Q$1.$ Develop a chemical engineering problem that involves modeling a chemical process or reaction system using linear algebra and numerical methods. Your task is to outline the problem scenario, formulate the system of equations representing the process, and describe how you would apply eigen$-$decomposition and the Newton$-$Raphson or Quasi$-$Newton Raphson methods to analyze and solve the system. Your answer should not include numerical solutions but should detail the approach and steps you would take to apply the concepts learned in the lectures. Highlight the importance of each method in the context of the problem.

Example:

Design a batch reactor system where a reversible reaction $AB$ takes place. The forward and reverse reactions have rate constants $k_f$ and $k_r,$ respectively. The reactor starts with a known concentration of reactant $A,$ and no product $B.$ The goal is to determine the equilibrium concentration of A and $B$ in the reactor. Your solution steps might include

Problem Formulation

System of Equations

Eigen$-$Decomposition Application

Eigen$-$Decomposition Application

Analytical Approach Discussion