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 Lecture #14: Low and High-T Limits for qrot and qvib Reading: Hill, pp. 153-159, Maczek pp. 51-53 TEMPERATURE DEPENDENCE OF Erot AND CVrot Low T limit of Erot: T )= 0lim Erot = lim (6Nkθre−2θr T→0 T→0 e−2θrlim CVrot = lim ⎜⎛12Nkθr2 T ⎞⎟ = 0 T→0 T→0 ⎝T2 ⎠Low T Limit High T Limit rot 2 rot CV 12θre−2θrT Cv ≅ 1nR ≅ T 2 nR Erot ≅ 6θre−2θr T Erot ≅ TnR nR Note maximum in CRv ≅ 1.624 at T = 1.0 if we retain the two-term formula for the low-T θrot 1.0 CV rot nR Erot nR 1.0 T/θrot limit. Actual maximum, derived from the full qVrot , is CV/nR = 1.098 at T/θrot = 0.8. [Rapid5.62 Spring 2008 Lecture 14, Page 2 change in CV is a signal of a gap in the level spacing measured in units of kT. What gap wouldbe relevant here? At what value of T/θrot would you expect the most rapid change in CV?] QUANTUM BEHAVIOR VIBRATIONAL MOLECULAR PARTITION FUNCTION qvib U (R ) = (k / 2) R − Re( )2 Using harmonic approximation: 1 1 ⎛⎜⎝ ⎞⎟⎠ ⎛⎜⎝ ⎞⎟⎠ hν = hcωe( ) =ε v v + v +2 2 zero point energy — when v = 0 ε(v = 0)= 1hν2 revised 3/10/08 2:17 PM5.62 Spring 2008 Lecture 14, Page 3 Calculate qvib ∞ ∞ ( ) kT ∑ e−hcωe (v+1/2 ) kT=qvib = ∑ e−ε vv=0 v=0 Define θvib = hcωe k “vibrational temperature” [K] ∞ = ( T)θvib qvib ∑ e− v+1/2 v=0 For vibration, θvib > T almost always. Must sum over each vibrational level. ∞ qvib = e−θvib 2T ∑ e−vθvib T pull zero point energy out of sum v=0 Let x = e−θvib T qvib = x1/2 ∞∑ xv v=0 Now ∑∞ xv = 1 + x + x2 + … = 1 converges for |x| < 1, but 0 ≤ e−θvib T <1 for1 − xv=0 all T, thus we have qvib valid for all T. qvib = x1/2 1− x = e−θvib 2T 1− e−θvib T Molecular Vibrational Partition Function Zero of Evib is set at minimum of potential energy curve Define q*vib ∞ 2T ∑ e−θvib Tqvib = e−θvib v=0  * qvib revised 3/10/08 2:17 PM5.62 Spring 2008 Lecture 14, Page 4 ∞ So q*vib = ⎡⎣ (( ) − ε (v = 0)∑ exp − ε v) kT⎤⎦ v=0 ∞ ∞ q*vib = ∑ e−vθvib T = ∑ xv = 1− 1xv=0 v=0 qvib * = 1 1− e−θvib T all values of θvib/T Put this result aside. We will see how it is useful later in redefining our zeros of energy. q*vib effectively shifts the zero of Evib to the energy of the v = 0 level. High Temp Limit of q*vib θvib  T or εvib  kT * 1− e−θvib T1 =qvib T~ +1 −θvib + θvib 2 If θvib  T, then e−θvib T 2T2 − … So q*vib = 1− e−θvib ⎡⎣ 2 2T2 ⎤⎦ 1T  1 1− 1− θvib T + θvib * θTvib hckT ωe high temperature limit=qvib  When is high temperature limit form useful? For molecules, not often … EXACT EXACT MOLECULE θvib[K] q*(T=300K) (300/θvib) q*(3000K) (3000/θvib)H2 6328 0.0474 1.138 0.474 HCl 4302 1 + 6 × 10–7 0.0697 1.313 0.697 CO 3124 1 + 3 × 10–5 0.0961 1.546 0.961 Br2 465 1.269 0.645 6.964 6.45 I2 309 1.556 1.029 1 + 7 × 10–10 10.22 10.29 Cs2 60.4 5.481 4.926 50.14 49.64 1% error revised 3/10/08 2:17 PM5.62 Spring 2008 Lecture 14, Page 5 Only for very heavy molecules at very high T is the high temperature limit form for q*vib useful. VIBRATIONAL CONTRIBUTIONS TO THERMODYNAMIC FUNCTIONS 2T = e−θvib =qVIB e−θvib 1− e−θvib T 2Tq*vib N = e−Nθvib *N 2Tqvib QVIB = qvib lnQVIB = −Nθvib / 2T + N ln q*vib Evib = kT2 ⎜⎛∂lnQvib ⎞⎟ = kT2 ⎡∂ −( Nθv / 2T )+ N∂ln q*vib ⎤ ⎝∂T ⎠N,V ⎣⎢ ∂T ∂T ⎦⎥ T )−1 ⎤(NkT2θvib + NkT2 ⎡⎢∂ln 1− e−θv ⎥2T2 ⎣∂T ⎦ ) ⎡∂(1− e−θv = Nk 1− e−θv T ⎢ ⎥= 2 θvib + NkT2 ( ⎣∂TT )−1 ⎦⎤ T T ) θvib e−θv =Evib Nk θvib + NkT2 (1 − e−θv T )22 T2 (1 − e−θv eθvT − 1Nkθvib (E − E0 = Nkθvibe−θTv T =)vib 1 − e−θv ↑ zero point energy (energy of v = 0 above minimum of potential curve) E0 = Nk = Nhcωe reference all energies with respect to2 θvib 2 zero point energy NkTx Define x ≡ θvib/T (E − E0 )vib = ex − 1 Einstein Function(E − E0 x RT )vib = ex − 1 plotted vs. x in handout revised 3/10/08 2:17