Simple mathematical equations for calculating oxidation number of organic carbons, number of transferred electrons, oxidative ratio, and mole of oxygen molecule in combustion reactions

Introduction

The balancing of chemical equations, stoichiometry, and redox reactions are some of the basic contents in chemistry curriculum (Chang & Goldsby, 2013; Tro, 2020). Redox reaction is an electron-transfer reaction, and it emerges with the oxidation number change (∆ON). The oxidation number (ON) and the number of transferred electrons (Te⁻) are two paramount parameters in the study of redox reactions (IUPAC, 2019). The oxidation number of organic carbons (ON_C) is widely used in the fields of organic and biological chemistry (Bentley, Franzen, & Chasteen, 2002; Hanson, 1990; Halkides, 2000; Yuen & Lau, 2022a), environmental chemistry (Kroll et al., 2011; Kroll, Lim, Kessler, & Wilson, 2015), and biogeochemistry (Chadwick, Masiello, Baldock, Smernik, & Randerson, 2007; Chadwick, Masiello, Baldock, Smernik, & Randerson, 2007; Dick, Yu, Tan, & Lu, 2019; Masiello, Hockaday, & Gallagher, 2007).

The net primary production (NPP) reactions and combustion reactions are two opposite redox reactions. Examples are shown as follows:

• Net primary production reaction: CO₂ + H₂O + N-resource → C _x H _y O _z N _v  + O₂

• Combustion reaction: C _x H _y O _z N _v  + O₂ → CO₂ + H₂O + N-product

The oxidative ratio (OR) is defined as the ratio of moles of O₂ to moles of CO₂ in NPP and combustion reactions (Gallagher et al., 2014; Randerson et al., 2006; Worrall, Clay, Masiello, & Mynheer, 2013). The relationship between OR and ON_C has been established in the study of NPP reactions (Hockaday et al., 2009, 2015; Masiello, Gallagher, Randerson, Deco, & Chadwick, 2008), and the relationships among OR, ON_C, and Te⁻ have been found in the study of organic combustion reactions (Yuen & Lau, 2022b). Although the stoichiometric amount of oxygen molecule (nO₂) plays a significant role in NPP and combustion reactions, the relationships among nO₂, OR, Te⁻, and ON_C have not been revealed.

Assigning ON and counting Te⁻ are important tasks in the study of redox reactions and they often pose difficulties for teachers and students (Brandriet & Bretz, 2014; De Jong, Acampo, & Verdonk, 1995; Garnett & Treagust, 1992). The purpose of this article is to explore a method which can solve the aforesaid problems. A combustion model is established to understand the relationships among ON_C, Te⁻, nO₂, and OR. Three pillars are needed. First, an organic combustion reaction acts as a model reaction. Second, the method of balancing and deducting chemical equations is used as a tool. Third, the stoichiometric coefficients (SC) in the balanced combustion reaction are used as connectors. Three sets of relations: (i) OR, ON_C, and Te⁻; (ii) ON_C and Te⁻; and (iii) nO₂ and Te⁻ are integrated. Consequently, new interrelationships among nO₂, OR, Te⁻, and ON_C are established. The Te⁻ and ON_C of organic compounds can also be determined by the derived mathematical equations.

Procedures for counting OR, Te⁻, and ON_C of organic molecules

The ON_C of organonitrogen are seldom studied (Kauffman, 1986; Jurowski, Krzeczkowska, & Jurowska, 2015; Yuen & Lau, 2022c). The general formula of organonitrogen compound, C _x H _y O _z N _v , is chosen as an example. Even though , , , and have the same molecular formula of C₃H₇O₃N, these four structural formulas have different ON_N.

Example 1. Balancing the combustion reaction of C₃H₇O₃N.

Step 1: Starting from the structural formula of C₃H₇O₃N.

Counting ON for non-carbon atoms:

• All hydrogen atoms: ON_H = +1.

• All oxygen atoms: ON_O = −2.

