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Unformatted text preview: 7.1. ISOTHERMAL, ISOCHORIC KINETICS 231 Equations (7.247.25) with Eqs. (7.22) represent two nonlinear ordinary differential equa tions with initial conditions in two unknowns ρ O and ρ O 2 . We seek the behavior of these two species concentrations as a function of time. Systems of nonlinear equations are generally difficult to integrate analytically and gen erally require numerical solution. Before embarking on a numerical solution, we simplify as much as we can. Note that d ρ O dt + 2 d ρ O 2 dt = 0 , (7.26) d dt ( ρ O + 2 ρ O 2 ) = 0 . (7.27) We can integrate and apply the initial conditions (7.22) to get ρ O + 2 ρ O 2 = hatwide ρ O + 2 hatwide ρ O 2 = constant. (7.28) The fact that this algebraic constraint exists for all time is a consequence of the conservation of mass of each O element. It can also be thought of as the conservation of number of O atoms. Such notions always hold for chemical reactions. They do not hold for nuclear reactions. Standard linear algebra provides a robust way to find the constraint of Eq. (7.28). We can use elementary row operations to cast Eq. (7.16) into a rowechelon form. Here our goalcan use elementary row operations to cast Eq....
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 Fall '11
 ParkSou
 Dynamics, Linear Algebra, Invertible matrix, J. M. Powers, standard linear algebra

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