  We derived a two-dimensional map of the molecular fraction, fmol, (the ratio
of molecular-gas density to that of total gas) in the spiral galaxy M51, and
examined the behavior of molecular fronts (MF), where MF represents the place
where fmol changes drastically from nearly zero to unity and vice versa. We
show that the MF phenomenon occurs not only radial, but also in the azimuth
direction through the spiral arms, and fmol changes rapidly in the
arm-to-inter-arm transition regions. The existence of the azimuthal MF
indicates that the atomic gas (HI) is quicklytransformed to molecular gas (H_2)
during the passage through spiral arms. We performed a numerical simulation of
MF based on an HI-to-H_2 phase transition theory, and reproduced the
observations. We estimated a azimuthal scale length of the transition to be
less than 200 pc, corresponding to a time scale of about 2 Myr for H_2 gas
formation. The azimuthal width of a molecular arm is estimated to be at most
2.5 kpc, where the gas can remain in molecular phase for about 25 Myr.
