{i}{K}_{i}}{1+{\alpha}_{g}\left({K}_{i}-1\right)}$$ (12.7)

$$\text{LiquidPhaseComposition:}{x}_{i}=\frac{{z}_{i}}{1+{\alpha}_{g}\left({K}_{i}-1\right)}$$ (12.11)