By Ilya Prigogine, Stuart A. Rice
The Advances in Chemical Physics sequence presents the chemical physics and actual chemistry fields with a discussion board for severe, authoritative reviews of advances in each sector of the self-discipline. jam-packed with state-of-the-art examine pronounced in a cohesive demeanour no longer stumbled on in other places within the literature, each one quantity of the Advances in Chemical Physics sequence serves because the excellent complement to any complex graduate type dedicated to the research of chemical physics.
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19). Hence each of the polynomials on the right-hand side is a linear combination of the factors (xi - l), for i = 1, . . , tt. Hence each term of these polynomials is divisible by one of the factors (xi - l), for i = 1, . . , n. 7. 5. If x, y and I are the dimensionless concentrations, (diag E h p ( x'ln x) = ( j 2 y ,j l x u , (j l + j 2 ) x , j 2 x 2 ,(jl+j2)z)' and the nonlinear equations are obtained by mutliplying on the left by _v. We write the 36 B. L. CLARKE equations so that every term contains a factor of ( x Y 5;- = -jlJ+ - I), ( y - I), or (I - 1) to get - 1) - ( j , +j2)(y - 1) + ( j l + j 2 ) ( z - 1) i - = 2(j, +j&X - 1) hz - (I - I)] In factorizing, we divided each polynomial by x - 1 to obtain a quotient and a remainder.
The existence of multiple steady states for this j means that these equations have other solutions, which we call a’(j),a”(j), . . If the solution a’@ corresponds to the steady state (h’,j’),whose concentration at steady state is Xi”= hi-’, the dimensionless concentration at steady state is therefore h / h and this is a’@. 51) using x = a’(i). Hence for j = 1, . . 57) 37 STABILITY OF COMPLEX REACTION NETWORKS If this sytem of equations is solved for j' as a function of j, the solution will be unique only if II, is a simplex because the redundant parameters have been used.
Physical Limitations to the Accessible States Molecules occupy a nonzero volume; hence one mole of particles of type i cannot be packed into a less than h,? (say) liters. Thus hi > h,? 58) is a physical constraint equivalent to an upper limit on the solubility of X i . On the average, molecules in a dense medium at room temperature cannot collide more frequently than lOI4 times per second. 59) The part of parameter space satisfying two such restrictions for arbitrary ho E R and jo E R :-d is the potentially accessible region corresponding to ho and jo.