EPR line shifts due to spin exchange of perdeuterated 2,2,6,6-tetramethyl-4-oxopiperidine-1-oxyl (N-14-PDT) in aqueous solutions and the same probe isotopically substituted with N-15 (N-15-PDT) were measured from 293 to 338 and 287 to 353 K, respectively. Nonlinear least-squares fits of the EPR spectra yielded the resonance fields of the nitrogen hyperfine lines to high precision from which the shifts were deduced. The shifts are described by two terms: one linear and the other quadratic in the electron spin-exchange frequency, omega(e). The quadratic term is due to spin exchange that occurs when two spin probes diffuse together and collide. A linear term is predicted for spin exchanges that occur upon re-encounter of the same two probes while they occupy the same "cage" before diffusing apart. The quadratic term has no adjustable parameters, while the linear term has one: the mean time between re-encounters, tau(RE). The theory is cast in terms of the spin-exchange-induced line broadening that can be measured from each spectrum independently of the line shifts, thereby removing the explicit dependence of omega(e) on the temperature and the spin-probe concentration. In this form, theoretically, the value of the linear term is about a factor of 2 larger for N-15-PDT than for N-14-PDT for all temperatures; however, THE must be the same. Experimentally, we find that both of these expectations are fulfilled, providing strong support that the linear term is indeed due to re-encounter collisions. Values of THE derived from N-14-PDT and N-15-PDT are of the same order of magnitude and show the same trend with temperature as a hydrodynamic estimate based on the Stokes-Einstein equation.