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CHAPTER 10  ORDINARY DIFFERENTIAL EOUATIONS. PART I                  219



                   The differential equation for the change in concentration of the species A as a
               function of time is
                                            d[ A] ldt = -k[ A]                    (1 0-4)

                   Expressing this  in  terms  of finite  differences, the  change  in  concentration
               A[A] that occurs during the time interval from t = 0 to t = At is

                                            A[A] = -k[A],  At                     (1 0-5)
                   Thus, if the concentration of A at t = 0 is 0.2000 My then the concentration at
                t = (0 + At) is [A] = 0.2000 - (5 x  lO")(O.2OOO)(2O)  = 0.1800 M.  The calculation,
                known as Euler's method, is illustrated in Figure 10-1. The formula in cell 87 is

                   =BG-k*BG*DX.
                   The concentrations at  subsequent time  intervals  are calculated  in the  same
                way.  In general, the formula is

                                         Yfl +I  = Yfl + hF(x,,, Yfl)             ( 10-6)
                where h = xfl  - x,.
























                         Figure 10-1.  Simulation of first-order kinetics by Euler's method.
                        (folder 'Chapter 10 Examples', workbook 'ODE Examples', worksheet 'Euler')



                   The advantage of Euler's  method  is that it can be easily expanded to handle
                systems of any complexity.  It is not particularly useful, however, since the error
                introduced by  the  approximation  d[A]ldt = A[A]/At is  compounded  with  each
                additional calculation.  Compare the Euler's method result in column B of Figure
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