Polyelectrolyte solutions are modeled as linear tangent-jointed charged hard-sphere chains and counterions embedded in a continuum medium; the stickiness between the polyions and the counterions beyond the territorial binding is taken into account as the short-range non-Coulombic perturbation. Expressions of thermodynamic properties derived are then consisted of two contributions. The chain contribution concerns the formation of polyion chains from the monomers. The sticky contribution accounts for the additional non-Coulombic stickiness, which is derived by solving the Ornstein-Zernike integral equation through the mean-spherical approximation (MSA) and the hypernetted-chain approximation (HNC). For the model solutions without stickiness between the polyions and the counterions, effects of polyion chain lengths, counterion sizes, counterion charges, and the dielectric constants on the thermodynamic properties are extensively studied. Comparisons of the osmotic pressures for the model solutions with those predicted from Manning＇s limiting law (LL), and where possible, with computer simulations are presented. For practical synthetic polyelectrolyte solutions, involving an adjustable parameter responsible for the additional stickiness, experimental thermodynamic properties can be described satisfactorily.