How ions affect RNA folding thermodynamics and kinetics is an important but a vexing problem that remains unsolved. Experiments have shown that the free-energy change, Delta G(c), of RNA upon folding varies with the salt - concentration (c) as, Delta G(c) = k(c)ln c + const, where the coefficient k(c) is proportional to the difference in the ion preferential coefficient, Delta Gamma. We performed simulations of a coarse-grained model, by modeling electrostatic interactions implicitly and with explicit representation of ions, to elucidate the molecular underpinnings of the relationship between Delta G and Delta Gamma. The simulations quantitatively reproduce the heat capacity for a pseudoknot, thus validating the model. We show that Delta G(c), calculated directly from Delta Gamma, varies linearly with ln c (c < 0.2 M), for a hairpin and the pseudoknot, demonstrating a molecular link between the two quantities. Explicit ion simulations also show the linear dependence of Delta G(c) on In c at all c with k(c)= 2k(B)T, except that Delta G(c) values are shifted by similar to 2 kcal/mol higher than experiments. The discrepancy is due to an underestimation of Gamma for both the folded and unfolded states while giving accurate values for Delta Gamma. The predictions for the salt dependence of Delta Gamma are amenable to test using single-molecule pulling experiments. The framework provided here can be used to obtain accurate thermodynamics for other RNA molecules as well.