Problem 2

Kinetic isotope effect (KIE) and zero-point vibrational energy (ZPE). The harmonic oscillator model gives the vibrational frequency

\nu = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{\mu}}

, where

k

is the force constant and

\mu = \dfrac{m_1 m_2}{m_1 + m_2}

is the reduced mass. Vibrational energies are

E_n = (n + \tfrac{1}{2})h\nu

(

n = 0, 1, 2, \dots

), and the zero-point energy is

\text{ZPE} = \tfrac{1}{2}h\nu

. (a) Calculate the reduced masses

\mu_{\ce{CH}}

and

\mu_{\ce{CD}}

in atomic mass units (u), taking

m(\ce{C}) = 12.00

m(\ce{H}) = 1.008

u, and

m(\ce{D}) = 2.014

u. (b) Given

k_{\ce{CH}} = k_{\ce{CD}}

and the

\ce{C-H}

stretching wavenumber

\tilde{\nu}_{\ce{CH}} = 2900\ \text{cm}^{-1}

, calculate the

\ce{C-D}

stretching wavenumber

\tilde{\nu}_{\ce{CD}}

(cm

^{-1}

). (c) Calculate

\text{ZPE}_{\ce{CH}}

and

\text{ZPE}_{\ce{CD}}

in kJ mol

^{-1}

. (d) Calculate the difference in bond dissociation energies

\Delta \text{BDE} = \text{BDE}_{\ce{CD}} - \text{BDE}_{\ce{CH}}

(kJ mol

^{-1}

). (e) Assuming

E_a \approx \text{BDE}

and identical Arrhenius pre-exponential factors, calculate the theoretical primary KIE

k_{\ce{CH}}/k_{\ce{CD}}

25\ ^{\circ}\text{C}

. (f) In the chromic acid oxidation of diphenylmethanol, measured first-order rate constants are

k_{\ce{CH}} = 0.012\ \text{min}^{-1}

and

k_{\ce{CD}} = 0.0018\ \text{min}^{-1}

. Compare this experimental ratio with the theoretical value and determine whether

\ce{C-H}

bond cleavage is rate-determining.

Step 6 of 6: Experimental comparison and mechanistic conclusion

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\left(\dfrac{k_{\ce{CH}}}{k_{\ce{CD}}}\right)_{\text{exp}} = \dfrac{0.012\ \text{min}^{-1}}{0.0018\ \text{min}^{-1}} = 6.67 \approx 6.7 \Rightarrow \ce{C-H} \text{ cleavage is rate-determining}

Analysis

The measured ratio $k_{\ce{CH}}/k_{\ce{CD}} = 0.012 / 0.0018 = 6.7$ is in excellent agreement with the theoretical primary KIE (6.5). If $\ce{C-H}$ cleavage occurred after the rate-determining step, a secondary KIE close to 1 would be observed; therefore, the $\ce{C-H}$ bond-breaking step is indeed the rate-determining step.

Common pitfall. Do not confuse primary KIE (cleavage of the bond directly involved in the reaction, ratio 6–8) with secondary KIE (isotopic substitution on an adjacent atom, ratio usually 0.8–1.4).