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.

Introduction

Chemical equilibrium is important in most chemical reactions. Furthermore, equilibration of a chemical system has significance in all branches of chemistry, biochemistry, biology, geology, and environmental sciences and disciplines in engineering such as chemical engineering and metallurgical and materials engineering.¹⁻⁵

For a number of reasons, chemical equilibrium remains a difficult concept for some to master. The predecessor to chemical equilibrium is chemical kinetics wherein reactions proceed to completion;²⁻⁵ i.e., the reactant is completely consumed to form product. However, if one of the reactants in a multireactant reaction is limiting in concentration, then the reaction goes to completion only with respect to that limiting reagent.⁶ Furthermore, in the study of chemical kinetics, the emphasis is on unidirectionality. A one-way arrow denotes the direction of reaction progress (from reactants to products) in the forward direction. For example, a typical generic reaction used as an example in chemical kinetics is

Here, A is consumed, and B is generated. The rate of this reaction, when first-order in A, is given by

where k is the rate constant and [A] is the concentration of the reactant expressed in molarity (M). Students of chemical kinetics are taught that the reactant is consumed completely in a finite amount of time, and no more product is produced once the reactant is completely exhausted. This is an outcome of the law of conservation of matter.

Chemical Equilibrium

In the succeeding topic involving equilibrium, they are introduced to the concept of reversibility and reversible reactions. Here, the conversion of A to B is not unidirectional. The heretofore “product”, B, acts as a “reactant” and forms A, the “product”, with respect to “reactant” B. As a result, A is never completely consumed. The “undoing” of the complete consumption of A in the preceding kinetics chapter, via the reversibility of B, is a necessary and first step toward understanding what chemical equilibrium is all about.

The reaction A → B is now transformed into

Simultaneously, a new rate constant is introduced, and the unidirectional arrow is replaced by half-headed arrows, indicating the direction of each reaction. The back-conversion of B to A is governed by the reverse rate constant k_r. The forward rate constant, reserved for the process A → B, is henceforth referred to as k_f.

Relation between K_eq, k_fwd, and k_rev

The equilibrium constant K_eq for the aforementioned reaction of the type A ⇋ B is given by K_eq = [B]_e/[A]_e where the subscripts denote the concentrations of the reactants and products when the system has reached equilibrium.

A popular exercise for students involves arriving at the equality

A

from the initial reaction conditions, i.e., from initial concentrations of the reactants and the products at the start of the reaction and their respective associated rate constants, viz., k_f for A → B and k_r for B → A.^1,7−13 For convenience, we introduce a term, Q, defined as [B]_i/[A]_i; i.e., Q is the ratio of the initial concentrations of reactant to the product when t = 0. The exercise then involves the transformation of the system from its initial conditions, i.e., at Q, to that at equilibrium (K_eq).

This transformation from Q to K_eq takes place because the initial reactant and product concentrations are more often than not different from their concentrations when the system reaches equilibrium. The students are encouraged in the exercise to “manually convert” reactants to products and vice versa until Q = K_eq, i.e., until the system attains equilibrium and the concentrations of the reactants and products are transformed from their initial concentrations, to the concentrations found at equilibrium.

By conducting the manual transformation (discussed below) of reactants into products and vice versa, the students derive an understanding of how the equality, k_f[A]_e = k_r[B]_e, is achieved. This process is iterative. That is, the reactants are consumed to form products and vice versa until Q tends to K_eq (or ([B]_i/[A]_i) → [B]_e/[A]_e). The expert will recognize that the equality follows from the principle that at equilibrium the forward reaction rate equals the reverse reaction rate.

To reiterate, a typical exercise provided to students and researchers involves understanding how the system, A ⇋ B, proceeds to equilibrium by sequential transferring of units of A to B and vice versa. Each cycle in which reactants become products and the products become reactants is defined as the transfer cycle (TC). Repetitive TCs permit the reaction, from its arbitrary starting conditions such as [A]_i = 100 M, [B]_i = 0 M, k_f = 0.01 s^–1, and k_r = 0.005 s^–1, to reach equilibrium.

Frequently, objects such as uncooked beans, tokens, or cardboard cut-outs are employed in a laboratory setting for students to “feel” the process of chemical transfer in a safe manner.⁷⁻¹³ The stress on units is less important at this stage. For example, one “uncooked bean” may represent 1 M without taking into consideration the “volume” that the bean resides in. In practice, it is convenient to use, say, 100 beans to represent [A]_i and “0 beans” to represent [B]_i. The process by which A chemically converts to B is also not important. Lastly, the uncooked bean represents both A and B. Therefore, the chemical conversion of A to B, and vice versa, is achieved by physically moving (transferring) the bean from position “A” on a desk or a lab bench to another position, viz., B. The number of beans moved in the forward direction, every transfer cycle, is given by Rate = k_f[A]_i or, in this case, 0.01x[100] = 10 beans in the first TC. The number of beans back-transferred (in the direction B → A) in the first TC is given by k_r[B]_i = 0.005[0] = 0.

