Introduction

The topic of chemical equilibrium is usually discussed early on in the teaching of chemistry. The topic may be described in terms Gibbs free energy and chemical potential (Raff, 2014a–c; Raff & Cannon, 2016). These descriptions include the concept of the extent of reaction (ξ) which is defined in (1)–(2). The extent of reaction is an extensive property, has units of moles and ranges between 0 and very large numbers depending on the initial number of moles of reactants. A pedagogically more accessible approach to describe the progress of reaction has been suggested earlier (Dumon et al. 1993; Peckham 2001) and utilizes reaction advancement ratio (χ). χ is a dimensionless, intensive property (3) which ranges between 0 and 1 and is thus intuitively much easier to grasp for students than the extent of reaction or equilibrium constant (K). χ expressed as percentage (100χ) gives the reaction efficiency i.e. the percent of reactants converted to products at any time during the reaction. In the traditional teaching of equilibrium one uses Le Chatelier’s principle to get qualitative predictions or mass action quotient (Q) and K to obtain quantitative predictions of equilibrium state. Comparison of Q and K gives the direction along which the system currently not in equilibrium will proceed to reach the equilibrium state. K can then be used to calculate composition of the system at equilibrium quantitatively. This is because the initial reactant and/or product concentrations uniquely determine subsequent equilibrium concentrations (Weltin 1990). Using K to calculate equilibrium concentrations may involve solving polynomial equations which taxes the student’s mathematical ability and for equations of order >3 may even require the use of numerical methods. Furthermore, no consideration is usually given to % of reaction completed (reaction efficiency) once the equilibrium is reached. An additional calculation to extract reaction efficiency from the equilibrium concentrations is required.

Our aim in this work was to develop a quantitative approach which provides χ or reaction efficiency directly and is mathematically simple to apply regardless of the degree of resulting polynomials (i.e. without recourse to numerical methods).

χ is equivalent to the fraction of reaction completed at any time after the start of reaction. For irreversible, “complete reactions” χ = 0 at the start of reaction and χ = 1 when the reaction has been completed. Nevertheless, many chemical reactions are reversible and lead to the establishment of equilibrium state with 0 < χ < 1. The definitions of ξ are given in (1–2) where ν is the stoichiometric number of a particular reactant, n ₀ the initial number of moles of that reactant and n _t the number of moles of the reactant remaining after time (t) has elapsed since the beginning of reaction. The subscript e refers to equilibrium state.

The definition of χ is given in (3) and its value ranges between 0 (at the beginning) and 1 at the completion of irreversible reaction. For reversible reaction when the system assumes equilibrium state χ _e < 1 i.e. only a fraction of reactants would have been converted to products. ξ e   and   ξ max refer to changes in the number of moles of limiting reactant (LR) which took place upon reaching the equilibrium state and change in the number of moles which upon hypothetical complete conversion of LR into products, respectively.

The χ is a useful quantity not only in teaching chemical kinetics where it tells us how far has the reaction advanced towards completion at any given time, but also in teaching chemical equilibrium. In case of reversible reactions the value of χ e states what fraction of LR has been converted to products by the time the system has reached equilibrium state or what % of reaction has been completed (efficiency of reaction). The maximization of product yield is often the goal of chemical reaction and can be achieved (in general) by variation of temperature, pressure, volume, and concentration. However, in teaching laboratory, student often works with reversible reaction in solution where temperature, pressure and volume remain approximately constant and only the number of moles of reactants is varied to generate more products. Therefore the question arises: how will the change in concentrations (moles) of LR affect χ e ?

The quantitative relationships between χ e , the initial number of moles of reactants (concentrations) and K can be derived in only few special cases as described earlier (Novak 2020). χ e allows us to predict the maximum number of moles of product (and reaction efficiency) we can get in a reversible reaction under certain temperature, pressure and volume which is encompassed in the value of K. For irreversible reactions χ _max = 1 and the maximum amount of product attainable follows directly from the stoichiometric equation, but this is not true for reversible reactions.

Discussion

We shall discuss briefly several typical assessment problems used by chemistry teachers when teaching chemical equilibrium. The examples illustrating how to calculate equilibrium concentrations are presented in most textbooks, often using ICE tables. However, there is no explicit textbook description of how χ e can be obtained. Furthermore, textbook problems are selected in such a way that calculations involve solving (or can be reduced to) polynomial equations of the third order at the most. This is due to the necessity of using numerical methods for solving higher order polynomial equations. We present a graphical method which can be used to obtain χ _e for polynomials of any order. The χ _e can in turn be used to calculate equilibrium concentrations. Our approach is the reverse of the traditional one: we obtain χ _e first and then use it to get equilibrium concentrations instead of the other way around (using K to get equilibrium concentrations from which we extract χ _e).