• One nitrogen atom: ON_N = −3.

Step 2: Setting up the combustion reaction.

The N-product plays a critical role in understanding the stoichiometric relationship in a combustion reaction. The ON_N of the reactant () is identified as −3, and the NH₃ (ON_N = −3) is selected as the N-product accordingly.

The combustion reaction is shown as: C₃H₇O₃N → CO₂ + H₂O + NH₃.

Step 3: Balancing the combustion reaction.

The combustion reaction is balanced by balancing atoms in the sequence from C atoms to N atoms to H atoms to O atoms.

Step 4: Identifying the stoichiometric coefficients of nO₂ and nCO₂.

Based on the balanced combustion reaction, the coefficients of nO₂ =  5 2 and nCO₂ = 3 can be identified.

Step 5: Counting OR from the ratio of nO₂ to nCO₂.

Based on the ratio of nO₂ to nCO₂ ( n O 2 n C O 2 ) , the OR =  5 6 can be determined.

Step 6: Counting Te⁻ from nO₂.

O₂ is the oxidizing agent in the combustion reactions. Each O atom gains 2 electrons (O + 2e⁻ → O⁻²) and each O₂ molecule gains 4 electrons (O₂ + 4e⁻ → 2O⁻²).

Step 7: Counting ON_C from Te⁻.

The relationship between Te⁻ and ∆ON_C has been established in the half redox reactions (Yuen & Lau, 2022d). The mathematical equations and calculation are shown as follows:

The mean ON_C of (C₃H₇O₃N) equals to + 2 3 .

Balancing combustion reactions: counting nO₂, nCO₂, OR, Te⁻, and ON_C

Nitrogen atoms have nine oxidation numbers (ON_N), the values of which lie between −3 to +5. By using the same procedure as shown in Example 1 (ON_N = −3), different N-products are assigned by different ON_N. The chemical formulas of N-products are combined with H-atom, O-atom, or both O-atom and H-atom. The resulted values of nO₂, nCO₂, OR, Te⁻, and ON_C are given in Table 1.

When Equations (1) and (2) are compared, NH₃ and N₂H₄ are found to have different ON_N, and they produce different nO₂, OR, and Te⁻.

When Equations (3) and (4) are compared, N-products of HNO₃ and N₂O₅ are found to contain the same ON_N even though they have different chemical formulas. Their nO₂, OR, and Te⁻ remain the same.

Balancing general combustion reactions: counting nO₂, nCO₂, OR, Te⁻, and ON_C by using SC

C _x H _y O _z N _v is used for balancing and deducting the combustion equation. NH₃ (ON_N = −3) is chosen as a N-product in Example 2.

Example 2. Balancing the general combustion reaction of C _x H _y O _z N _v .

1. Balancing the combustion reaction: C _x H _y O _z N _v  + O₂ → CO₂ + H₂O + NH₃ .

1. Expressing nO₂, nCO₂, OR, Te⁻, and ON_C by using SC.

By using the same procedure shown in Example 2, different N-products are assigned by different ON_N. Their values of ON_N, nO₂, OR, Te⁻, and ON_C are given in Table 2.

Formulating nO₂, nCO₂, OR, Te⁻, and ON_C of C _x H _y O _z N _v molecule

Any non-carbon ON (ON_H, ON_O, ON_N) of C _x H _y O _z N _v can be determined by its structural formula (Yuen & Lau, 2022c). Its atomic coefficients can also be identified.

• non-carbon ON: ON_H, ON_O, ON_N.

• atomic coefficient: x for carbon, y for hydrogen, z for oxygen, v for nitrogen.

By balancing the combustion reaction of C _x H _y O _z N _v  + O₂ → CO₂ + H₂O + [N-product], the general mathematical equations of C _x H _y O _z N _v molecules are given as follows:

The values of nO₂, Te⁻, nCO₂, OR, and ON_C can be calculated by using non-carbon ON of ON_H, ON_O, and ON_N, and atomic coefficients of x, y, z, and v.