By repeating the TCs, the students recognize that the flux of beans from one direction eventually equals that in the other direction. This is because, even though the magnitude of the rate constants (k_f and k_r) does not change, the concentrations of the reactants and the products do. Thus, with each TC, the forward rate decreases because the initial concentration of A depletes rapidly because k_f[A] is large given that there is 100% A in the beginning of the reaction. Conversely, with each TC, the concentration of B increases, and the reverse rate, k_r[B], increases. The system reaches equilibrium when the forward rate (k_f[A]) equals the reverse rate (k_r[B]), and there is no more change in the concentration of the reactant or the product.

Thus, the equality in (A) is eventually used to derive the relationship between the rate constants and the equilibrium constant (K_eq). Through the “transfer” exercise, students learn that at equilibrium, given that k_f[A]_e = k_r[B]_e

B

The students can then proceed toward analyzing more complex chemical reactions such as

Transfer Cycles (TCs) and the Ratio of the Rate Constants (k_f/k_r)

Here, we examine and review the number of TCs (#TCs) required for a simple system, A ⇋ B, to reach equilibrium as a function of the ratio of forward to reverse rate constants (k_f/k_r). We also explore the said relation, viz., #TC vs k_f/k_r, as a function of the initial concentrations chosen for the reactants and products ([A]_i and [B]_i, respectively).

By permitting both students and instructors to choose appropriate values of the rate constants and starting concentrations of the reactants and products (various Q’s), this study facilitates an understanding of the relation between k_f, k_r, [A]_i, and [B]_i. In more general terms, we advance an understanding of the relation between k_f, k_r, with [A]_t and [B]_t, where the subscript “t” denotes the concentration of the reactant and product at any point during the reaction progress. For example, [A]_t = [A]_i and [B]_t = [B]_i when t = 0, and [A]_t → [A]_e and [B]_t → [B]_e as t → ∞.¹

Our analysis reveals a logarithmic growth-type relationship between the #TCs and k_f/k_r. The work underscores several special scenarios that are encountered in the relation between these variables which further enable students to understand the physics behind the conclusions drawn.

We also provide an online link that permits students to explore this relation by allowing the parameters ([A]_i, [B]_i, k_f, and k_r) to float.

Practical Details and Requirements

Paper, pencil, calculator, and graph paper can be used. A computer with a plotting program could be used in lieu of graph paper as it provides a more rapid mechanism to plot #TCs vs k_f/k_r. Access to the Internet is required to use the online program that allows users to input variables ([A]_i, [B]_i, k_f, and k_r). The program is available freely at https://repl.it/@jaehayi/K-and-Cycles. The “run” command executes the program. The initial reactant and product concentration and the rate constants can be input in the windows provided. They can be varied, and the program can be executed once more as follows: input values of [A]_i, [B]_i, k_f, and k_r into the program and execute it; record the number of transfer cycles required to achieve equilibrium and the ratio of k_f to k_r; and repeat the procedure choosing a larger ratio. For example, keeping [A]_i and [B]_i fixed at 100 and 0, respectively, use the program to compute the number of transfer cycles to reach equilibrium for k_f to k_r varying from, say, 2 to 10 in increments of 1 ratio unit. Record the number of transfer cycles required to achieve equilibrium for every k_f to k_r selected. Either using graph paper or a graphing program, plot the number of transfer cycles required to achieve equilibrium (y-axis) vs the k_f to k_r ratio (x-axis).

For calculating the number of transfer cycles across large spans of the k_f/k_r ratio (such as 0.1–10000), a graphing program becomes necessary. Furthermore, it may be advantageous to divide the #TCs vs the k_f/k_r plot into smaller ranges of ratios such as 0.1–1; 2–10; 2–100; and 10–10 000 to better appreciate the processes involved and to avoid data compression.

Discussion

Relation between #TCs and k_f/k_r

All discussions pertain to the simple reaction type A ⇋ B for which the following equality applies:

Units (M and s^–1) have been omitted for simplicity and to better focus on understanding the physics behind the processes involved.¹⁴⁻¹⁷

Figure 1 shows the number of transfer cycles required to reach equilibrium when only reactant is present ([A]_i = 100 and [B]_i = 0) and the forward rate constant at fixed at 0.1. As the reverse rate constant is increased 9-fold from 0.01 to 0.09 (Table 1), the #TCs required to reach equilibrium decrease from 55 (for k_r = 0.01; k_f/k_r = 10) to 23 (k_r = 0.09; k_f/k_r = 1.1111); i.e., as the ratio k_f/k_r decreases from 10 to 1.11, the #TC required for the system to converge decreases (55 → 23). This is because the product k_f[A] is much larger than the product k_r[B] when the difference between k_f and k_r is the largest (k_f/k_r = 10) compared to when the ratio between them is the smallest (1.111). Thus, the #TCs required to converge to equilibrium are correspondingly large since k_r[B] has to “catch up” to k_f[A] and requires the largest number of cycles to do so when the ratio between k_f and k_r is the largest (for a fixed value of initial reactant and product concentrations). Lastly, the plot reveals a logarithmic growth-type relation between k_f/k_r vs the number of transfer cycles; i.e., as the ratio increases, the #TCs required for the system to converge to equilibrium increase. These results suggest that the number of transfer cycles required to reach the equilibrium ratio of reactant and product concentration, viz., [B]_e/[A]_e, in a simple reaction of the type A ⇋ B is dependent on the magnitude of the forward to reverse rate constant ratio (k_f/k_r).

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

Table 1. k_f/k_r vs #TC When [A]_i = 100, [B]_i = 0, k_f = 0.1, and k_r = 0.01 → 0.09.

kf/kr	no. of transfer cycle
10	55
5	45
3.3	39
2.5	35
2	31
1.7	29
1.4	26
1.25	24
1.1	23

Figure 2 shows the same system as before with the ratio now further reduced from a previous minimum of 1.1 (in Figure 1) to 0.11 (Table 2). A further smooth decrease in the number of transfer cycles required to achieve equilibrium is observed as k_f/k_r recedes from 1.1 to 0.11. Overall, the #TC vs k_f/k_r plot mimics that in Figure 1. An interesting observation can be made while comparing the data in Tables 1 and 2. An ∼9-fold increase in k_f/k_r from 1.1 to 10 results in an ∼2.4-fold increase in the #TCs required for the system to equilibrate (Table 1). However, the same increase (from 0.11 to 1) results in a 21-fold increase (from 1 to 21) in the #TCs required to reach equilibrium (Table 2). This is consistent with the logarithmic growth nature of the relation between the #TCs required to reach equilibrium vs k_f/k_r, which initially rapidly rises before starting to flatten. Furthermore, the #TC required for equilibration of the system when k_f/k_r is reduced from 1.11 to 0.11 also decreases, a finding consistent with the previous conclusions (pertaining to Figure 1) where the #TC increased as the k_f/k_r ratio increased.

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.

Table 2. k_f/k_r vs #TC When [A]_i = 100, [B]_i = 0, k_f = 0.1, and k_r = 0.01 → 0.9.

[A]i = 100; [B]i = 0
kf/kr	#TC
10	55
5	45
3.33	39
2.5	35
2	31
1.67	29
1.42	26
1.25	24
1.11	23
1	21
0.5	13
0.33	9
0.25	6
0.2	5
0.16	4
0.14	3
0.12	2
0.11	1

Note: While it is easy to manually test scenarios where 1 or 2 transfer cycles alone suffice for equilibrium to be achieved, a true appreciation of how the system reaches equilibrium is realized when more transfer cycles are required. Such an approach helps a student examine how the concentration of the reactant progressively decreases and reactant increases (for k_f > k_r), while k_f and k_r remain constant throughout.

Figure 3 simulates a scenario where only the initial concentration of the reactant changes relative to the system in Figure 2. Here, the reactant concentration is 50 (Table 3) which is half the value of system 2. However, the k_f and k_r values (and hence the ratio k_f/k_r) are identical to the previous scenario. As can be predicted, the number of cycles required for the system to equilibrate as a function of k_f and k_r is the same. The student may infer that “smaller amounts” are being transferred since the initial concentration of the reactant is half that in the previous system.

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.

Table 3. k_f/k_r vs #TC When [A]_i = 50, [B]_i = 0, k_f = 0.1, and k_r = 0.01 → 0.9.

[A]i = 50; [B]i = 0
kf/kr	#TC
10	55
5	45
3.33	39
2.5	35
2	31
1.67	29
1.42	26
1.25	24
1.11	23
1	21
0.5	13
0.33	9
0.25	6
0.2	5
0.16	4
0.14	3
0.12	2
0.11	1

The data (#TC) obtained are again self-consistent and consistent across scenarios in Figures 1 and 2. The initial value of [A], as long as [B]_i is 0, has no impact on the #TCs required for the system to equilibrate as the “flux” being transferred is proportional to the concentration of reactant (and product).

