5.62 Lecture #14: Low and High-T Limits for qrot and qvib

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: lim E rot = lim ( 6Nkθr e−2θr T→0 T→0 T )=0 ⎛ 12Nkθ2r −2θ T ⎞ lim CVrot = lim ⎜ e r ⎟=0 T→0 T→0 ⎝ ⎠ T2 1.0 E rot nR Crot V nR 1.0 Low T Limit rot V 2 r C 12θ ≅ 2 e−2θr nR T E rot ≅ 6θr e−2θr nR Note maximum in T T/θrot High T Limit T Crot v ≅1 nR E rot ≅T nR Cv T ≅ 1.624 at = 1.0 if we retain the two-term formula for the low-T R θrot limit. Actual maximum, derived from the full q Vrot , is CV/nR = 1.098 at T/θrot = 0.8. [Rapid change in CV is a signal of a gap in the level spacing measured in units of kT. What gap would be relevant here? At what value of T/θrot would you expect the most rapid change in CV?] QUANTUM BEHAVIOR VIBRATIONAL MOLECULAR PARTITION FUNCTION qvib 2 U (R ) = (k / 2)(R − Re ) Using harmonic approximation: ⎛ 1⎞ ⎛ 1⎞ ε (v) = ⎜ v + ⎟ hν = ⎜ v + ⎟ hcωe ⎝ 2⎠ ⎝ 2⎠ zero point energy — when v = 0 ε (v = 0 ) = 1 hν 2 Calculate qvib ∞ q vib = ∑ e −ε ( v) kT ∞ = ∑ e−hcωe v=0 θvib = Define ( v+1/2 ) kT v=0 hcωe k “vibrational temperature” [K] ∞ q vib = ∑ e− v+1/2 θvib ( ) T v=0 For vibration, θvib > T almost always. Must sum over each vibrational level. −θvib 2 T q vib = e ∞ ∑ e−vθvib T pull zero point energy out of sum v=0 Let x = e−θvib T ∞ q vib = x 1/2 ∑ xv v=0 ∞ Now ∑ x v = 1+ x + x 2 +… = v=0 1 1− x converges for |x| < 1, but 0 ≤ e−θvib T <1 for all T, thus we have qvib valid for all T. q vib x1/2 e−θvib 2 T = = 1− x 1− e−θvib T Molecular Vibrational Partition Function Zero of Evib is set at minimum of potential energy curve Define q*vib q vib = e −θvib 2 T ∞ ∑ e−θvib T v=0    q*vib ∞ So q * vib = ∑ exp ⎡⎣− (ε (v) − ε (v = 0 )) kT⎤⎦ v=0 ∞ q * vib =∑ e −vθvib T = ∑ xv = v=0 q*vib = ∞ v=0 1 1 − e−θvib 1 1− x all values of θvib/T 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. θvib  T or εvib  kT High Temp Limit of q*vib q*vib = If 1 1 − e−θvib T then e−θvib θvib  T, q*vib = So 1 1 − e−θvib q*vib  T T  ~ +1 − θvib θ2vib + −… T 2T2 1 ⎡ 1 − ⎣1 − θvib T + θ2vib 2T2 ⎤⎦ T kT = high temperature limit θvib hcωe When is high temperature limit form useful? For molecules, not often … θvib[K] 6328 EXACT q*(T=300K) 1 + 7 × 10–10 (300/θvib) 0.0474 EXACT q*(3000K) 1.138 (3000/θvib) 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 10.22 10.29 Cs2 60.4 5.481 4.926 50.14 49.64 1% error MOLECULE H2 Only for very heavy molecules at very high T is the high temperature limit form for q*vib useful. VIBRATIONAL CONTRIBUTIONS TO THERMODYNAMIC FUNCTIONS e−θvib 2 T = e−θvib 2 Tq*vib −θvib T 1− e Q VIB = q Nvib = e−Nθvib 2 Tq*N vib q VIB = ln Q VIB = −Nθvib / 2T + N ln q*vib ⎡ ∂ (−Nθv / 2T) N∂ln q*vib ⎤ ⎛ ∂ln Q vib ⎞ E vib = kT2 ⎜ + ⎟ = kT2 ⎢ ⎥ ⎣ ⎝ ∂T ⎠N,V ∂T ∂T ⎦ ⎡ −θv T −1 ⎤ ( ) NkT2θvib 2 ∂ln 1 − e ⎥ = + NkT ⎢ 2 ⎣ ⎦ 2T ∂T −1 ⎤ ⎡ ( Nk ∂ 1 − e−θv T ) ⎥ 2 −θv T ( ) ⎢ = θvib + NkT 1 − e ⎣ ⎦ 2 ∂T E vib (E − E 0 )vib Nk e−θv T 2 −θv T θ vib ( ) = θvib + NkT 1 − e 2 T2 (1 − e−θv T )2 Nkθvib e−θv T Nkθvib = = θv T 1 − e−θv T e −1 ↑ zero point energy (energy of v = 0 above minimum of potential curve) E0 = Nk Nhcωe θvib = 2 2 (E − E 0 )vib = Define x ≡ θvib/T (E − E 0 )vib RT reference all energies with respect to zero point energy = x x e −1 NkTx e x −1 Einstein Function plotted vs. x in handout