Procedures for determining Te⁻, nO₂, nCO₂, OR, and ON_C of C _x H _y O _z N _v molecule and balancing its corresponding combustion reaction

Example 3. Given methyl red, .

1. Determining all non-carbon ON and atomic coefficients from the structural formula.

1. Calculating Te⁻, nO₂, nCO₂, OR, and ON_C by using mathematical equations.

1. Balancing the combustion reaction by using the SC of nO₂ and nCO₂.

The unbalanced combustion reaction of “C₁₅H₁₅O₂N₃ + O₂ → CO₂ + H₂O + N₂H₂ + NH₃” can be set up and balanced as follows:

Using the calculated SC of nO₂ and nCO₂:

Balancing N₂H₂ and NH₃:

Balancing H₂O:

In Example 3, the values of Te⁻, nO₂, OR, and ON_C can be calculated by the derived mathematical equations, which are based on the balanced organic combustion reactions. Reversely, the combustion reaction of C _x H _y O _z N _v molecules can also be balanced by the resulted nO₂ and nCO₂.

Interrelationships among Te⁻, nO₂, OR, and ON_C in C _x H _y O _z X _w N _v S _u P _t molecule

The strategy, which is used for balancing and deducting C _x H _y O _z N _v , can be extended to work on C _x H _y O _z X _w N _v S _u P _t molecule (X, N, S, P stand for halogen, nitrogen, sulfur, and phosphorus respectively). For C _x H _y O _z X_wN _v S _u P _t , all possible non-carbon ON are shown as: ON_H = +1, ON_O = −1 to −2, ON_X = −1 to +7, ON_N = −3 to +5, ON_S = −2 to +6, and ON_P = −3 to +5.

The general combustion reaction: C x H y O z X w N v S u P t + O 2 → C O 2 + H 2 O + [ O ‑ p r o d u c t ] + [ X ‑ p r o d u c t ] + [ N ‑ p r o d u c t ] + [ S ‑ p r o d u c t ] + [ P ‑ p r o d u c t ] .

1. ON and atomic coefficients for calculating Te⁻, nO₂, nCO₂, OR, and ON_C.

   • non-carbon ON: ON_H, ON_O, ON_X, ON_N, ON_S, ON_P.

   • atomic coefficients: x, y, z, w, v, u, t.

The derived general mathematical equations for C _x H _y O _z X _w N _v S _u P _t are shown as follows:

1. Relationships among SC, nO₂, OR, Te⁻, and ON_C.

By using combustion reactions as models, the relationships among nO₂, OR, Te⁻, and ON_C are derived. The tetrahedral model with SC in the center as connectors can be established and is shown in Figure 1.

Figure 1:

Tetrahedral relationships among nO₂, OR, Te⁻, and ON_C with SC in the center.

By integrating three sets of relationships of (i) OR and Te⁻; (ii) ON_C and Te⁻; and (iii) nO₂ and Te⁻, the interrelationships among four parameters of nO₂, OR, Te⁻, and ON_C are established. As a result, there are six sets of relationships among four parameters: nO₂, OR, Te⁻, and ON_C, which are derived and summarized in Table 3.

1. Triangular relationships among Te⁻, OR, and ON_C.

The Te⁻, OR, or ON_C can be determined by nO₂. The triangular relationship among Te⁻, OR, and ON_C with nO₂ in the center is shown in Figure 2.

1. Triangular relationships among nO₂, ON_C, and Te⁻.

Figure 2:

Triangular relationships among Te⁻, OR, and ON_C with nO₂ in the center.

The nO₂, ON_C, and Te⁻ can be determined by OR. The triangular relationship among nO₂, ON_C, and Te⁻ with OR in the center is shown in Figure 3.

1. Equivalence between the gain of electrons and the loss of electrons.

Figure 3:

Triangular relationships among nO₂, ON_C, and Te⁻ with OR in the center.

