Principle of detailed balance

Question

$$\ce{A -> B -> C -> A}$$

In Alberty's Physical Chemistry (7th ed.), the author says that this reaction mechanism is not possible. The reason given is:

Although this mechanism at first sight appears reasonable, no actual process follows this mechanism because it violates the principle of detailed balance. According to this principle, at equilibrium the forward rate of each step is equal to the reverse rate of that step. Thus equilibrium cannot be maintained by this mechanism.

I understand the statement of the principle ("at equilibrium the forward rate of each step is equal to the reverse rate of that step"), but I don't get how it implies that equilibrium cannot be maintained, or that this reaction is not possible.

Answer 1

In this triangular scheme, taking it literally, it is clear that all the population will end up in species C. If $A \rightarrow B$ is mathematically irreversible (rate constant back is zero) it implies that B is at an energy infinitely below that of A, and the energy of C infinitely below that of B, so clearly C cannot revert to A without the input of an infinite amount of energy. Such an argument hardly makes sense in chemistry, however.

If we now assume that the reaction is reversible but with a very small back rate constant a similar conclusion is obtained, i.e. that the overwhelming (but not total) population ends up in C.

Suppose, without loss of generality and for simplicity, that the activation energy A/B and B/C are the same at $E_A$ and that B is at energy E below A, and C at E below B. Now let the forward rate constant A to B be $k_f=k_0\exp(-E_A/RT) $ and the back rate constant is $k_{BA}=k_0\exp(-E_A/RT)\exp(-E/RT)$ and which is going to be smaller, say it is $10^{-3}$ smaller. The rate constant from C to A is therefore $k_{CA}=k_0\exp(-E_A/RT)\exp(-2E/RT)$ which is now $10^{-6}$ times smaller that the rate constant A to B.