📄 chemistry

Chemical equilibrium constants of diatomic molecules from a q-deformed hyperbolic interaction potential

This study employs a q-deformed hyperbolic interaction potential solved via the parametric Nikiforov-Uvarov method to calculate and validate the chemical equilibrium constants for the formation of BrF, ClF, IF, and SO against NIST-JANAF data, demonstrating high accuracy within specific temperature ranges.

Original authors: E. S. Eyube, D. Festus, A. Zakariya, Saïdou Alioum, P. A. Njidda, I. Joel, T. G. Nyagong, J. Abe

Published 2026-09-28✓ Author reviewed ⓘ

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: E. S. Eyube, D. Festus, A. Zakariya, Saïdou Alioum, P. A. Njidda, I. Joel, T. G. Nyagong, J. Abe

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the invisible world of atoms, molecules are not static beads but dynamic systems held together by invisible forces that stretch and compress like springs. To understand how these tiny structures behave, scientists rely on mathematical maps called potential energy functions. These maps describe how the energy between two atoms changes as they move closer together or drift apart, dictating everything from how tightly they hold on to how they vibrate when heated. By knowing these energy levels, researchers can predict how molecules will act in large groups, calculating the temperature-dependent properties that drive chemical reactions. This connection between the microscopic dance of individual atoms and the macroscopic behavior of gases is essential for fields ranging from atmospheric science to combustion engineering, allowing us to predict whether a reaction will happen and how far it will proceed under different conditions.

A team of researchers recently set out to refine these maps using a specific mathematical model known as a q-deformed hyperbolic interaction potential. This model offers a flexible way to describe the complex, non-linear forces that bind atoms together, particularly for diatomic molecules, which consist of just two atoms. The team focused on four specific molecules: bromine fluoride, chlorine fluoride, iodine fluoride, and sulfur monoxide. These are not just random choices; the first three are interhalogen compounds formed from different halogen atoms, while sulfur monoxide is a key player in combustion and atmospheric chemistry. The researchers wanted to see if their specific mathematical model could accurately predict the chemical equilibrium constants for these molecules. An equilibrium constant is essentially a number that tells scientists how much of a product will exist compared to its starting ingredients at a given temperature, a crucial piece of information for understanding chemical stability and reaction yields.

To do this, the team first solved the fundamental equations governing the motion of the atoms within these molecules. They used a specialized mathematical technique to find the exact energy levels associated with the vibration of the atoms, treating the bond between them as a vibrating system. Once they had these energy levels, they built a statistical description of the molecules, known as a partition function, which sums up all the possible ways the molecules can store energy. This calculation included the vibrations of the atoms, their rotation in space, and their movement through the air. Using these components, they computed the Gibbs free energy, a thermodynamic quantity that determines the direction and extent of a chemical reaction. To make the model fit reality as closely as possible, the researchers applied a small, molecule-dependent correction factor to the calculated Gibbs free energy, ensuring the values matched known reference data. They then used these results to derive the equilibrium constants for the formation of each molecule from its homonuclear reactants.

The results of this work were compared against a trusted database of experimental values known as NIST-JANAF, which serves as the gold standard for thermochemical data. The comparison revealed that the new model is remarkably accurate, but its performance depends heavily on the specific molecule and the temperature range being studied. For bromine fluoride, the model produced results that were incredibly close to the reference data across a vast temperature span, from 300 Kelvin up to 6000 Kelvin, with an average difference of less than one percent. For chlorine fluoride, the model worked very well up to 4600 Kelvin, but the accuracy began to slip at higher temperatures. Iodine fluoride showed a similar pattern, with the largest discrepancies appearing at the lowest temperatures, while sulfur monoxide presented a unique case where the model struggled significantly below 900 Kelvin but aligned almost perfectly with reference data at higher temperatures.

These findings demonstrate that the q-deformed hyperbolic interaction potential is a powerful tool for predicting chemical behavior, but it is not a one-size-fits-all solution. The study shows that while the model can reproduce reference equilibrium data with high precision over broad temperature intervals for some molecules, it has specific limits where the deviations become noticeable. For bromine fluoride, the model is reliable across the entire tested range. For the others, the researchers identified the specific temperature windows where the predictions remain trustworthy and where they begin to diverge from established values. This work provides a clear, quantitative assessment of where this mathematical approach succeeds and where it falls short, offering a refined method for calculating the equilibrium constants of diatomic molecules without needing to rely solely on experimental measurements for every new condition.

Technical Summary: Chemical Equilibrium Constants of Diatomic Molecules from a q-Deformed Hyperbolic Interaction Potential

Problem Statement
The prediction of chemical equilibrium constants for gaseous systems relies fundamentally on accurate molecular energy levels, which are derived from interaction potentials. While numerous analytical and semi-analytical potential energy models exist (e.g., Morse, Kratzer, Deng-Fan), there is a need to extend these models beyond spectroscopic constants to macroscopic thermodynamic quantities, specifically Gibbs free energies and equilibrium constants. Although q-deformed potentials have been investigated for molecular spectra and thermodynamic functions, their application to the statistical-mechanical determination of chemical equilibrium constants over extended temperature intervals remains underexplored. This study addresses the gap in applying a q-deformed hyperbolic interaction potential (qHIP) to predict the equilibrium constants for the formation of specific diatomic molecules (BrF, ClF, IF, and SO) from their homonuclear reactants.

