Match List I with List II:
Choose the correct answer from the options given below:
Components \( A \) and \( B \) form an azeotrope. The saturation vapor pressures of \( A \) and \( B \) at the boiling temperature of the azeotrope are 87 kPa and 72.7 kPa, respectively. The azeotrope composition is _________ mol% \( A \) (rounded off to the nearest integer).
% Given GIVEN: \[ \ln \left( \frac{\gamma_A}{\gamma_B} \right) = 0.9 \left( x_B^2 - x_A^2 \right) \] where \( x_i \) and \( \gamma_i \) are the liquid phase mole fraction and activity coefficient of component \( i \), respectively.
A binary \(A\)-\(B\) liquid mixture containing 30 mol% \(A\) is subjected to differential (Rayleigh) distillation at atmospheric pressure in order to recover 60 mol% \(A\) in the distillate. Assuming a constant relative volatility \(\alpha_{AB} = 2.2\), the average composition of the collected distillate is __________ mol% \(A\) (rounded off to the nearest integer).
If $ X = A \times B $, $ A = \begin{bmatrix} 1 & 2 \\-1 & 1 \end{bmatrix} $, $ B = \begin{bmatrix} 3 & 6 \\5 & 7 \end{bmatrix} $, find $ x_1 + x_2 $.