By using the combustion model, the gain of electrons from oxygen molecules (O₂) and the loss of electrons from carbon atoms of C _x H _y O _z X _w N _v S _u P _t molecule must be equal. It means that for all other atoms, their ON do not change in the combustion reaction.

Determining Te⁻, nO₂, OR, and ON_C of C _x H _y O _z X _w N _v S _u P _t molecule by using mathematical equations

Example 4. Given Celecoxib, .

• C _x H _y O _z X _w N _v S _u  = C₁₇H₁₄O₂F₃N₃S.

• ON_H = +1, ON_O = −2, ON_F = −1, ON_N = −3, ON_N = −2, ON_S = +4 

• x = 17, y = 14, z = 2, w = 3, v ₁ = 1, v ₂ = 2, u = 1

Example 5. Given Cangrelor, .

• C _x H _y O _z X _w N _v S _u P_t = C₁₇H₂₅O₁₂Cl₂F₃N₅S₂P₃

  • ON_H = +1, ON_O = −2, ON_F = −1, ON_Cl = −1, ON_N = −3, ON_S = −2, ON_P = +5 

  • x = 17, y = 25, z = 12, w ₁ = 2, w ₂ = 3, v = 5, u = 2, t = 3 

In Examples 4 and 5, the mathematical equation for counting Te⁻ of organic compounds in the general chemical formula of C _x H _y O _z X _w N _v S _u P_t is used and then nO₂, OR, and ON_C can be calculated consequently.

Procedures for balancing combustion reactions of C _x H _y O _z X _w N _v S _u P_t molecule and determining its corresponding Te⁻ and ON_C from nO₂ or OR

Example 6. Given Celecoxib, , C₁₇H₁₄O₂F₃N₃S.

1. Determining all non-carbon ON and all atomic coefficients.

1. Balancing the combustion reaction

The unbalanced combustion reaction of “C₁₇H₁₄O₂F₃N₃S + O₂ → CO₂ + H₂O + HF + H₂NNH₂ + NH₃ + SO₂” can be set up and balanced as follows:

Using the coefficient of nCO₂ (n_C = nCO₂):

Balancing HF:

Balancing H₂NNH₂ and NH₃:

Balancing SO₂:

Balancing H₂O:

Balancing O₂:

1. Calculating Te⁻ and ON_C from nO₂ or OR

Example 7. Given Cangrelor, , C₁₇H₂₅O₁₂Cl₂F₃N₅S₂P₃.

1. Determining all non-carbon ON and all atomic coefficients.

1. Balancing the combustion reaction.

The unbalanced combustion reaction of “C₁₇H₂₅O₁₂Cl₂F₃N₅S₂P₃ + O₂ → 17CO₂ + H₂O + HCl + HF + NH₃+H₂S + H₃PO₄” can be set up and balanced as follows:

Using atomic coefficient n_C = nCO₂:

Balancing HCl:

Balancing HF:

Balancing NH₃:

Balancing H₂S:

Balancing H₃PO₄:

Balancing H₂O:

Balancing O₂:

1. Calculating Te⁻ and ON_C from nO₂ or OR.

In Examples 6 and 7, firstly the combustion reactions of C _x H _y O _z X _w N _v S _u P_t molecules are balanced; secondly the nO₂, and nCO₂ can be identified; and thirdly their corresponding Te⁻ and ON_C can be determined by using nO₂ or OR.

Conclusions

The concepts of ON_C and Te⁻ play significant roles in both pure and applied chemistry. However, the tasks of assigning ON_C and counting Te⁻ are challenging. In this article, the interrelationships among nO₂, OR, Te⁻, and ON_C are established by using an organic combustion reaction as a model. Consequently, six sets of mathematical equations among these four parameters of organic compounds in the general chemical formula of C _x H _y O _z X _w N _v S _u P_t are formulated. Furthermore, ON_C and Te⁻ of a given organic compound can be determined by the derived mathematical equations.