= Solution
\b[<water> stabilizes the planar <amide> more strongly relative to the twisted rotational transition region, increasing the free-energy barrier.] <nitrogen> lone-pair donation into the carbonyl gives the C–N bond partial double-bond character through <amide resonance>, represented by
$$
\mathrm{O=C{-}NR_2\ \leftrightarrow\ {}^-O{-}C= N^+R_2}.
$$
<amide bond rotation> between the two <amide> <conformations> requires loss of favourable overlap. Near a strongly twisted geometry the <nitrogen> <lone pair> is less conjugated with C=O, and the charge-separated <resonance in chemistry> contribution is reduced. Polar <water>, a <hydrogen bond donor>, preferentially stabilizes the planar starting structures, including their polar carbonyl/resonance distribution, compared with this less conjugated rotational region. <chloroform> provides less of that differential stabilization. What controls the rate is the \b[difference between the starting minimum and the <transition state>], not the absolute stabilization of either one in isolation.
For a first-order interpretation of the <half-lives>, the characteristic <rate constants> are
$$
k_{\rm CHCl_3}=\frac{\ln2}{2\,\mathrm s}=0.3466\,\mathrm{s^{-1}},\qquad
k_{\rm water}=\frac{\ln2}{50\,\mathrm s}=0.01386\,\mathrm{s^{-1}}.
$$
Thus the interconversion is \b[25 times slower in <water>]. With comparable <Eyring equation> prefactors at the same temperature,
$$
\boxed{\Delta G^\ddagger_{\rm water}-\Delta G^\ddagger_{\rm CHCl_3}=RT\ln25,}
$$
which is about $8.0\,\mathrm{kJ\,mol^{-1}}$ at $298\,\mathrm K$. This temperature is illustrative because none is specified for the measurements.
\Image[/past-exam-of-the-chemistry-b-course-of-the-university-of-cambridge/2023/paper-2-free-energy.png]
{title=Relative amide-rotation barriers in two solvents, with an illustrative water increment RT ln 25}
The two endpoint <conformations> remain minima separated by a twisted barrier in each <solvent>. The profiles are schematic, with arbitrary absolute barriers; the <half-life> ratio supports the relative kinetic comparison. If the quoted <half-lives> describe relaxation of a reversible cis/trans mixture, $k_{\rm obs}=k_f+k_r$, so the ratio directly concerns relaxation rates. Assigning the full $RT\ln25$ difference to one microscopic forward barrier additionally requires comparable equilibrium populations or directional-rate information.
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