Graphical method

In our method we were influenced by Weltin (1990, 1991 and assume that during the course of reaction temperature, pressure and volume remain constant. Weltin (1990) did not use graphical approach and used ξ rather than χ in his work.

Consider a general single-equation reversible reaction:

(4) aA + bB + ⇌ uU + vV +

We can define two polynomials pertaining to reactants (R) and products (P) as

(5) P ( χ ) = ( u 0 + u a χ a 0 ) u ( v 0 + v a χ a 0 ) v

(6) R ( χ ) = ( a 0 − χ a 0 ) a ( b 0 − b a χ a 0 ) b

where a ₀ is the initial concentration of limiting reactant. b ₀ u ₀ v ₀ …. are initial concentrations of other reactants and products. The two polynomials express how the concentrations of products and reactants change depending on χ. If no limiting reactant is present i.e. if reactants are present in stoichiometric amounts and if reversible reaction is initiated with reactants only (5) and (6) can be simplified to

(7) P ( χ ) = ( u a χ a 0 ) u ( v a χ a 0 ) v

(8) R ( χ ) = ( a 0 − χ a 0 ) a ( b 0 − χ a 0 ) b

At equilibrium K = P(  χ e )/R(  χ e ) or P(  χ e ) = K*R(  χ e ). Graphical expression of these equations leads to two polynomials plots of P(  χ e ) and K*R(  χ e ) versus χ which are represented by two curves which intersect uniquely at the value of χ e . The uniqueness of curve intersection is due to constraint that concentrations (numbers of moles of reactants and products) cannot have negative values. This approach constitutes a graphical method for solving polynomial equations (5–8) regardless of whether analytical solution exists or not. The polynomial equation P(  χ e ) = K*R(  χ e ) can of course be solved in χ numerically e.g. using bisection algorithm (Weltin 1990,1991), but the graphical method is easier for students to understand even though plotting may require the use of computer software.

Type 1

In this type of textbook problem the equilibrium concentrations are given and the quantity to be calculated is K. We only show how to calculate χ _e which is not usually required in textbook problems.

Example:

For reversible reaction C₂H₄ (g) + H₂O (g) ⇌ C₂H₅OH(g) at given temperature, the equilibrium concentrations are: [C₂H₄] = 0.0148, [H₂O] = 0.0336 , [C₂H₅OH] = 0.18. What is the advancement ratio at equilibrium if formula weights (FW) of ethene, water and ethanol are 28, 18 and 46, respectively?

The advancement ratio can be calculated according to the conservation of mass principle by considering total mass of products and total mass of reactants and products at equilibrium. Recalling that mass = number of moles*FW we obtain χ _e for the reaction above using (9)

(9) χ e = total   mass ( products ) total   mass   ( products + reactants ) = 0.18 ∗ 46 0.0148 ∗ 28 + 0.0336 ∗ 18 + 0.18 ∗ 46 = 0.8904

The value of 0.8904 is the reaction advancement ratio and it means that the reaction above is 89% complete (has 89% efficiency) at equilibrium. The linear relationship between the extent of reaction ξ and number of moles of individual reactants has been discussed by Vandezande et al. (2013). However, the pedagogical usefulness of ξ is inferior to χ as was pointed out earlier (Dumon et al. 1993; Peckham 2001).

Type 2

In this type of problems the initial concentrations of reactants and products as well as K are given. The required quantities are equilibrium concentrations of reactants and products.

Example:

In the reaction: Br₂(g) + Cl₂(g) ⇌ 2BrCl(g) initial concentrations were [Br₂] = 0.0305, [Cl₂] = 0.0243, [BrCl] = 0.0425 and K_c = 6.9 so that the limiting reactant is Cl₂. What is the advancement ratio χ e and % completion (efficiency) of the reaction in the equilibrium state? We first calculate mass action quotient Q = [BrCl]²/[Br₂]*[Cl₂] which equals 2.44 and which indicates that the system will move towards equilibrium state by converting some of Br₂ and Cl₂ into BrCl.

We use graphical method based on (5–8) and P(  χ e ) = K*R(  χ e ) to plot the two polynomials using Mathematica software with code given below. “f” in the code designates χ. The result is shown in Figure 1.

Figure 1:

Plots of P(χ) and K*R(χ) as a function of advancement ratio for reaction.

r=6.9*(0.0243-f*0.0243)*(0.0305-f*0.0243)

p=(0.0425+2*0.0243*f)ˆ2

Plot[{r,p},{f,0,1}, AxesLabel->{advancement, "[M]²"}]

Br₂(g) + Cl₂ ⇌ 2BrCl. The intersection point in the plot is at χ _e = 0.255 i.e. reaction in equilibrium state is 25.5% complete. The right plot is the expanded part of the left plot to allow more accurate readout of advancement ratio.

