5.62 Physical Chemistry II Spring 2008

Transcription

1 MI OpenCourseWare Physical Chemistry II Spring 008 For information about citing these materials or our erms of Use, visit:

2 5.6 Spring 008 Lecture Summary Page HERMODYNAMICS OF SOLIDS: EINSEIN AND DEBYE MODELS Reading: Hill, pages For the next few lectures we will discuss solids, in particular crystalline solids, in which the particles are arranged in a regular lattice. he lattice could consist of single atoms or atomic ions, such as Ar or Na + Cl arranged in something like a face-centered-cubic FCC) or body-centered-cubic BCC) crystalline array. Or the lattice could be a crystal of more complex molecules in a lattice, such as CO, CO, H O, penicillin, hemoglobin, etc. OAL DEGREES OF FREEDOM 3N where N of atoms in the crystal) 3 correspond to overall translation! of whole crystal 3 correspond to overall rotation remaining 3N 6 correspond to internal vibrations within crystal In this treatment the crystal is viewed as a giant polyatomic molecule undergoing simple harmonic motion in each of its 3N-6 vibrational normal modes. he behavior of such a harmonic molecular crystal is described by the normal modes of vibration. here are 3N 6 harmonic oscillators that can be treated independently a convenient idealization) to describe the motions and energies within the crystal. here are many kinds of vibrations in a crystal. Viewed along a particular direction, there will be periodic distortions of alternating extension and compression, analogous to the stretching modes of a linear molecule. here will also be alternating displacements of atoms above and below the specified direction, analogous to the bending modes of a linear molecule. hese longitudinal and transverse modes of a crystal can have wavelengths ranging from as short as a bond length high frequency) and as long as the macroscopic crystal itself low frequency). he distribution of frequencies, directions relative to unit cell axes), and types of vibrations can be very complicated. he simplest models for crystalline solids are based on assumptions about the crystal vibrations that simplify calculation of Q vib and the derivation of thermodynamic properties from Q vib. hese models also permit inferences about the nature of the vibrations in a crystal based on the small number much smaller than 3N 6) of possible experimental observations of the macroscopic properties in the crystal. revised 3//08 8:5 AM

3 5.6 Spring 008 Lecture Summary Page he normal modes for a violin string correspond to one-dimensional particle-in-a-box solutions: From Lectures 4 and 5 we recall that for a single harmonic oscillator and excluding the zero-point energy) q vib! e vh k v0 0,!q e h k!,!q! and for a set of independent i.e. uncoupled), harmonic oscillators Q vib q vib )q vib )q vib 3) q vib 3N! 6)! e!h i k Energy U vib k!lnq vib! E 0 + k!lnq vib! revised 3//08 8:5 AM

4 5.6 Spring 008 Lecture Summary Page 3 k Q vib U vib! E 0 k Q vib Q vib lnq vib )! e!h k k Q vib [ ]!!)! h [ ]! h )! e!h k h e!h k )! e!h k k k e!h k [ ]! Q vib e!h k,! e!h k i+! Q vib [ ]! [ ]!,! e!h i k h i e!h i k! e k!h i k x i e x i! Einstein Function where x i h i k NB: his derivation treated all oscillators as harmonic and uncoupled. Heat Capacity C vib V!U vib! ) )! U E, vib 0 +! -.!! 3N 6 3N 6 0 h/ e h/ k [ ] 0 h/ ) e h/ k 3N 6 0 k h/ k 3N 6 k 0 x e x e x [ ] ) h/ e h/ k e h/ [ k ] [ ] k eh/ k Free Energy Helmholtz) A vib! E 0 )!k lnq vib +k!k ln! e!h i k ln! e!h i k ) k ln! e!x i ) ) Einstein Function revised 3//08 8:5 AM

5 5.6 Spring 008 Lecture Summary Page 4 Entropy U! E vib 0 ) S vib U! A vib vib k x i e x i!! ln! e!x i ) add and subtract the same quantity! A! E ) vib 0 So, we should be able to calculate all properties of solids, but it seems as though we need to know the frequencies of all of the normal modes. Classical reatment Equipartition Principle Put k of energy into each degree of energy storage, where each normal mode of vibration has WO degrees for storage of energy one for kinetic energy and the second for potential energy). U vib E class vib ) k + P.E. + k K.E.,. 3N! 6) k ) / 3Nk - E class vib 3R for a mole of AOMS in crystal C V class!u! )N,V 3R per mole his is correct at HIGH EMPERAURE, and is known as the LAW OF DULONG AND PEI ~89) Heat capacity per mole is roughly the same for all substances Measure heat capacity per gram different for all substances) he ratio is grams/mole molecular mass! he Classical reatment, however, turns out to be incorrect at low temperature. revised 3//08 8:5 AM

6 5.6 Spring 008 Lecture Summary Page 5 Einstein reatment Use quantum theory as opposed to classical) Assume all ν i equal all x i equal) his makes it easy to evaluate the sum over vibrations. C Einstein V k x i e x i e x i!) k 3N! 6 ) x e x e x!) 3R x e x per mole N N e x!) a,kn a R ) where x h E k E his approach provides a significant improvement over the classical equipartition) result, because C V 0 for 0. lim C Einstein V lim C Einstein e x V lim!0 x! x! 3Rx e x ) 0 he success of the Einstein model gave important early support to quantum theory: it showed that quantization of vibrational energy could account for low- heat capacity. revised 3//08 8:5 AM