Learnings from the Relation between the Number of Forward and Reverse Reactions (Transfer Cycles) Required to Converge to Equilibrium and the Ratio of the Forward to the Reverse Rate Constants in Simple Chemical Reactions

Abstract

In simple, reversible, chemical reactions of the type A ⇋ B, chemical equilibrium is related to chemical kinetics via the equality between the equilibrium constant and the ratio of the forward to the reverse rate-constant, i.e., K _eq = k _f/k _r, where K _eq is the equilibrium constant and k _f and k _r denote the rate constants for the forward (A → B) and reverse (B → A) reactions, respectively. We review and examine the relation between the number of forward and reverse reactions required to take place for the aforementioned system to reach equilibrium and the ratio of the forward to the reverse rate constant. Each cycle of reactants becoming products and the products becoming reactants is defined as the transfer cycle (TC). Therefore, we underscore the relation between the number of TCs required for the system to equilibrate and k _f/k _r. We also vary the initial concentrations of the reactants and products to examine their dependency of the relation between the number of TCs required to reach equilibrium and k _f/k _r. The data reveal a logarithmic growth-type relation between the number of TCs required for the system to achieve equilibrium and k _f/k _r. The results of this relation are discussed in the context of several scenarios that populate the trajectory. We conclude by introducing students and researchers in the area of chemistry and biochemistry to physical phenomena that relate the initial concentrations of the reactants and products and k _f/k _r to the number of TCs necessary for the system to equilibrate.

Figure 1

#TCs vs k_f/k_r when [A]_i = 100, [B]_i = 0, k_f = 0.1, and k_r = 0.01 → 0.09

Figure 2

#TCs vs k_f/k_r when [A]_i = 100, [B]_i = 0, k_f = 0.1, and k_r = 0.01 → 0.9.

Figure 3

#TCs vs k_f/k_r when [A]_i = 50, [B]_i = 0, k_f = 0.1, and k_r = 0.01 → 0.9.

Figure 4

#TCs vs k_f/k_r when [A]_i = 100, [B]_i = 25 (A), 50 (B), 75 (C), 100 (D), 200 (E), and 400 (F); k_f = 0.1 and k_r = 0.01 → 0.9.

Figure 5

#TCs vs k_f/k_r.

Figure 6

k_f/k_r vs #TC, when [A]_i = 100, [B]_i = 0 → 400, and k_f/k_r = 1 → 20.

Figure 7

#TCs vs k_f/k_r when [A]_i = 100, [B]_i = 0–400, and k_f/k_r = 1 → 20. Bottom to top: [B]_i = 0, 25, 50, 75, 100, 200, 400.