You hook the liquid up to a heat bath, via a
super-efficient heat exchanger!
Now everything is in equilibrium during the
process of evaporation: liquid, vapor, heat
bath, everything. The latent heat is supplied
from the heat bath, so energy is conserved,
which is good. The speed distribution in the
liquid is the same before and after, so you
can infer -- sorta maybe -- that the particles
that left must have had the same distribution
as those that remained. Everything fits the
specification of the problem AFAICT.
The liquid evaporates without cooling, which
fits the qualitative part of the answer.
HOWEVER I am still not buying one word of
Hewitt's explanation. It strikes me as one
of those "canceling the sixes" explanations.
Causation is a tricky concept, but if you ask
me, if you repeat the experiment without the
heat bath, the liquid will cool ... not
because of the distribution of speeds, but
because you didn't supply the latent heat.
It seems to me that the energy-conservation
argument is more grade-appropriate. Also it
is guaranteed to make correct predictions, in
this situation and innumerable others.
It would be nice if somebody could parlay this
into a believable lesson about the microscopic
mechanisms, but I haven't seen that, not even
close. The fact that canceling the sixes is
consistent with the observed right answer in
one chosen example doesn't mean it's right in
general, or worth learning, or worth teaching.