We now consider scenarios where product is initially present (values represented in Table 4).

Table 4. k_f/k_r vs #TC When [A]_i = 100, [B]_i = 25–400, k_f = 0.1, and k_r = 0.01 → 0.9.

[A]i=100	number of transfer cycles
kf/kr	[B]i = 25	[B]i = 50	[B]i = 75	[B]i = 100	[B]i = 200	[B]i = 400
10	53	51	49	48	43	36
5	43	41	39	38	33	20
3.33	37	35	33	32	25	16
2.5	33	31	29	27	17	21
2	29	27	25	23	0	21
1.67	26	24	22	19	13	21
1.42	24	22	19	16	15	21
1.25	22	20	17	13	16	20
1.11	20	18	15	8	16	19
1	19	16	12	0	16	19
0.5	10	0	9	11	13	14
0.33	6	7	8	9	10	10
0.25	0	6	7	7	8	8
0.2	3	5	5	6	6	6
0.16	3	4	4	5	5	5
0.14	3	3	4	4	4	4
0.12	2	3	3	3	3	3
0.11	1	1	1	1	1	1

Figure 4 depicts conditions wherein, for [A]_i = 100, [B]_i varies as follows: 25, 50, 75, 100, 200, and 400. In each of these cases, since some [B] is initially present, it requires fewer TCs to reach equilibrium as compared to when [B]_i = 0 (Figure 3), for every k_f/k_r ratio tested. Furthermore, even though the profiles of #TC vs k_f/k_r appear to follow logarithmic growths, closer inspection reveals that there is a deviation from such behavior at each [B]_i tested. For example, in 4(A), at k_f/k_r = 0.25 for [B]_i = 25, the number of transfer cycles required is 0. This is because k_f[A]_i = 25, which is the same as k_b[B]_i; i.e., the system is already at equilibrium under the supplied conditions of k_f, k_r, [A]_i, and [B]_i. Similar “0” TCs are observed for [B]_i = 50 (4B), 100 (4D), and 200 (4E). This is because the initial conditions are such that the system is already at equilibrium. For [B]_i values of 75 (4C) and 400 (4F), a downward curvature toward the abscissa is seen. Nevertheless, the “0” TC point is not realized, and the TC vs k_f/k_r profile inflects upward thereafter. This is simply because the “0” TC condition (i.e., K = Q) is not coincidentally met with any of the choices of k_f, k_r, [A]_i, and [B]_i. However, if we had selected a more closely spaced ratio of k_f/k_r, a “0” point would have been realized.

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.

Lastly, as the initial concentration of B appreciates from 0 to 400, the #TCs required to reach equilibrium depreciate as a function of the k_f/k_r ratio (Figure 5; Table 5 inset).

Figure 5.

#TCs vs k_f/k_r.

Table 5. k_f/k_r vs #TC.

[B]i	#TC
0	55
25	53
50	51
75	49
100	48
200	43
400	36

However, the “sensitivity” to the concentration appears to vanish as the k_f/k_r ratio subsides from 10 to 0.11. This is an outcome of the plot itself which arises from the nature of the demarcations used in the graph vis-à-vis the number of data points obtained per interval. A large number of y-values within the same excursion on the x-axis provide a skewed view of the data.

We also examined purely integral values of k_f/k_r ratios where k_f ≥ k_r. For this relation, we chose a fixed value of [A]_i = 100 and varied [B]_i from 0 to 400 as before. For every [B]_i selected, we varied k_f/k_r from 1 to 20. The data (Figure 6) reveal that as the k_f/k_r ratio increases from 1 to 20 the number of transfer cycles increases from 21 (k_f/k_r = 1) to 63 (k_f/k_r = 20) (Table 6). While the explanation for this has already been discussed, we reiterate that such a behavior is to be expected since a large ratio between the forward and reverse rate constants would require “many steps for the process B → A to catch up with A → B”. Nevertheless, in conclusion, the trend seen in the plot is consistent with that observed in Figures 1–3 since it is the numerical value of k_f/k_r that dictates the #TCs required for the system to equilibrate.

Figure 6.

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

Table 6. k_f/k_r vs #TC When [A]_i = 100, [B]_i = 0 → 400, and k_f/k_r = 1 → 20.

kf/kr	#TC
1	21
2	31
3	38
4	42
5	45
6	48
7	50
8	52
9	53
10	55
11	56
12	57
13	58
14	59
15	60
16	61
17	61
18	62
19	63
20	63

As the concentration of B is increased from 0 to 400 (Figure 7; Table 7), for a fixed value of [A]_i = 100, the number of cycles required to achieve equilibrium is attenuated (as previously seen for fractional values of the same ratio). In three selected conditions, the system is already at equilibrium as evidenced by the number (0) of cycles required for transfer.

