On the road to equilibrium

Equilibrium was an important theme last quarter, forming the basis of the ideas of reversibility and nearly all of the rest of thermodynamics. This quarter equilibrium has shown itself as the state towards which transport properties and chemical kinetics approach. As we solve the mechanism of opposing/reversible reactions, we are bridging the worlds of thermodynamics and kinetics.

For the reaction A ↔ B, we can readily write the differential equations for A and B:

dA/dt = - k_fA + k_rB
dB/dt = + k_fA - k_rB

Assuming that there is only A initially, we can state that A_o = A+B, or B = A_o - A, which, when plugged into equation 1 above, makes it integrable (two variables only). The solutions are:

A = A_o[ (k_r + k_fe^-(kf+kr)t)/(k_r + k_f) ]
B = A_o[ 1 - (k_r + k_fe^-(kf+kr)t)/(k_r + k_f) ]

To connect to equilibrium, we recognize that A(∞)=A_eq and that B(∞)=B_eq:

A_eq = A_o[ k_r/(k_r + k_f) ]
B = A_o[ k_f/(k_r + k_f) ]

Cool, from just kinetics we can predict equilibrium quantities of A and B. But not only that, we can take their ratio and obtain the equilibrium constant:

K = k_f/k_r

Remarkably, this is the result for all reversible reactions ( 2A ↔ B, 3A ↔ 2B, etc) and is called the principle of microscopic reversibility.

More complicated mechanisms can be understood as combinations of branching, sequential and opposing reactions. When intermediates are reactive, we can often use the steady-state approximation, which says that dI/dt ≅ 0. This allows us to more easily obtain rate laws from mechanisms without solving elaborate series of differential equations that take lots of time and make your soul cry.