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.� MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF CHEMISTRY Physical Chemistry 5.62 EINSTEIN AND DEBYE SOLIDS 1. Einstein model The basic assumption is 3N harmonic vibrational modes all of the same frequency, νE. The ther-modynamic functions are given in terms of the reduced parameter, x ≡ hνE/kT . Thermodynamic function Molar value for Einstein solid (E − E0)/3RT x(ex − 1)−1 (A − E0)/3RT ℓn(1 − e−x) S/3R x(ex − 1)−1 − ℓn(1 − e−x) CV /3R x2ex(ex − 1)−2 2. Debye Model In the Debye model there are 3N harmonic modes, whose distribution in frequencies is given by ρ(ν): ρ(ν) = 9Nν2 νmax 3 for ν < νmax ρ(ν) = 0 for ν > νmax where c is the velocity of sound in the solid, V is the volume, and νmax = c(3N/4πV)1/3. The thermody-namic functions for a Debye solid are expressed in terms of the dimensionless parameter y = θD/T , where the Debye temperature, θD, is θD = hνmax/k The thermodynamic functions are often written in terms of the Debye function, D(y): D(y) = 3y−3 � yu3(eu − 1)−1du 0 where u = hν/kT . Thermodynamic function Molar value for Debye solid (E − E0)/3RT 3y−3 � yu3(eu − 1)−1du = D(y)0 (A − E0)/3RT 3y−3 � yu2ℓn(1 − eu)du = ℓn(1 − e−y) − 1 D(y)03 S/3R (E − A)/3RT = 34 D(y) − ℓn(1 − e−y) CV /3R 3y−3u4eu(eu − 1)−2du = 4D(y) − 3y(ey − 1)−15.62 Lecture # 13 Supplement Page 2 Einstein Functions x (ex−1) x −ln(1 − x−x)(exx 2−e1)x 2 −ln(1 − e−x)(exx −1) 0.05 3.99584 0.99979 3.02063 0.97521 0.10 3.30300 0.99917 2.35217 0.95083 0.15 2.89806 0.99813 1.97118 0.92687 0.20 2.61110 0.99667 1.70777 0.90333 0.25 2.38889 0.99481 1.50869 0.88020 0.30 2.20771 0.99253 1.35023 0.85749 0.35 2.05491 0.98985 1.21972 0.83519 0.40 1.92293 0.98677 1.10963 0.81330 0.45 1.80690 0.98329 1.01508 0.79182 0.50 1.70350 0.97942 0.93275 0.77075 0.55 1.61035 0.97517 0.86026 0.75008 0.60 1.52569 0.97053 0.79587 0.72982 0.65 1.44820 0.96552 0.73824 0.70996 0.70 1.37684 0.96015 0.68634 0.69050 0.75 1.31079 0.95441 0.63935 0.67144 0.80 1.24939 0.94833 0.59662 0.65277 0.85 1.19209 0.94191 0.55759 0.63450 0.90 1.13844 0.93515 0.52184 0.61661 0.95 1.08807 0.92807 0.48897 0.59910 1.00 1.04065 0.92067 0.45868 0.58198 1.05 0.99592 0.91298 0.43069 0.56523 1.10 0.95363 0.90499 0.40477 0.54886 1.15 0.91358 0.89671 0.38073 0.53285 1.20 0.87560 0.88817 0.35838 0.51722 1.25 0.83952 0.87937 0.33758 0.50194 1.30 0.80520 0.87031 0.31818 0.48702 1.35 0.77253 0.86102 0.30008 0.47245 1.40 0.74139 0.85151 0.28315 0.45824 1.45 0.71168 0.84178 0.26732 0.44436 1.50 0.68331 0.83185 0.25248 0.43083 1.55 0.65620 0.82173 0.23857 0.41762 1.60 0.63027 0.81143 0.22552 0.40475 1.65 0.60546 0.80096 0.21326 0.39221 1.70 0.58171 0.79035 0.20173 0.37998 1.75 0.55895 0.77958 0.19089 0.368065.62 Lecture # 13 Supplement Page 3 Einstein Functions x (ex−1) x −ln(1 − x−x)(exx 2−e1)x 2 −ln(1 − e−x)(exx −1) 1.80 0.53714 0.76869 0.18068 0.35646 