An analysis is presented of the equilibrium composition of syngas mixtures parametrized in terms of the syngas ratio and the hydrogen mole fraction, for fixed values of pressure and temperature. Conditions for the existence of single or multiple carbon-free solutions are determined by analysis of the carbon affinity equation, formulated by the method of atomic potentials. The system is characterized by two curves describing the maximum possible mole fraction of hydrogen, and the mole fraction of hydrogen at unit carbon affinity. These curves are tangent to each other at a single contact point. The unit carbon affinity curve consists of two branches, corresponding either to the higher or to the lower carbon affinity root becoming equal to one. These branches join together at the contact point of the maximum bounding and unit carbon affinity curves. The syngas ratio at the contact point is pressure-independent and is expressed as a function of temperature. A single carbon-free solution is obtained in the region below both branches of the unit carbon affinity curve. It is shown, however, that two distinct carbon-free solutions can be constructed in the region enclosed between the maximum bounding curve and the right branch of the unit carbon affinity curve.