  We present observations of ammonia emission lines toward the interstellar
filament WB~673 hosting the dense clumps WB~673, WB~668, S233-IR and
G173.57+2.43. LTE analysis of the lines allows us to estimate gas kinetic
temperature ($\lesssim$ 30~K in all the clumps), number density
($7-17\times10^3$~cm$^{-3}$), and ammonia column density ($\approx 1-1.5\times
10^{15}$~cm$^{-2}$) in the dense clumps. We find signatures of collapse in WB
673 and presence of compact spatially unresolved dense clumps in S233-IR. We
reconstruct 1D density and temperature distributions in the clumps and estimate
their ages using astrochemical modelling. Considering CO, CS, NH$_3$ and
N$_2$H$^+$ molecules (plus HCN and HNC for WB~673), we find a chemical age of
$t_{\rm chem}=1-3\times 10^5$~yrs providing the best agreement between the
simulated and observed column densities in all the clumps. Therefore, we
consider $t_{\rm chem}$ as the chemical age of the entire filament. A long
preceding low-density stage of gas accumulation in the astrochemical model
would break the agreement between the simulated and observed column densities.
We suggest that rapid star formation over a $\sim 10^5$~yrs timescale take
place in the filament.
