Exercise:
In a simple chemical reaction a chemical A turns o chemical B: A xlongrightarrowk B The rates of change of the concentrations of A and B are given by the reaction rate k: fracdAdt -kA fracdBdt +kA A more complicated reaction A xrightleftharpoonsk_k_ B xrightarrowk_ C with chemicals A B and C which also takes o account the reverse reaction between A and B can be described by fracdAdt -k_A + k_ B fracdBdt k_A - k_B - k_B fracdCdt k_B abcliste abc Derive the matrix describing the system of differential s. abc Calculate the eigenvalues and eigenvectors for k_ k_ k_ abc Find the solution for the initial conditions A A_ B C abcliste

Solution:
abcliste abc The matrix is M leftmatrix-k_ & k_ & k_ & -k_+k_ & & k_ & matrixright abc The eigenvalues of the matrix M are lambda_ lambda_ --sqrt lambda_ -+sqrt and the corresponding eigenvectors bf v_ leftmatrix matrixright bf v_ leftmatrix+sqrt --sqrt matrixright bf v_ leftmatrix-sqrt -+sqrt matrixright For the initial conditions we have to find coefficients a_ a_ and a_ such that leftmatrix matrixright a_bf v_+a_bf v_+a_bf v_ The three s are a_ +sqrt + a_ -sqrt a_ --sqrt + a_ -+sqrt a_ + a_ + a_ We find that a_ a_ frac-+sqrt a_ frac--sqrt so the concentrations are A frac-+sqrt+sqrt e^--sqrt t+frac--sqrt-sqrt e^-+sqrtt resultfrac e^--sqrt t+frace^-+sqrtt B frac-+sqrt--sqrt e^--sqrt t+frac--sqrt-+sqrt e^-+sqrtt result-fracsqrte^--sqrt t+fracsqrte^-+sqrtt C e^ t+frac-+sqrt e^--sqrt t+frac--sqrt e^-+sqrtt result+frac-+sqrt e^--sqrt t+frac--sqrt e^-+sqrtt The solution has the expected behaviour see figure below. Asymptotically the concentrations of A and B t towards while the concentration of C reaches %. center includegraphicswidthtextwidth#image_path:chemical-reactions# center abcliste