Various isotopes of a given atom obey the same chemical principles, but their different masses cause — Organic Chemistry Chemistry Question
Kinetic isotope effects
Various isotopes of a given atom obey the same chemical principles, but their different masses cause different behaviour in a dynamic sense. The kinetic isotope effect refers to a phenomenon wherein isotopically substituted molecules react at different rates. It was postulated in 1933 by Eyring and Polanyi and since then kinetic isotope effects have provided detailed information about mechanisms of many organic and biochemical reactions.
Vibrational modes are quantized and we can use the harmonic oscillator approximation for the description of the stretching modes. The energy level Ev can be calculated as:
where v = 0, 1, 2, ... is the vibrational quantum number and 𝜈 the frequency which depends on the force constant k and the reduced mass µ:
Note for the following calculations: unless stated otherwise, round the isotopic mass in amu to the nearest integer.
Let us start with a simple diatomic molecule 1HF. Calculate its harmonic vibrational wavenumber in cm–1 and energies of the first two vibrational levels in J. The value of the harmonic force constant is k = 968 kg s−2.
Model Answer
Reduced mass:
Wavenumber = 4.159 × 10^5 m^-1 = 4159 cm^-1
Energies: Eo = ½ h c × wavenumber = ½ × 6.6261×10^-34 × 2.9979×10^8 × 4.159×10^5 = 4.13×10^-20 J
E1 = 3/2 h × c × wavenumber = 1.24×10^-19 J
Isotopic substitution does not change the potential energy surface of a molecule. Therefore, k remains unaffected. Given the vibrational wavenumbers of 1HAX (2 439.0 cm-1) and 2DA+2X (1 734.8 cm-1), determine the unknown element X.
Model Answer
We are going to determine the atomic mass A of the lighter isotope of the element X.
The second root of the quadratic equation 2.155, which would correspond to A = 2 and A + 2 = 4, is unphysical.
A = 79.4 amu
A = 79; A + 2 = 81; X = Br
Zero-point vibrational energy is the key contributor to the kinetic isotope effect. If we assume that the bond is fully broken at the transition state and only the ground vibrational state is populated, the difference in activation energies has the same absolute value as the difference in zero-point vibrational energies. The wave numbers of the C−H and C−D stretches are 2 900 cm-1 and 2 100 cm-1, respectively. Calculate the ratio of the rate constants k(C−H)/k(C−D) for the cleavage of the C−H/D bond at 300 K, taking into account only the difference in zero-point vibrational energies.
Model Answer
The difference of the activation energies Ea(H−C) − Ea(D−C) is equal to the negatively taken difference of zero-point vibrational energies: −E0(H−C) + E0(D−C)
Kinetic isotope effects provide insight into the rate-determining step of a reaction mechanism. The ratio of kH/kD for the formation of propene from 1-bromopropane and 1-bromo-2,2-dideuteriopropane in basic solution is 6.5. Does the reaction proceed by E1 or E2 mechanism? E1 takes place in two steps: formation of the carbocation intermediate followed by loss of H+. E2 occurs in a single step involving removal of the halide at the same time as the neighbouring hydrogen.
Model Answer
E2 elimination. The value of the kinetic isotope effect of 6.5 indicates that the C−H/D bond is broken in the rate-determining step of the reaction.
Let us consider the formation of the corresponding alkene from 2-bromo-3,3-dideuterio-2-methylbutane and its light-hydrogen analogue upon heating in ethanol. How significant would the kinetic isotope effect be in this case?
Model Answer
For a tertiary substrate, we can expect E1 elimination, where the C−H/D bond is not broken during the rate-determining step. Therefore, we observe only a small secondary kinetic isotope effect with the kH / kD ratio slightly larger than 1.0.