Gas entropy in a representative sample of nearby X-ray galaxy clusters
  (REXCESS): relationship to gas mass fraction
  (Abridged) We examine the radial entropy distribution and its scaling using
31 nearby galaxy clusters from the Representative XMM-Newton Cluster Structure
Survey (REXCESS). The entropy profiles are robustly measured at least out to
R_1000 in all systems and out to R_500 in 13 systems. Compared to theoretical
expectations, the observed distributions show a radial and mass-dependent
excess entropy that is greater and extends to larger radii in lower mass
systems. At R_500, the mass dependence and entropy excess are both negligible
within the uncertainties. Mirroring this behaviour, the scaling of gas entropy
is shallower than self-similar in the inner regions, but steepens with radius,
becoming consistent with self-similar at R_500. The dispersion in scaled
entropy in the inner regions is linked to the presence of cool cores and
dynamical activity; at larger radii the dispersion decreases by a factor of two
and the dichotomy between subsamples disappears. Parameterising the profiles
with a power law plus constant model, there are two peaks in central entropy
K_0; however, we cannot distinguish between a bimodal or a left-skewed
distribution. The outer slopes are correlated with system temperature; their
distribution is unimodal with a median value of 0.98. Renormalising the
dimensionless entropy profiles by the gas mass fraction profile f_gas(< R),
leads to a remarkable reduction in the scatter, implying that gas mass fraction
variations with radius and mass are the cause of the observed entropy
properties. We discuss a tentative scenario to explain the behaviour of the
entropy and gas mass fraction in the REXCESS sample, in which extra heating and
merger mixing maintains an elevated central entropy level in the majority of
the population, and a smaller fraction of systems develops a cool core.
