  We report on the lifetime measurement of the $6\,^1\Sigma_g^+ (7,31)$ state
of Na$_2$ molecules, produced in a heat-pipe oven, using a time-resolved
spectroscopic technique. The $6\,^1\Sigma_g^+ (7,31)$ level was populated by
two-step two-color double resonance excitation via the intermediate
$A\,^1\Sigma_u^+ (8,30)$ state. The excitation scheme was done using two
synchronized pulsed dye lasers pumped by a Nd:YAG laser operating at the second
harmonics. The fluorescence emitted upon decay to the final state was measured
using a time-correlated photon counting technique, as a function of argon
pressure. From this the radiative lifetime was extracted by extrapolating the
plot to collision-free zero pressure. We also report calculated radiative
lifetimes of the Na$_2$ $6\,^1\Sigma_g^+$ ro-vibrational levels in the range of
$v=0-200$ with $J=1$ and $J=31$ using the LEVEL program for bound-bound and the
BCONT program for bound-free transitions. Our calculations reveal the
importance of the bound-free transitions on the lifetime calculations and a
large difference of about a factor of three between the $J=1$ and $J=31$ for
the $v=40$ and $v=100$, due to the wavefunction alternating between having
predominantly inner and outer well amplitude.
