  The new mathematical framework based on the free energy of pure classical
fluids presented in [R. D. Rohrmann, Physica A 347, 221 (2005)] is extended to
multi-component systems to determine thermodynamic and structural properties of
chemically complex fluids. Presently, the theory focuses on $D$-dimensional
mixtures in the low-density limit (packing factor $\eta < 0.01$). The formalism
combines the free-energy minimization technique with space partitions that
assign an available volume $v$ to each particle. $v$ is related to the
closeness of the nearest neighbor and provides an useful tool to evaluate the
perturbations experimented by particles in a fluid. The theory shows a close
relationship between statistical geometry and statistical mechanics. New,
unconventional thermodynamic variables and mathematical identities are derived
as a result of the space division. Thermodynamic potentials $\mu_{il}$,
conjugate variable of the populations $N_{il}$ of particles class $i$ with the
nearest neighbors of class $l$ are defined and their relationships with the
usual chemical potentials $\mu_i$ are established. Systems of hard spheres are
treated as illustrative examples and their thermodynamics functions are derived
analytically. The low-density expressions obtained agree nicely with those of
scaled-particle theory and Percus-Yevick approximation. Several pair
distribution functions are introduced and evaluated. Analytical expressions are
also presented for hard spheres with attractive forces due to K\^ac-tails and
square-well potentials. Finally, we derive general chemical equilibrium
conditions.
