MIT OpenCourseWare http://ocw.mit.edu 5.62 Physical Chemistry II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.5.62 Spring 2008 Lecture Summary #21 Page 1 THERMODYNAMICS OF SOLIDS: EINSTEIN AND DEBYE MODELS Reading: Hill, pages 86-98 For the next few lectures we will discuss solids, in particular crystalline solids, inwhich the particles are arranged in a regular lattice. • The 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 asCO, CO2, H2O, penicillin, hemoglobin, etc. TOTAL # DEGREES OF FREEDOM = 3N (where N = # of atoms in the crystal) 3 correspond to overall translation3 correspond to overall rotation!"# of whole crystalremaining 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. The behavior of such a harmonic molecular crystal is described by the normal modes of vibration. There are 3N–6 harmonic oscillators that can be treated independently (a convenient idealization) to describe the motions and energies within thecrystal. There are many kinds of vibrations in a crystal. Viewed along a particulardirection, there will be periodic distortions of alternating extension and compression, analogous to the stretching modes of a linear molecule. There will also be alternatingdisplacements of atoms above and below the specified direction, analogous to thebending modes of a linear molecule. These longitudinal and transverse modes of a crystal can have wavelengths ranging from as short as a bond length (high frequency) andas long as the macroscopic crystal itself (low frequency). The distribution of frequencies,directions (relative to unit cell axes), and types of vibrations can be very complicated.The simplest models for crystalline solids are based on assumptions about the crystalQvib*vibrations that simplify Qvib*calculation of and the derivation of thermodynamic properties from . These models also permit inferences about the nature of thevibrations 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 3/21/08 8:51 AM5.62 Spring 2008 Lecture Summary #21 Page 2 The normal modes for a violin string correspond to one-dimensional particle-in-a-box solutions: From Lectures 14 and 15 we recall that for a single harmonic oscillator (and excludingthe zero-point energy) qvib*=v= 0!"e#vh$ kT=11 # e#h$ kTT % 0,!q* % 1T % !,!q* % !and for a set of independent (i.e. uncoupled), harmonic oscillators Energy Qvib*= qvib*(1)qvib*(2)qvib*(3)…qvib*(3N ! 6) =i=13N !6"11! e!h#ikTUvib= kT2!lnQvib!T"#$%&'N ,V= E0+ kT2!lnQvib*!T"#$%&'N ,Vrevised 3/21/08 8:51 AM5.62 Spring 2008 Lecture Summary #21 Page 3 Uvib! E0= kT2"lnQvib*"T#$%&'(N ,V= kT21Qvib*"Qvib*"T#$%&'(N ,V"Qvib*"T#$%&'(N ,V=j)1 ! e!h*jkT[ ]!2(!1) !h*jkT2#$%&'(e!h*jkTi + j3N !6,1 ! e!h*jkT[ ]!1=j)1 ! e!h*jkT[ ]!1h*jkT2#$%&'(e!h*jkTi =13N !6,1 ! e!h*ikT[ ]!1Qvib*! "### $###kT21Qvib*"Qvib*"T#$%&'(N ,V= h*je!h*jkTj)1 ! e!h*jkT[ ]!1=i=13N !6"h#ie!h #ikT1 ! e!h#ikT= kTi =13 N !6"xiexi!1where xi=h#ikTEinstein FunctionNB: This derivation treated all oscillators as harmonic and uncoupled. Heat Capacity CVvib=!Uvib!T"#$%&'N ,V=! Uvib( E0( )!T)*+,-.N ,V=!!Th/jj =13N (60eh/jkT(1[ ](1= h/jj =13N (60((1) eh/jkT(1[ ](2((1)h/jkT2eh/jkT= kj =13N (60h/jkT"#$%&'2eh/jkTeh/jkT(1[ ](2= k xj2j =13N (60exjexj(1[ ](2Free Energy (Helmholtz) Avib! E0( )= !kT ln Qvib*= !kT lni=13N !6"11 ! e!h#ikT$%&'()= +kTi=13N !6*ln 1 ! e!h#ikT( )= kTi=13N !6*ln 1 ! e!xi( )Einstein Functionrevised 3/21/08 8:51 AM1 5.62 Spring 2008 Lecture Summary #21 Page 4 Entropy Svib=Uvib! AvibT=Uvib! E0( )T!Avib! E0( )T= ki =13N !6"xiexi!1! ln 1! e!xi( )#$%&'(add and subtract thesame quantitySo, we should be able to calculate “all” properties of solids, but it seems as though weneed to know the frequencies of all of the normal modes. Classical Treatment — Equipartition Principle Put 12 kT of energy into each “degree of energy storage”, where each normalmode of vibration has TWO degrees for storage of energy (one for kinetic energy and the second for potential energy). Uvib= Evibclass=i =13N !6"12kT#$%&'(P.E.+12kT#$%&'(K .E.)*+,-.= 3N ! 6( )kT( )/ 3NkTEvibclass= 3RT for a mole of ATOMS in crystalCVclass=!U!T( )N ,V= 3R per moleThis is correct at HIGH TEMPERATURE, and is known as the LAW OFDULONG AND PETIT (~1819) • Heat capacity per mole is roughly the same for all substances • Measure heat capacity per gram (different for all substances) • The ratio is grams/mole = molecular mass! The Classical Treatment, however, turns out to be incorrect at low temperature. revised 3/21/08 8:51 AM2 5.62 Spring 2008 Lecture Summary #21 Page 5 Einstein Treatment Use quantum theory (as opposed to classical) This makes it easy to evaluate the sum overvibrations. Assume all νi equal (all xi equal) CVEinstein= ki =13N !6"xi2exiexi!1( )2= k 3N ! 6( )x2exex!1( )2# 3Rx2exex!1( )2per mole N = Na, kNa= R( )where x = h$EkT=%ETThis approach provides a significant improvement over the classical (equipartition)result, because CV → 0 for T → 0. limT !0CVEinstein= limx !"CVEinstein= limx !"3Rx2e# x1 # e# x( )2= 0The success of the Einstein model gave important early support to quantum theory: itshowed that quantization of vibrational energy could account for low-T heat capacity. revised 3/21/08 8:51