Using χ _e = 0.255 from the plot we calculate the equilibrium concentrations obtaining:

• [BrCl] = (0.0425–2*0.255*0.0243) = 0.0551

• [Br₂] = (0.0305–0.255*0.0243) = 0.0243

• [Cl₂] = (0.0305–0.255*0.0243) = 0.0181

As a check, when inserting these equilibrium concentrations into K = [BrCl]²/[Br₂]*[Cl₂] we do obtain K = 6.9.

Other computer software (Maple, Matlab, Excel) can also be used to plot requisite polynomials. The graphical method allows students to calculate advancement ratio and % reaction completion (efficiency) easily and from these arrive at equilibrium reactant and product concentrations (numbers of moles). Changing reactant concentrations leads to direct visualization of changes in χ. This visualization is useful when co-presenting qualitative Le Chatelier’s principle with quantitative advancement ratio (or % completion) of reaction.

Example:

Esterification reaction below has K = 0.11 at 373 K. What are the equilibrium concentrations of four species if initially there was 0.81 mol of ethanol and 0.81 mol of acetic acid present?

C 2 H 5 OH ( aq ) + CH 3 COOH ( aq ) ⇌ CH 3 COOC 2 H 5 ( aq ) + H 2 O   ( l )

Using graphical method and Mathematica code we obtain plots in Figure 2.

r=0.11*(0.81-f*0.81)ˆ2

p=(0.81*f)ˆ2

Plot[{r,p},{f,0,1}, AxesLabel->{advancement, "[M]²"}]

Figure 2:

Plots of P(χ) and K*R(χ) as a function of χ for reaction.

C₂H₅OH(aq) + CH₃COOH(aq) ⇌ CH₃COOC₂H₅ (aq) + H₂O (l). The intersection point is at χ _e = 0.249 i.e. reaction in equilibrium state is 24.9% complete. The right plot is the expanded part of the left plot to allow more accurate readout of advancement ratio.

Using χ _e = 0.249 from the plot we calculate the equilibrium concentrations obtaining:

• [C₂H₅OH] = [CH₃COOH] = (0.81–0.249*0.81) = 0.60831

• [CH₃COOC₂H₅] = [H₂O] = (0.249*0.81) = 0.20169

As a check, inserting these equilibrium concentrations into

K = [CH₃COOC₂H₅][H₂O]/[C₂H₅OH][CH₃COOH we do obtain K = 0.11

Type 3

In this type of problems the effect of change in concentration (moles) of reactants on equilibrium state is discussed.

Example:

Isomerization reaction: butane (g) ⇌ isobutane (g) has K = 2.5 and equilibrium concentrations are [butane] = 0.5 and [isobutane] = 1.25. At equilibrium we use plot to get χ _e = 0.7143 which gives 71.43% reaction efficiency. What happens to χ _e when we add 1.5 mol of butane to the equilibrium mixture?

Using graphical method and Mathematica code we obtain plots in Figure 3.

r=2.5*(2-f*2)

p=(1.25+2*f)

Plot[{r,p},{f,0,1}, AxesLabel->{advancement, "[M]"}]

Figure 3:

Plots of P(χ) and K*R(χ) as a function of advancement ratio for reaction butane (g) ⇌ isobutane (g). The intersection point is at χ _e = 0.536 i.e. reaction in equilibrium state is 53.63% complete. The right plot is the expanded part of the left plot to allow more accurate readout of advancement ratio.

An interesting observation is that the absolute increase in the moles of reactants does not necessarily lead to increased advancement ratio or reaction efficiency. In this example adding extra butane brings efficiency down from 71.43 to 53.63%. The new equilibrium numbers of moles are [butane] = 0.928 and [isobutane] = 2.322. As a check K = 2.322/0.928 equals K = 2.5 as it must.

Conclusions

Our graphical approach provides general, quantitative method for solving equilibrium problems in reversible single-equation reactions (in a closed system). It can be used by students concurrently with the well-known qualitative Le Chatelier’s principle and may be seen as alternative to ICE tables. The approach also introduces students of general chemistry to the concept of the advancement ratio of reaction χ and % efficiency of reaction.

It would be beneficial to students if the discussion on the reaction advancement ratio was to take place simultaneously with the discussion of Le Chatelier’s principle to highlight the differences and similarities. For example, the general prediction by Le Chatelier’s principle that increase in concentration of reactants in system at equilibrium gives more products does not necessarily imply increase in the advancement ratio of reaction (reaction efficiency)!

χ _e is easier for students to grasp intuitively than K, since the former quantity is restricted to 0–1 numerical range while K values span perhaps 60 orders of magnitude! The reason why K is taught more often is that K is independent of initial reactant and product concentrations while χ _e is uniquely defined by them (Weltin 1990). Nevertheless, for any reversible reaction, χ _e can be readily calculated from (9) using known concentrations (moles of reactants and products) in equilibrium state.