Question #44436

In the Kiliani-Fischer chain extension, the nucleophilic addition of \ce{CN^-} to the acyclic carbonyl form of a sugar is the rate-determining step. For a heavily branched aldose derivative, 2-C-methyl-D-arabinose, the approach of the cyanide ion along the Bürgi-Dunitz trajectory ($107^\circ$) is sterically hindered. This causes a significant increase in torsional strain during the $sp^2 \to sp^3$ rehybridization, mimicking 1,3-diaxial interactions in the transition state. If this steric inhibition reduces the reaction rate constant by a factor of $10^4$ at $298\ K$ compared to unbranched D-arabinose, what is the corresponding increase in the activation free energy ($\Delta \Delta G^\ddagger$) for the addition step? (Assume identical pre-exponential factors and $2.303 RT = 5.7\ kJ\ mol^{-1}$ at $298\ K$)