Using Mean Oxidation Number of Organic Carbons to Quantify Buswell’s Equation

,

AD has been given less attention than aerobic digestion in biochemistry textbooks (Shen & Chen, 2021) and BEq is rarely mentioned in physical sciences and life sciences.Many students are unable to understand and manipulate the stoichiometric BEq.Although the concept of mean oxidation number of organic carbons has been applied in organic chemistry and biochemistry (Hanson, 1990;Halkides, 2000;Bentley et al., 2002;Yuen & Lau, 2022a), environmental chemistry (Kroll et al., 2011), biogeochemistry (Masiello et al., 2008;LaRowe & Van Cappellen, 2011;Dick et al., 2019), water treatment (Vogel et al., 2000), and organic combustion reaction (Yuen & Lau, 2023a), the relationship between ON C and nCH 4 has not been studied.
The purpose of this article is to explore the BEq model which can help unpack the redox nature of organic compounds and establish the mathematical relationship between mean oxidation number of organic carbons (ON C ) and theoretical quantity of biomethane (nCH 4 ).The H-atom method is used for balancing and deducting BEqs (Yuen & Lau, 2023b).The mathematical equation between ON C and nCH 4 is then derived.Lastly, the Buswell's ratio, biogas ratio, biomethane content percentage, and theoretical biochemical methane potential (TBMP) can be calculated by using ON C as a metric for any given organic compound.

Procedures for Counting ONN, ON C , and Balancing BEq
Organonitrogen compounds are rich in nature and nitrogen is an essential element in life (Bada, 1998).The general formula of organonitrogen compound, C x H y O z N v , is chosen to be the model molecule for balancing and deducting BEqs.Step 3. Balance the BEq by using the H-atom method (Yuen & Lau, 2021;2022b) Step 3.1 Divide an overall equation into two half reactions Step 3.2 Balance two half reactions (from C→N→O→H) Step 3.3 Combine two half reactions with equivalence of H-atom (LCM = 30) The ON C of (C 2 H 5 O 2 N) equals +1.

Relationships among ONN, ONC, and Stoichiometric Coefficients of C2H5O2N Molecules in BEq
By using the same procedure shown in Example 1 (ON N = −3), different N-products are assigned by different ON N .
Nitrogen atoms have nine individual oxidation numbers (ON N ) and their values lie between the integer range of −3 to +5.
The chemical formulas of N-products are combined with H-atom, O-atom, or both O-atom and H-atom.The resulting oxidation numbers of ON N and ON C , stoichiometric coefficients of nH 2 O, nCH 4 , and nCO 2 are summarized in Table 1.
When the value of nH 2 O is positive, it is on the reactant's side.When the value is negative, it is on the product's side.
With reference to Table 1, when Equation 1 and Equation 3 are compared, NH 3 (ON N = −3) and N 2 H 2 (ON N = −1) have different ON N , and their ON C are +1 and 0 respectively.Consequently, they produce different stoichiometric coefficients of nCH 4 and nCO 2 .
Equation 1: When Equation 8and Equation 9 are compared, N-products of HNO 2 and N 2 O 3 have identical ON N but different chemical formulas.Their ON N produce the same ON C of −2.Their stoichiometric coefficients of nCH 4 and nCO 2 remain the same in Equations 8 and 9.
Equation 8: BEq is a biochemical redox reaction, in which the carbon atoms of an organic molecule disproportionate to CH 4 and CO 2 .An ideal BEq model indicates that electron transfer occurs in sole organic carbons of organic compound, and other noncarbon atoms are not involved in the redox process.
Step 2. Deduct the BEq by using the H-atom method Step 2.1 Divide an overall equation into two half reactions Step 2.2 Balance two half reactions The value of ON for hydrogen is either −1 or +1, for oxygen it is either −1 or −2, and for nitrogen it is between the integer range of −3 to +5.By using the above derived general mathematical equations, the balanced BEq of C x H y O z N v molecule can be found, and is shown in the following example.

Mathematical Relationships among nCH4, nCO2, and ONC in CxHyOzXwNvSuPt Molecule
The strategy, which is used for balancing and deducting C x H y O z N v , can be extended to work on organic molecules containing the molecular formula C x H y O z X w N v S u P t (C, H, O, X, N, S, P stand for carbon, hydrogen, oxygen, halogen, nitrogen, sulfur, and phosphorus element respectively, and x, y, z, w, v, u, t stand for their corresponding atomic coefficients).For C x H y O z X w N v S u P t , all possible ON of non-carbon atoms are shown as: ON H = +1 or −1, ON O = −1 or −2, ON X = integer between −1 and +7, ON N = integer between −3 and +5, ON S = integer between −2 and +6, and ON P = integer between −3 and +5.
The general unbalanced BEq is shown as: The     ) can be determined by using the sole ON C , or ON C to complement x and/or μ.These parameters and their mathematical equations are summarized in Table 3.
When ON C for any given molecule is determined, the parameters of BEq can be counted.According to the molecules in Examples 3 -7, all the resulting values of parameters of BEq are exhibited in Table 4.According to the given molecules in Examples 3 − 7, all resulting values are summarized in Table 5.The TBMP can be calculated by ON C , x, and molecular mass (μ) for any given organic molecule.Their ON C values are +1, 0, and −2 respectively.All parameters of BEq are calculated and summarized in Table 6.Among these isomers of C 2 H 5 O 2 N, the isomer carrying the highest reduced ON C can produce the most TBMP.

