Assume relative volatilities are constant.

a. Show that mole fractions for a bubble-point calculation are given by

\[
\begin{equation*}
\mathrm{y}_{\mathrm{i}, \mathrm{j}}=\left(\mathrm{x}_{\mathrm{i}, \mathrm{j}} \alpha_{\mathrm{i}-\mathrm{ref}}\right) / \sum\left(\mathrm{x}_{\mathrm{i}, \mathrm{j}} \alpha_{\mathrm{i} \text {-ref }}\right) \tag{5-26}
\end{equation*}
\]

and temperature is found from estimated \(\mathrm{K}\) value of the reference component:

\[
\begin{equation*}
K_{\text {ref.j }}=1 / \sum\left(x_{i, j} \alpha_{i-\text {-ref }}\right) \tag{5-27}
\end{equation*}
\]

b. For dew-point calculations show that

\[
\begin{equation*}
x_{i, j}=\left(y_{i, j} / \alpha_{i-r e f}\right) / \Sigma\left(y_{i, j} / \alpha_{i-r e f}\right) \tag{5-28}
\end{equation*}
\]