International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 11 Issue: 09 | Sep 2024
p-ISSN: 2395-0072
www.irjet.net
Mathematical Description of Energies of Oscillating Atoms Arnav Mody1, Dr. Soumya Sri-Bhattacharya2 1Student at PPSIJC, Mumbai, Maharashtra, India -400056 2Research Professor
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Abstract - This research explores atomic vibrations using
Where : V(r) represents the Electric Potential Energy when separation is r ϵ represents the Binding Strength of the Particular Bond σ represents the Zero of the function. Since the σ and ϵ are unique characters, the relationship between
the Lennard-Jones Potential (LJP) to model the balance of attractive and repulsive forces between atoms. Focusing on energy exchange between kinetic and potential forms, we examine how atoms oscillate around equilibrium points. By applying these principles to hydrogen gas, we calculate energy distributions and investigate how atomic vibrations affect material properties like thermal and electrical conductivity. The study provides a theoretical foundation for understanding atomic behavior, with potential applications in materials science and thermodynamics. Limitations of the LJP are noted, such as its neglect of multi-body interactions.
1.INTRODUCTION Everything in the world around us is due to the effect of two main types of forces, the Gravitational and the Electromagnetic force. The macroscopic world works mainly on the principles of the gravitational force. However, in the microscopic world, everything is governed by electromagnetic forces that dictate the behavior of atoms and molecules, ultimately shaping the properties of matter. One of the fundamental concepts in understanding these forces is the Lennard-Jones Potential (LJP), which models the interaction between a pair of neutral atoms or molecules. The LJP describes how the potential energy between two particles changes with respect to the separation and how the electric force transitions from being attractive at longer distances to repulsive as they approach each other, preventing them from occupying the same space. This interconnection between potential energy and force can be explained by a simple mathematical equation that shows that the force exerted is the negative gradient of the potential energy distance graph. The Lennard-Jones Potential uses the parameters of the well depth and the zero of the function to describe how the Electric Potential Energy of two atoms in a bond changes with respect to the distance between the two atoms. The Lennard-Jones Equation is as follows: [1]
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Fig -1: Diagram showing how the LJP graphs are different for different bonds the Electric Potential Energy and the distance is unique to each bond. The forces acting on the atoms partaking in the bond will also be bond-specific, resulting in a certain aspect of peculiarity. The interplay of forces between the atoms, represented by the Lennard-Jones Potential equation, leads to an equilibrium state where the net force between particles is zero. Oscillations around this equilibrium position play a crucial role in determining the properties of substances. Understanding these oscillations is essential as they underlie several important phenomena including spectroscopic signatures, thermal and electrical conductivity, and reaction kinetics. This research aims to delve deeper into the oscillatory behavior of atoms as they transition between attractive and repulsive forces. We explore the energy distribution within these oscillating systems, considering the exchange between kinetic and potential energy as atoms vibrate around their equilibrium positions. By developing mathematical models and differential equations, we seek to describe these energy fluctuations and their impact on the overall behavior of materials. Through this investigation, we aim to provide a more comprehensive understanding of the internal energy in microscopic systems, offering insights into how energy is conserved and distributed at the atomic level. This research contributes to the theoretical understanding of molecular interactions and has potential
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