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.

It is to be noted that irrespective of the ratio of the rate constants being integral or fractional the function relating it to the number of transfer cycles remains similar in all cases (log growth) as long as the coincidental choice of Q = K is not met within the series of ratios chosen. As the concentration of [B]_i is increased (from 0 to 25 to 400), the number of cycles to reach K_eq reduces as previously seen. Again, deviations from the logarithmic growth are encountered whenever Q = K_eq, but the lag in the deviations becomes smaller and approaches the origin as expected when [B]_i increases from 0 to 400.

Conclusions

Using a simple chemical reaction, viz., A ⇋ B, as an example,^18,19 we have examined the number of forward and reverse reactions (TC) between reactants and products that is required for the reaction to reach equilibrium from its initial condition as a function of the forward and reverse rate constant ratio (k_f/k_r). Our analyses reveal that a plot of TC vs k_f/k_r follows a logarithmic growth curve, the reasons for which have been extensively discussed in this article. The exercise can be utilized by students of general chemistry to explore the nature of the components involved in the system, viz., the initial concentrations of reactant and product and the dependency of TC on these factors and the forward and reverse rate constant. Importantly, both the beginner student as well as researchers not familiar with the principles governing chemical equilibrium can potentially us this exercise to advance their understanding of the physics behind the chemistry or biology in any “real-world” scenarios. These include the role of chemical equilibria and their treatment in biological, environmental, chemical, and metallurgical engineering systems.²⁰⁻²⁴ The findings are also useful to design experiments in the laboratory for individual research applications and for teaching structured classes.

A user-friendly program that generates the number of TCs as a function of the rate constant ratio is made freely available for students and educators to peruse.

Biographies

Gyan M. Narayan is an entering freshman at UCLA having recently graduated from Silva Magnet High School in El Paso, TX. He is majoring in Biochemistry and eventually plans to join the medical field. A black-belt in Taekwondo, he enjoys reading and playing the violin.

Agustin Valles was born in El Paso, Texas, on August 28, 2003. His parents are Agustin Valles and Myrna Armenta, and he has a younger sibling. He lived in Juarez, Chihuahua (Mexico), until the age of 12 when his family moved to El Paso, Texas. He is currently a senior at Coronado High School and is on track to graduate by June 2021. He has a developed a deep interest in Chemistry throughout his high school career and plans on pursuing a chemistry degree in university. He also hopes to one day complete a doctoral degree in a science-related field and regularly participate in research.

Felix Venegas is a 12th grade student at Coronado High School, in El Paso, TX. He was in the 11th grade and taking Advanced Placement Chemistry when he researched about patterns in reaction kinetics and chemical equilibria. Felix has always been intrigued by the physical sciences as they describe natural phenomena. Their foundation in math is something that he believes he can use to his advantage. He wishes to continue studying in the STEM fields beyond high school. Currently, he is also involved in his school’s Orchestra and Student Council.

Jaeha Yi is a fourth-year student in the Coronado High School IB Program (El Paso, TX) and has been computer programming since age 12. He has received several recognitions in the STEM field including USA Computing Olympiad gold division, US National Chemistry Olympiad participant, and American Invitation Math Examination qualifier. He has published two android apps implementing computer vision. He is interested in multidisciplinary research, in real-world applications of computer science, and in machine learning.

Mahesh Narayan is a Professor in the Department of Chemistry and Biochemistry at the University of Texas at El Paso. The Narayan lab embraces a spectrum of research thrusts including advancing an atomic-level understanding of elements that drive protein folding and misfolding; defining amyloid–cross-toxic interactions that provoke comorbid outcomes; and developing and promoting carbon nanomaterial scaffolds and halogen bonding interactions as tools to ameliorate neuronal toxicity. The PI himself is also interested in chemical education and in promoting a physics-based understanding of chemical processes. He has authored and/or coauthored ∼85 research articles, reviews, and book chapters in the aforementioned areas plus another 10 works of education via lab manuals and work books. He remains on the editorial board of PLOS One and CBBI and has served as the guest editor for several issues of other publishing forums.

Author Contributions

^∥ G.M.N., A.V., F.V., and J.Y. contributed equally. M.N. conceived the topic. G.M.N., A.V., F.V., and J.Y. performed the calculations, generated the data, and contributed to the discussions. J.Y. compiled the program. All authors tested the program. M.N. wrote and reviewed the manuscript.

The authors declare no competing financial interest.