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 3 of 6: Zero-point vibrational energies

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\text{ZPE} = \tfrac{1}{2}hc\tilde{\nu}N_A \Rightarrow \text{ZPE}_{\ce{CH}} = 17.35\ \text{kJ mol}^{-1},\quad \text{ZPE}_{\ce{CD}} = 12.73\ \text{kJ mol}^{-1}

Analysis

Multiplying by $hcN_A = (6.626\times 10^{-34}\times 2.998\times 10^{10}\times 6.022\times 10^{23}\times 10^{-3}) = 0.01196$ kJ mol $^{-1}$ cm gives $\text{ZPE}_{\ce{CH}} = \frac{1}{2}\times 0.01196\times 2900 = 17.35$ kJ mol $^{-1}$ and $\text{ZPE}_{\ce{CD}} = 12.73$ kJ mol $^{-1}$ .