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 X-ray surface brightness and 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 applied a recently developed technique for the deprojection and PSF-deconvolution of X-ray surface brightness profiles to obtain non-parametric gas-density profiles out to distances ranging between 0.8 R500 and 1.5 R500. We scaled the gas density distributions to allow for the systems' differing masses and redshifts. The central gas densities differ greatly from system to system, with no clear correlation with system temperature. At intermediate radii (~ 0.3 R500), the scaled density profiles show much less scatter, with a clear dependence on system temperature. We find that the density at this radius scales proportionally to the square root of temperature, consistent with the presence of an entropy excess as suggested in previous literature. However, at larger scaled radii this dependence becomes weaker: clusters with kT > 3 keV scale self-similarly, with no temperature dependence of gas-density normalisation. The REXCESS sample allows us to investigate the correlations between cluster properties and dynamical state. We find no evidence of correlations between cluster dynamical state and either the gas density slope in the inner regions or temperature, but do find some evidence of a correlation between dynamical state and outer gas density slope. We also find a weak correlation between dynamical state and both central gas normalisation and inner cooling times, but this is only significant at the 10% level. We conclude that, for the X-ray cluster population as a whole, both the central gas properties and the angle-averaged, large-scale gas properties are linked to the cluster dynamical state. We also investigate the central cooling times of the clusters. While the cooling times span a wide range, 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, finding that , 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 consistent with earlier work on relaxed cluster samples. We measure a logarithmic intrinsic scatter in this relation of ~, which should be a good measure of the intrinsic scatter in the -T relation for the cluster population as a whole
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