Domain: Solution kinetics

At 298.15 K in MeCN \[(\varepsilon \approx 36)\], estimate the second-order rate constant \[k_2\] \[(\mathrm{M}^{-1} \mathrm{s}^{-1})\] for the bimolecular reaction     

\[
\mathrm{A + B \rightarrow P}
\]                      
          
using solution-phase transition state theory with a Wigner tunneling correction.
You are given (gas-phase, \[1\ \mathrm{atm}\], \[298.15\ \mathrm{K}\]) per-species free energies and data (in \[\mathrm{kJ\ mol}^{-1}\] unless noted):      
• \[G_\mathrm{gas}(A)= -120.0\], \[G_\mathrm{gas}(B)= -80.0\]       
• \[G^\ddagger_\mathrm{gas} = -180.0\] for the TS (includes ZPE & thermal)                    
• TS has one imaginary frequency \[\tilde\nu^\ddagger = 700\ \mathrm{cm^{-1}}\]              
• Symmetry numbers: \[\sigma(A)=1\], \[\sigma(B)=2\], \[\sigma^\ddagger=1\]                   

Simple SMD-like solvation free energies \[(1\ \mathrm{M}\] standard state) at \[298.15\ \mathrm{K}\]:        

• \[\Delta G_\mathrm{solv}(A)=-12.0\]                       
• \[\Delta G_\mathrm{solv}(B)=-8.0\]                   
• \[\Delta G_\mathrm{solv}^\ddagger=-15.5\]                      

Reaction mixture is \[0.20\ \mathrm{M}\] in A and \[0.50\ \mathrm{M}\] in B (MeCN). Assume ideal solution (activity \[a_i=\frac{\gamma_i c_i}{c^\circ}\], take \[\gamma_i=1\], \[c^\circ=1\ \mathrm{M}\]). Gas constant \[R=8.314462618\ \mathrm{J\ mol^{-1}\ K^{-1}}\], \[k_B T/h = 6.208\times 10^{12}\ \mathrm{s^{-1}}\] at \[298.15\ \mathrm{K}\], \[1\ \mathrm{kJ\ mol^{-1}}/RT = 0.40341\].

Grading rubric:
criterion: Correctly identifies the energetic penalty as "desolvation" or "differential solvation" due to the transition state being less stabilized than the reactants.
weight: 10
criterion: Correctly attributes the symmetry factor of 2 specifically to the statistical indistinguishability of the reaction pathways or faces.
weight: 20
criterion: Correctly performs dimensional analysis to justify the standard state concentration term converting frequency units to second-order rate units.
weight: 20
criterion: Correctly integrates the symmetry factor, tunneling coefficient, and standard state correction into a single symbolic master equation before numerical substitution. 
weight: 10
criterion: Correctly interprets the magnitude of the final rate constant as approaching the diffusion-controlled limit or being kinetically fast.
weight: 20
criterion: Correctly calculates the final rate constant value within the range of 9.2e8 to 9.4e8.
weight: 20

Answer:
9.3 × 10^8 M^-1 s^-1