A quadratic free energy relationship (FER) between the kinetic activation free energy DeltaG(double dagger) and the thermodynamic reaction asymmetry DeltaG(RXN) is derived for acid-base ionization proton-transfer reactions AH...B --> A(-)...HB+ in a polar environment in the proton adiabatic regime, in which the proton is treated quantum mechanically, but does not tunnel. The description differs from traditional treatments in both the proton quantization and the identification of a solvent coordinate as the reaction coordinate, The key coefficients in the FER are analyzed analytically for the simplified case, where the proton donor-acceptor distance is held fixed (a restriction removed in the following paper). In particular, the intrinsic barrier is shown to be the sum of an intrinsic solvent barrier, largely determined by solvent reorganization, and the zero point energy difference of the proton between the reactant and the transition state in a solvent coordinate. The Bronsted coefficient is related to the quantum proton-averaged solute electronic structure at, and the position of, this transition state along this reaction coordinate. Similarities and differences of the FER with the well-known Marcus relation are discussed.