from @Wiesenfeld2013 who consider H$_2$CO collisions with H$_2$ based on the high accuracy potential energy surface introduced by Troscompt et al. (). These rates differ significantly from the scaled rates of H$_2$CO in collision with He (@1991ApJS...76..979) used in previous analyses. At the same time we also took improved distances for our sources to evaluate more accurately the previously reported gradient of the carbon isotope ratio as a function of Galactocentric distance. In Section \[sec:obser\], the observations of our large sample are described, encompassing Galactocentric radii from 0 to 10.5 kpc. In Section \[sec:DataReduction\], we first perform the analysis of spectra and the radio continuum, followed by modelling work based on radiative transfer calculations. Section \[sec:discussions\] provides discussions on the isotope ratio gradient, derived from our observational data alone as well as also including previously obtained data. Our main results are summarized in Section \[sec:summary\].

Observations {#sec:obser}
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The $1_{10}-1_{11}$ and $2_{11}-2_{12}$ transitions were observed toward our sources in March and October 2016 as well as in May and July 2017 with the Shanghai Tianma Radio Telescope (TMRT). We used two cryogenically cooled receivers covering the frequency ranges of 4.0-8.0 GHz (C-band) and 12.0-18.2 GHz (Ku-band). The rest frequencies of 