Buswell’s Equation for Quantifying Biohydrogen

Biohydrogen is widely generated by dark fermentation and two-stage anaerobic digestion. Anaerobic digestion can be chemically represented by the Buswell’s equation, which is based on elemental composition of organic matter. When compared to Buswell’s equation for biomethane, its equation for biohydrogen has been given less attention. In this article, the nature of Buswell’s equation for biohydrogen is introduced by using the H-atom method. The mean oxidation number of organic carbons is employed as a metric for counting theoretical quantity of biohydrogen and parameters of Buswell’s equation for biohydrogen. Based on an empirical formula and mean oxidation number of organic carbons, the general Buswell’s equation for biohydrogen is developed. The mathematical relationships among mean oxidation number of organic carbons, empirical formula, quantity of biohydrogen, theoretical biohydrogen potential, and biohydrogen percent yield are also established. Biowastes and bio-substrates are chosen to demonstrate this notion.

In this article, the H-atom method (Yuen & Lau, 2021) is used to introduce the nature of BEqH through balancing, quantifying, deducting, and defining its stoichiometric equation.The mean oxidation number of organic carbons (ONc) acts as a metric for developing the ONc-BEqH model, in which the mathematical relationships among ONc, empirical formula, quantity of biohydrogen (nH 2 ), theoretical biohydrogen potential (TBHP), and theoretical biohydrogen yield (TBHY) can be established.The nH 2 , nCO 2 , ratio of nH 2 to nCO 2 , TBHP, and TBHY can also be determined.Biowastes and bio-substrates are chosen to demonstrate the counting of BEqH's parameters, biodegradability index (BDI) and biohydrogen percent yield (BHPY).

H-atom Method for Understanding BEqH
The H-atom method can be used to study Buswell's equations (Yuen & Lau, 2023a;2023b) and to analyze the nature of BEqH.

Balancing BEqH of Organic Matters
Example 1. Balancing the BEqH for an empirical formula, C 3 H Step 1. Divide overall equation into two half reactions Step 3. Combine two half reactions with equivalence of H-atom (LCM = 10) Step 1. Divide overall equation into two half reactions Step 2. Balance two half reactions

Defining Redox Terms of BEqH
By using the H-atom model (IUPAC, 2019), the redox terms of Example 2 are demonstrated in Figure 1.

Figure 1. The H-atom method for BEqH
The redox terms of Example 2 are summarized in Table 1.C x H y O z X w N v S u P t molecules not only gain H-atom (from H atoms) to form hydrogen gas, but also lose H-atom (from C atoms) to produce carbon dioxide.In addition, the H-atom of H 2 O gains one H-atom from C x H y O z X w N v S u P t to form hydrogen gas.There are three sets of redox couples in BEqH.Assumptions of the model: • BEqH is a redox reaction for pure or mixed organic matters in the form of an empirical formula.
• Organic carbons are fully converted to CO 2 .
• All halogen, nitrogen, sulfur, and phosphorous (X, N, S and P) atoms are converted to HX, NH 3 , H 2 S and H 3 PO 4 , respectively.

ONc-BEqH Model
The ONc-BEq model for biomethane was established by integrating the concept of ONc and BEq model (Yuen & Lau, 2024a;2024b).The same method is applied to construct the analogous ONc-BEqH model.The mathematical equations for the parameters of BEqH are shown below.

ONc of Empirical Formula, C x H y O z X w N v S u P t
The ONc can be determined by using the non-carbon atom method (Yuen & Lau, 2023b).Regarding empirical formulas C x H y O z X w N v S u P t of organic matters, individual oxidation number of hydrogen, oxygen, halogen, nitrogen, sulfur, and phosphorous (H, O, X, N, S, and P) atoms are assumed to be +1, −2, −1, −3, −2 and +5 respectively. x

BEqH and ONc-BEqH Models: Stoichiometric Chemical Equations
In the conversion of BEqH to ONc-BEqH, their stoichiometric chemical equations are shown as follows: C

, and TBHP
In Figure 2, the working scheme shows how empirical formula is converted to nH 2 ,  ))

Operating Scheme
When the mass% of elements are identified, organic matter's empirical formula and ONc can be determined.Consequently, the empirical formula mass (μ), nH 2 and theoretical biohydrogen potential (TBHP) can be counted.Figure 3 shows the operating scheme.The biodegradability index for nH 2 in BEqH is analogous to biodegradability index for nCH 4 in BEq (Nielfa et al., 2015).The biodegradability index for biohydrogen (BDI) and biohydrogen percent yield (BHPY) can then be calculated.

Data Collection
The C/H/O/X/N/S/P contents of most organic matters are attained by elemental analysis.The selected ultimate analysis of dried biowastes which include the six elements of C/H/O/Cl/N/S (Achinas & Euverink, 2016) and C/H/O/N/S/P (Zaher et al., 2010) are retrieved.The empirical formulas (C x H y O z Cl w N v S u P t ) of selected biowastes are calculated from mass% of elements (Yuen & Lau, 2023b;2024a).The data are listed in Table 2.

× 100%
When the experimental biohydrogen potential (EBHP) of bio-substrate is identified, the BDI can be counted.Three sets of calculated μC x H y O z , ONc, nH 2 , theoretical biohydrogen potential (TBHP), and BDI are summarized in Table 4.

× 100%
When the experimental biohydrogen yield (EBHY) of bio-substrate is identified, the BHBY can be counted.The value of EBHY is equal to nH 2 .Two sets of calculated μC x H y O z , ONc, theoretical biohydrogen yield (TBHY), and BHPY are summarized in Table 5.

Conclusion
The nature of Buswell's equation for biohydrogen (BEqH) is explored by using the H-atom method.

Table 1 .
The redox terms of the Buswell's equation for biohydrogen x H y O z X w N v S u P t +(2x−z+4t)H 2 O→ 2.5 Nature of BEqHBEqH is an overall microbial-mediated biochemical redox reaction.It is represented in the form of a stoichiometric molecular chemical equation.BEqH is composed of three sets of redox couples.All non-carbon and non-hydrogen atoms are non-redox atoms.Water is involved in the reaction.In the absence of O 2 , carbon atoms lose H atoms to form CO 2 and hydrogen atoms gain H atoms to form H 2 .H 2 O molecules also gain H atoms from organic matter to form H 2 molecules.C x H y O z X w N v S u P t +(2x−z+4t)H 2 O→

From Counting ONc to Determining nH2,
y O z X w N v S u P t 4.

Table 2 .
Selected biowastes: ultimate analysis and their calculated empirical formula

Table 4 .
Selected bio-substrates for molecular formula C x H y O z : calculated μC x H y O z , ONc, nH 2 , TBHP, and BDI

Table 5 .
Selected bio-substrates for molecular formula C x H y O z : calculated μC x H y O z , ONc, TBHY, and BHPY (%) The general ONc-BEqH is established as C x H y O z X w N v S u P t +(2x−z+4t)H 2 O→H 2 +xCO 2 +wHX+vNH 3 +uH 2 S+tH 3 PO 4 .For any given empirical formula of an organic matter, its BEqH parameters can be calculated by the following mathematical equations: