The decay rates of [A] and [B] are equal because each reaction consumes one particle of A and one of B, producing a particle of C. The rate depends on the product of the activities of the two reactants, since the product gives the probability of particles of the reactants encountering each other. Now, the first equation given only determines the decay of A owing to the reaction with B. This will be the net reaction when there is no C present, but after the reaction proceeds a while in a closed vessel, some C will accumulate and the decomposition of C back to A and B needs to be taken into account. Thus, when the activity is simply a concentration or density, the full system is governed by dA1 = -k+ • [A][B]+ k_ • [C] (8.4)
which will come into equilibrium when the left hand side is zero, namely when
The quantity on the right hand side is the equilibrium constant for the reaction. This equation constrains the relative proportion of the three substances once equilibrium has been achieved, but how one uses this information depends on what is specified in the setup of the problem. For example, if for some reason we know the activity [A], then we immediately know the ratio [B]/[C], though we don't know the absolute amounts unless something in addition is specified. As a slightly more complicated example, suppose we put 2Mole/m3 of A and 1 Mole/m3 of B into a closed vessel which initially contains no C. Then, after equilibrium has been reached xMole/m3 of C will have been produced, which depletes each of [A] and [B] by xMole/m3, since it takes one of each to produce a particle of C. The equilibrium equation then tells us that
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