  For small thermodynamic systems in contact with a heat bath, we determine the
free energy by imposing the following two conditions. First, the quasi-static
work in any configuration change is equal to the free energy difference.
Second, the temperature dependence of the free energy satisfies the
Gibbs-Helmholtz relation. We find that these prerequisites uniquely lead to the
free energy of a classical system consisting of $N$-interacting identical
particles, up to an additive constant proportional to $N$. The free energy thus
determined contains the Gibbs factorial $N!$ in addition to the phase space
integration of the Gibbs-Boltzmann factor. The key step in the derivation is to
construct a quasi-static decomposition of small thermodynamic systems.
