We present a study of the structural and scaling properties of the gas distributions in the intracluster medium (ICM) of 31 nearby (z < 0.2) clusters observed with XMM-Newton, which together comprise the Representative XMM-Newton Cluster Structure Survey (REXCESS). In contrast to previous studies, this sample is unbiased with respect to cluster dynamical state, and it fully samples the cluster X-ray luminosity function. The clusters cover a temperature range of 2.0 -- 8.5 keV and possess a variety of morphologies. The sampling strategy allows us to compare clusters with a wide range of central cooling times on an equal footing. We present non-parametric gas-density profiles out to distances ranging between 0.8 R_500 and 1.5 R_500. The central gas densities differ greatly from system to system, with no clear correlation with system temperature. At intermediate radii the scaled density profiles show much less scatter, with a clear dependence on system temperature, consistent with the presence of an entropy excess as suggested in previous literature. However, at large scaled radii this dependence becomes weaker: clusters with kT > 3 keV scale self-similarly, with no temperature dependence of gas-density normalisation. We find some evidence of a correlation between dynamical state and outer gas density slope, and between dynamical state and both central gas normalisation and cooling time. We find no evidence of a significant bimodality in the distributions of central density, density gradient, or cooling time. Finally, we present the gas mass-temperature relation for the REXCESS sample, which is consistent with the expectation of self-similar scaling modified by the presence of an entropy excess in the inner regions of the cluster, and has a logarithmic intrinsic scatter of ~10%.
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