  We consider here a dynamic model for a gas in which a variable number of
particles $N \in \mathbb{N}_0 := \mathbb{N} \cup \{0\}$ can be located at a
site. This point of view leads us to the grand-canonical framework and the need
for a chemical potential. The dynamics is played by the shift acting on the set
of sequences $\Omega := \mathcal{A}^\mathbb{N}$, where the alphabet is
$\mathcal{A} := \{1,2,...,r\}$. Introducing new variables like the number of
particles $N$ and the chemical potential $\mu$, we adapt the concept of
grand-canonical partition sum of thermodynamics of gases to a symbolic
dynamical setting considering a Lipschitz family of potentials $% (A_N)_{N \in
\mathbb{N}_0}$, $A_N:\Omega \to \mathbb{R}$. Our main results will be obtained
from adapting well-known properties of the Thermodynamic Formalism for IFS with
weights to our setting. In this direction, we introduce the
grand-canonical-Ruelle operator: $\mathcal{L}_{\beta, \mu}(f)=g$, when,
$\beta>0,\mu<0,$ and
  \medskip $\,\,\,\,\,\,\,\,\,\,\,\,\,\,g(x)= \mathcal{L}_{\beta, \mu}(f) (x)
=\sum_{N \in \mathbb{N}_0} e^{\beta \, \mu\, N }\, \sum_{j \in \mathcal{A}}
e^{- \,\beta\, A_N(jx)} f(jx). $ \medskip
  We show the existence of the main eigenvalue, an associated eigenfunction,
and an eigenprobability for $\mathcal{L}_{\beta, \mu}^*$. We can show the
analytic dependence of the eigenvalue on the grand-canonical potential.
Considering the concept of entropy for holonomic probabilities on $\Omega\times
\mathcal{A}^{\mathbb{N}_0}$, we relate these items with the variational problem
of maximizing grand-canonical pressure. In another direction, in the appendix,
we briefly digress on a possible interpretation of the concept of topological
pressure as related to the gas pressure of gas thermodynamics.
