3: Applying Particle Models to Matter
- Page ID
- 104098
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In this chapter, we introduce particle models to deepen our understanding of matter’s thermal properties. Beginning with the Particle Model of Matter, we examine atomic interactions, using the Lennard-Jones potential to explain forces between neutral particles. We then explore bond energy, understanding how energy is required to break atomic bonds and applying this to larger structures to estimate bond energies. The Particle Model of Thermal Energy describes thermal energy as the result of particle vibrations, connecting to concepts like equipartition and the distribution of energy across modes. This model links macroscopic measures like temperature and heat capacity with microscopic particle behavior. Looking ahead, we will use these particle models to explain and predict thermal and chemical phenomena, refining our understanding through thermodynamics and further energy interactions.
- 3.1: Where we are headed
- We will introduce particle models of matter to explain thermal properties across phases. Using the atomic hypothesis and the spring-mass model, atoms and molecules are viewed as oscillators with forces defining equilibrium, addressing melting points, heat capacities, and bond strengths. Quantum mechanics refines understanding, especially for energy discrepancies in specific heats at low temperatures. These models will help explain and predict properties, building on prior energy concepts.
- 3.2: Intro Particle Model of Matter
- We introduce the Particle Model of Matter, focusing on interactions between neutral atoms and molecules. Using the Lennard-Jones potential, we describe how particles attract at moderate distances and repel when too close. Key concepts include equilibrium separation, binding energy, and oscillation about equilibrium. These interactions form the basis for understanding phases, as bound particles form solids or liquids, while unbound particles form gases at higher energies.
- 3.3: Particle Model of Bond Energy
- We develop the Particle Model of Bond Energy, starting with the energy required to break bonds between two particles interacting via the Lennard-Jones potential. Extending to multi-atom structures, we calculate bond energy by summing pair-wise interactions. For macroscopic systems, we simplify by focusing on nearest-neighbor bonds, estimating the energy needed to separate particles. This model links to macroscopic measures, like heats of vaporization.
- 3.4: Particle Model of Thermal Energy
- We model thermal energy as random particle vibrations, where added energy increases both kinetic and potential energy. For solids and liquids, particles oscillate in three dimensions, leading to six energy modes per particle. Equipartition of Energy ensures thermal energy distributes equally across modes. In gases, monatomic gases have only translational modes, while diatomic and polyatomic gases also have rotational and vibrational modes, affecting heat capacity.
- 3.5: Looking Back and Ahead
- We review how bond and thermal energy models offer insight into matter's particulate nature, despite some approximations. These models let us predict thermal behaviors and temperature connections, simplifying complex phenomena. Thermodynamics will deepen this understanding, adding precision at the expense of simplicity. Unanswered questions, like bond strength determination, will require exploring forces and vibrations, which we’ll revisit later to advance our particle model.