Methodology
The research employs a theoretical framework based on the q-deformed hyperbolic interaction potential (qHIP), defined by interaction strength ( $U_0$ ), a screening parameter ( $\alpha$ ), and a deformation parameter ( $q$ ). The methodology proceeds through the following steps:

Solving the Schrödinger Equation: The radial Schrödinger equation containing the qHIP is solved using the parametric Nikiforov-Uvarov method (PNUS) to obtain analytical vibrational energy eigenvalues ( $E_\nu$ ). The centrifugal term is omitted to focus on vibrational motion.
Parameter Determination: The potential parameters are not assigned arbitrarily but are derived from measurable molecular constants: equilibrium bond length ( $r_e$ ), dissociation energy ( $D_e$ ), and vibrational wavenumber ( $\omega_e$ ). This is achieved by imposing equilibrium, dissociation, and curvature conditions on the potential, utilizing the Lambert W function to solve for the deformation parameter.
Partition Function Construction:
- Vibrational: The vibrational partition function ( $Q_a$ ) is constructed from the discrete energy eigenvalues. To account for higher-order anharmonicity and vibrational-rotational coupling not explicitly captured by the potential, a molecule-dependent dimensionless correction parameter ( $\eta$ ) is introduced. This parameter is optimized by minimizing the deviation between calculated and reference Gibbs free energies. The finite sum is evaluated using the Poisson summation method under a classical-limit approximation (retaining only the $j=0$ Fourier mode).
- Rotational and Translational: Standard analytical expressions for rotational ( $Q_b$ ) and translational ( $Q_c$ ) partition functions are incorporated.
Thermodynamic Calculation: The total molecular partition function is used to calculate the Gibbs free energy ( $G$ ). The change in Gibbs free energy ( $\Delta G$ ) for the formation reaction ( $X + Y \rightarrow XY$ ) is computed using standard enthalpies of formation and sensible molar enthalpies derived from the partition functions.
Equilibrium Constant Determination: The chemical equilibrium constant ( $K_p$ ) is derived from $\Delta G$ using the relation $\ln K_p = -\Delta G / RT$ .
Validation: The calculated values of $\log_{10} K_p$ are compared against NIST-JANAF thermochemical data. Accuracy is assessed using Relative Absolute Deviation (RAD) and Average Relative Absolute Deviation (ARAD) over specific temperature intervals.

Key Contributions

Extension of qHIP: The work extends the application of the q-deformed hyperbolic interaction potential from molecular energy spectra to the prediction of chemical equilibrium constants, establishing a direct link between the q-deformed model and reaction-equilibrium thermodynamics.
Analytical Framework: It provides a complete analytical formulation for the vibrational partition function of q-deformed potentials, incorporating a correction parameter to align theoretical results with reference thermodynamic data.
Systematic Assessment: The study systematically evaluates the performance of the qHIP for four distinct diatomic systems (BrF, ClF, IF, and SO), identifying specific temperature ranges where the model agrees closely with reference data and where deviations occur.

Results
The study calculated equilibrium constants for BrF, ClF, IF, and SO over broad temperature ranges and compared them with NIST-JANAF data. The results indicate species-specific and temperature-dependent performance:

BrF: Exhibits the highest consistency, with an Average Relative Absolute Deviation (ARAD) of 0.345% over the entire range of 300–6000 K. Deviations remain below 1% for most of this interval.
ClF: Shows good agreement with an ARAD of 0.751% over 300–4600 K. However, deviations increase significantly above 4600 K, reaching 19.063% at 6000 K.
IF: Displays a large deviation at the lowest temperature (8.075% at 300 K) but improves rapidly. Over the interval 400–6000 K, the ARAD is 0.871%, with deviations generally remaining small.
SO: Demonstrates substantial deviations at low temperatures (187.144% at 300 K), which decrease as temperature rises. The model achieves close agreement from 900 K onward, with an ARAD of 0.905% over the 900–6000 K interval.

Significance and Claims
The paper claims that the q-deformed hyperbolic interaction potential provides a viable statistical-mechanical framework for predicting chemical equilibrium constants. The significance of the work lies in its ability to reproduce reference NIST-JANAF equilibrium data with small average deviations within specific temperature intervals for the selected molecules.

The authors modestly conclude that while the formulation is effective within the identified ranges (e.g., 300–6000 K for BrF, 900–6000 K for SO), the observed departures at the extremes of these ranges (high temperatures for ClF, low temperatures for SO) define the temperature limits of the current formulation. The study does not claim universal applicability across all temperatures or molecules but rather specifies the intervals where the qHIP-derived constants agree closely with established thermochemical data, thereby validating the utility of the q-deformed model for reaction-equilibrium thermodynamics within those bounds.

More like this