Linear and nonlinear regression of rate data

For a Langmuir –Hinshelwood type rate equation , the value of rate law parameters k, K_A, K_B cannot be analyzed as simply as in power law kinetics. The rate law parameters can be determined by linear least square or nonlinear least square method. In linear least square method, the above LH rate equation can be linearized by dividing throughout by p_A and p_B and inverting it as shown below:

The parameters can be estimated by multiple regression technique using the following equation for N experimental runs. For the i^th run:

The best values of the parameters a₀ , a₁ and a₂ are found by solving following three equations

The above LH rate equation can also be solved by non-linear regression analysis. Usually linearized least square analysis is used to obtain the initial estimates of the rate parameters and used in non-linear regression. In non linear analysis the rate law parameters are first estimated to calculate the rate of reaction ‘r_c'. Then the values of rate law parameters that will minimize the sum of the squares of difference of the measured reaction rate r_m and the calculated reaction rate r_c, that is the sum of (r_m -r_c)² for all data points, are searched. If there are N experiments with K number of parameter values to be determined then the function to be minimized is given by

Here r_im and r_ic are the measured and calculated reaction rate for ‘i'^th run respectively. The parameter values giving the minimum of the sum of squares, , can be searched by various optimization techniques using software packages.