  The D/H ratios of carbonaceous chondrites, believed to reflect that of water
in the inner early solar system, are intermediate between the protosolar value
and that of most comets. The isotopic composition of cometary water has been
accounted for by several models where the isotopic composition of water vapor
evolved by isotopic exchange with hydrogen gas in the protoplanetary disk.
However, the position and the wide variations of the distribution of D/H ratios
in carbonaceous chondrites have yet to be explained. In this paper, we assume
that the D/H composition of cometary ice was achieved in the disk building
phase and model the further isotopic evolution of water in the inner disk in
the classical T Tauri stage. Reaction kinetics compel isotopic exchange between
water and hydrogen gas to stop at $\sim$500 K, but equilibrated water can be
transported to the snow line (and beyond) via turbulent diffusion and
consequently mix with isotopically comet-like water. Under certain simplifying
assumptions, we calculate analytically this mixing and the resulting
probability distribution function of the D/H ratio of ice accreted in
planetesimals and compare it with observational data. The distribution
essentially depends on two parameters: the radial Schmidt number Sc$_R$, which
ratios the efficiencies of angular momentum transport and turbulent diffusion,
and the range of heliocentric distances of accretion sampled by chondrites. The
minimum D/H ratio of the distribution corresponds to the composition of water
condensed at the snow line, which is primarily set by Sc$_R$. Observations
constrain the latter to low values (0.1-0.3), which suggests that turbulence in
the planet-forming region was hydrodynamical in nature, as would be expected in
a dead zone. Such efficient outward diffusion would also account for the
presence of high-temperature minerals in comets.