1.85 0.51623 0.75768 0.17107 0.34516 1.90 0.49617 0.74657 0.16201 0.33416 1.95 0.47692 0.73536 0.15347 0.32345 2.00 0.45845 0.72406 0.14541 0.31304 2.05 0.44071 0.71269 0.13781 0.30290 2.10 0.42367 0.70127 0.13063 0.29304 2.15 0.40731 0.68979 0.12385 0.28346 2.20 0.39158 0.67827 0.11744 0.27414 2.25 0.37647 0.66672 0.11138 0.26509 2.30 0.36194 0.65515 0.10565 0.25629 2.35 0.34797 0.64358 0.10023 0.24774 2.40 0.33454 0.63200 0.09510 0.23945 2.45 0.32163 0.62044 0.09025 0.23139 2.50 0.30921 0.60889 0.08565 0.22356 2.55 0.29727 0.59737 0.08130 0.21597 2.60 0.28578 0.58589 0.07718 0.20861 2.65 0.27473 0.57445 0.07327 0.20146 2.70 0.26410 0.56307 0.06957 0.19453 2.75 0.25387 0.55174 0.06606 0.18781 2.80 0.24403 0.54049 0.06274 0.18129 2.85 0.23456 0.52930 0.05958 0.17498 2.90 0.22545 0.51820 0.05659 0.16886 2.95 0.21669 0.50719 0.05376 0.16293 3.00 0.20826 0.49627 0.05107 0.15717 3.05 0.20014 0.48545 0.04852 0.15163 3.10 0.19234 0.47473 0.04610 0.14624 3.15 0.18482 0.46413 0.04380 0.14103 3.20 0.17760 0.45363 0.04162 0.13598 3.25 0.17065 0.44326 0.03955 0.13110 3.30 0.16396 0.43301 0.03758 0.12638 3.35 0.15752 0.42289 0.03571 0.12181 3.40 0.15133 0.41289 0.03394 0.11739 3.45 0.14537 0.40304 0.03226 0.11311 3.50 0.13964 0.39331 0.03066 0.108985.62 Lecture # 13 Supplement Page 4 Einstein Functions x (ex−1) x −ln(1 − x−x)(exx 2−e1)x 2 −ln(1 − e−x)(exx −1) 3.55 0.13413 0.38373 0.02915 0.10499 3.60 0.12883 0.37429 0.02770 0.10113 3.65 0.12373 0.36499 0.02633 0.09740 3.70 0.11883 0.35584 0.02503 0.09380 3.75 0.11411 0.34684 0.02380 0.09032 3.80 0.10958 0.33799 0.02262 0.08695 3.85 0.10522 0.32928 0.02151 0.08371 3.90 0.10102 0.32073 0.02045 0.08057 3.95 0.09699 0.31233 0.01944 0.07755 4.00 0.09311 0.30409 0.01849 0.07463 4.05 0.08939 0.29599 0.01758 0.07181 4.10 0.08580 0.28806 0.01671 0.06909 4.15 0.08236 0.28027 0.01589 0.06647 4.20 0.07905 0.27264 0.01511 0.06394 4.25 0.07587 0.26516 0.01437 0.06150 4.30 0.07281 0.25783 0.01366 0.05915 4.35 0.06987 0.25066 0.01299 0.05688 4.40 0.06705 0.24363 0.01235 0.05469 4.45 0.06433 0.23676 0.01175 0.05258 4.50 0.06172 0.23004 0.01117 0.05055 4.55 0.05922 0.22347 0.01062 0.04859 4.60 0.05681 0.21704 0.01010 0.04671 4.65 0.05450 0.21076 0.00961 0.04489 4.70 0.05228 0.20462 0.00914 0.04314 4.75 0.05014 0.19863 0.00869 0.04145 4.80 0.04809 0.19277 0.00826 0.03983 4.85 0.04613 0.18706 0.00786 0.03827 4.90 0.04424 0.18149 0.00747 0.03676 4.95 0.04242 0.17605 0.00711 0.03531 5.00 0.04068 0.17074 0.00676 0.03392 5.05 0.03901 0.16557 0.00643 0.03258 5.10 0.03740 0.16053 0.00612 0.03128 5.15 0.03586 0.15561 0.00582 0.03004 5.20 0.03438 0.15083 0.00553 0.02885 5.25 0.03296 0.14616 0.00526 0.027695.62 Lecture # 13 Supplement Page 5 Einstein Functions x (ex−1) x −ln(1 − x−x)(exx 2−e1)x 2 −ln(1 − e−x)(exx −1) Einstein Fns. 1.2 1.0 0.8 0.6 0.4 0.2 5.30 5.35 5.40 5.45 5.50 5.55 5.60 5.65 5.70 5.75 5.80 5.85 5.90