# Simultaneous equilibria

## Rephrasing:

The following reactions are a small example of my big system

$$(1)\ \ce{A + B <=> AB}$$ $$(2)\ \ce{AB -> CB}$$ $$(3)\ \ce{A -> C}$$ $$(4)\ \ce{C + B <=> CB}$$

with

$$\ce{\frac{[AB]}{[A][B]}} = 100 = K_e$$ $$\ce{\frac{[CB]}{[C][B]}} = 100 = K_e$$

I know the rate constants for reactions (2) and (3), and I know that $$\ce{[A] + [AB] + [C] + [CB] = T}$$

with T = some constant.

Now, although there is a fast equilibrium between $$\ce{A + B}$$ and $$\ce{AB}$$, and between $$\ce{C + B}$$ and $$\ce{CB}$$, I want to know how the concentration of each form changes with time.

But I dont know how to write down the differential equations of the system and include the equilibrium constant. Could someone please help me out? My only problem is how to include $$K_e$$ in the system of ordinary differential equations.

## 1 Answer

Ok. You have five equations, two are differential equations and three are algebraic. And you have the explicit assumption that the equilibria are fast enough so that the two equilibrium equations are always satisfied.

You also have five unknowns, A, B, C, AB, and BC. One method of solution is to use the algebraic equations to eliminate three of the variables. This should leave you with two differential equations with two unknowns. If you are very lucky (I've not worked out the algebra) you will get two equations that can be integrated independently. More likely you will have two simultaneous differential equations.

In either case the result is apt to be a mess. So I'd cheat. If this is a textbook problem or homework problem, there is probably a way to combine the equations to get simple enough expressions. Otherwise, you may have serious troubles solving the set.