  The mean area of a two-dimensional Gaussian ring of $N$ monomers is known to
diverge when the ring is subject to a critical pressure differential, $p_c \sim
N^{-1}$. In a recent publication [Eur. Phys. J. E 19, 461 (2006)] we have shown
that for an inextensible freely jointed ring this divergence turns into a
second-order transition from a crumpled state, where the mean area scales as
$<A> \sim N$, to a smooth state with $<A> \sim N^2$. In the current work we
extend these two models to the case where the swelling of the ring is caused by
trapped ideal-gas particles. The Gaussian model is solved exactly, and the
freely jointed one is treated using a Flory argument, mean-field theory, and
Monte Carlo simulations. For fixed number $Q$ of trapped particles the
criticality disappears in both models through an unusual mechanism, arising
from the absence of an area constraint. In the Gaussian case the ring swells to
such a mean area, $<A> \sim NQ$, that the pressure exerted by the particles is
at $p_c$ for any $Q$. In the freely jointed model the mean area is such that
the particle pressure is always higher than $p_c$, and $<A>$ consequently
follows a single scaling law, $<A> \sim N^2 f(Q/N)$, for any $Q$. By contrast,
when the particles are in contact with a reservoir of fixed chemical potential,
the criticality is retained. Thus, the two ensembles are manifestly
inequivalent in these systems.
