HeatCapacity

HeatCapacity - Theoretical calculation of the heat capacity...

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MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Theoretical calculation of the heat capacity ¾ Principle of equipartition of energy ¾ Heat capacity of ideal and real gases ¾ Heat capacity of solids: Dulong-Petit, Einstein, Debye models ¾ Heat capacity of metals ± electronic contribution Reading: Chapter 6.2 of Gaskell
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MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Degrees of freedom and equipartition of energy For each atom in a solid or gas phase, three coordinates have to be specified to describe the atom±s position ² a single atom has 3 degrees of freedom for its motion. A solid or a molecule composed of N atoms has 3N degrees of freedom. We can also think about the number of degrees of freedom as the number of ways to absorb energy. The theorem of equipartition of energy (classical mechanics) states that in thermal equilibrium the same average energy is associated with each independent degree of freedom and that the energy is ½ k B T. For the interacting atoms, e.g. liquid or solid, for each atom we have ½ kT for kinetic energy and ½ kT for potential energy - equality of kinetic and potential energy in harmonic approximation is addressed by the virial theorem of classical mechanics. Based on equipartition principle, we can calculate heat capacity of the ideal gas of atoms - each atom has 3 degrees of freedom and internal energy of 3/2k B T. The molar internal energy U=3/2N A kT=3/2RT and the molar heat capacity under conditions of constant volume is c v =[dU/dT] V =3/2R In an ideal gas of molecules only internal vibrational degrees of freedom have potential energy associated with them. For example, a diatomic molecule has 3 translational + 2 rotational + 1 vibrational = 6 total degrees of freedom. Potential energy contributes ½ k B Ton ly to th e energy of the vibrational degree of freedom, and U molecule = 7/2k B
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MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei Temperature and velocities of atoms At equilibrium velocity distribution is Maxwell-Boltzmann, ±² > @ z y x 2 z 2 y 2 x 2 3 dv dv dv 2kT v v v m exp kT 2 m T , v dN ° ¿ ° ¾ ½ ° ¯ ° ® ­ ³ ³ ´ ¸ ¹ · ¨ © § S T/m 3k v B i 2 If system is not in equilibrium it is often difficult to separate different contributions to the kinetic energy and to define temperature. Acoustic emissions in the fracture simulation in 2D model. Figure by
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HeatCapacity - Theoretical calculation of the heat capacity...

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