Conclusion
The ON C is an important redox concept in organic chemistry and biochemistry.In this article, the BEq model is developed by using the H-atom method for understanding the redox nature of organic compounds.ON C acts as a metric for quantifying parameters of BEq.The mathematical relationship between ON C and nCH 4 is established as nCH 4 = x(4−ON c )
Consequently, for any given structural formula of an organic molecule, the parameters of BEq, such as quantity of biomethane, Buswell's ratio, biogas ratio, biomethane content percentage and theoretical biochemical methane potential can be determined.
The chemical formula of C 2 H 5 O 2 N has different isomers.Three of them are selected and shown as follows: , , and .They have different ON N .The working procedure for assigning ON N , ON C , and balancing BEq is shown in Example 1. Example 1. Establishing the balanced Buswell's equation (BEq) of C 2 H 5 O 2 N Step 1. Choose the structural formula of as the reactant Count individual oxidation number (ON) of non-carbon atoms (Yuen & Lau, 2022a) for : all hydrogen atoms: ON H = +1 all oxygen atoms: ON O = −2 one nitrogen atom: ON N = −3 Step 2. Set up the BEq The ON N of the reactant ( ) is identified as −3.The NH 3 (ON N = −3) is selected to be the N-product accordingly.The unbalanced BEq is shown as: C 2 H 5 O 2 N+H 2 O→CH 4 +CO 2 +NH 3

4.
Deducting the General BEq of CxHyOzNv Molecules NH 3 (ON N = −3) is chosen as a N-product to balance BEq of C x H y O z N v in Example 2. Example 2. Deducting the BEq reaction of C x H y O z N v Step 1. Set up the BEq reaction by choosing [N-product] (ON N = −3) Identify the stoichiometric coefficients of nH 2 O, nCH 4 , and nCO 2 By using the atomic coefficients (AC) of the chemical formula, the stochiometric coefficients (SC) can be represented as follows: For [N-product] = NH 3 (ON N = −3) Count the ON C Count ON of non-carbon atoms: all hydrogen atoms: ON H = +1 all oxygen atoms: ON O = −2 one nitrogen atom: ON N = −3 By using the molecular formula method: ΣON i = 0 xON C + y − 2z − 3v = 0 xON C = − y + 2z + 3v ON C = −y+2z+3v x The ON C of C x H y O z N v is equal to AC, the assigned ON of ON H = +1, ON O = −2, and ON N , the SC of nCH 4 , nCO 2 , and ON C can be represented as follows: H y−ON O z−ON N v 8 = 4(9)−(1)(11)+(2)(3)−(−3)(1 4x−ON H y−ON O z−ON X w−ON N v−ON S u−ON P t 8 ON C = −ON H y−ON O z−ON X w−ON N v−ON S u−ON P t x By using these derived general mathematical equations, the balanced BEqs can be found by given structural formulas in Examples 4 to 7. Example 4. Given an example of taurine, , C 2 H 7 O 3 NS (a) Identify AC and ON of non-carbon atomsAtom Use mathematical equations to calculate nCH 4 , nCO 2 , and ON C Unbalanced BEq: C 10 H 16 O 13 N 5 P 3 +H 2 O→CH 4 +CO 2 +NH 3 +H 3 Use mathematical equations to calculate nCH 4 , nCO 2 , and ON C Unbalanced BEq: C 9 H 11 O 3 Cl 3 NSP+H 2 O→CH 4 +CO 2 +HCl+NH 3 +H 2 S+H 3

Table 1 .
Parameters of C 2 H 5 O 2 N molecules in the balanced BEqs

Table 2 .
1 Assigning Oxidation Numbers of ON N and ON C , and Balancing BEq for nH 2 O, nCH 4 , and nCO 2 By using the same procedure shown in Example 2, different N-products are assigned by different ON N .The unbalanced BEq is shown as: C x H y O z N v +nH 2 O H 2 O→nCH 4 CH 4 +nCO 2 CO 2 +v[N-product].Their ON N , nH 2 O, nCH 4 , nCO 2 , and ON C are given in Table 2. Stoichiometric coefficients of nH 2 O, nCH 4 and nCO 2 , and oxidation numbers of ON N , and ON C in the balanced BEq of C x H y O z N v 2 Mathematical Relationships among SC, AC, and ON of C x H y O z N v Molecules The values of nCH 4 , nCO 2 , and ON C can be calculated by using ON of ON H , ON O , and ON N , and AC of x, y, z, and v.By balancing the BEq: C x H y O z N v +H 2 O→CH 4 +CO 2 +[H-product]+[O-product]+[N-product], the general derived mathematical equations of C x H y O z N v molecules are given as follows: BEq by counting nH 2 O C 9 H 11 O 3 N+nH 2 O H 2 O→nCH 4 CH 4 +nCO 2 CO 2 +NH 3 C 9 H 11 O 3 N+nH 2 O H 2 O→ ON of non-carbon atoms and AC for calculating nCH 4 , nCO 2 , and ON C are shown as follows: Balance BEq by counting nH 2 O C 9 H 11 O 3 Cl 3 NSP+nH 2 O H 2 O→

Table 3 .
Mathematical equations for parameters of BEq and ON C

Table 4 .
ON C for determination of nCH 4 , nCO 2 , and parameters of BEq

Table 5 .
Counting TBMP by ON C , x, and μ With reference to Table 1, three structural formulas of C 2 H 5 O 2 N are , , and .

Table 6 .
ONc for determination of nCH 4 , nCO 2 , and BEq's parameters for C 2 H 5 O 2 N molecules