Consider a binary liquid mixture at equilibrium with its vapour at $25^{\circ}C$.

Antoine equation for this system is given as $log_{10}\:p_{t}^{sat}=A-\frac{B}{t+C}$ where $t$ is in $^{\circ}C$ and $p$ in Torr.

The Antoine constant ($A$, $B$ and $C$) for the system are given in the following table.

$$\begin{array}{|cl|cI|cI|cI|cI|c|}\hline &\textbf{Component} & \textbf{A} & \textbf{B} & \textbf{C} \\ \hline &\text{1} & \text{7.0} & \text{1210} & \text{230} \\ \hline &\text{2} & \text{6.5} & \text{1206} & \text{223} \\ \hline \end{array}$$

The vapour phase is assumed to be ideal and the activity coefficients ($\gamma_{t})$ for the non-ideal liquid phase are given by

$$ln\left ( \gamma _{1} \right )=x_{2}^{2}\left [ 2-0.6x_{1} \right ]$$

$$ln\left ( \gamma _{2} \right )=x_{1}^{2}\left [ 1.7+0.6x_{2} \right ]$$

If the mole fraction of component $1$ in liquid phase $(x_{1})$ is $0.11$, then the mole fraction of component $1$ in vapour phase $(y_{1})$ is ____________________