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Two types of Bi-2212 single crystals, one type having a low T c and the other having a high T¢ were annealed in air at different temperatures over the range 450°C to 750°C followed by furnace cooling. The transition temperature of the crystals with low T c (~ 56 K) increased with increase of annealing temperature, while the crystals with high Tc (~ 91 K) had just the opposite behaviour under the same annealing conditions. It is indicated that the quality of as-grown crystal has a significant effect on the superconducting properties of the annealed crystal. The c-axis values for both types of crystals decreased with increase of annealing temperature. According to the relationship between c-axis values and the oxygen content, our results confirmed the existence of a dome-shaped curve of T¢ with carrier concentration in Bi-2212 single crystal similar to that in Bi-2212 compound reported previously by Groen and de Leeuw [Physica C 159 (1989) 417]. By studying the increase of annealing time at a fixed temperature of 650°C, T¢ and ATe were found to change very little for both types of crystals. Thus, T c and AT~ were more sensitive to annealing temperature than to annealing time. XRD, SEM and EDS revealed that a small quantity of powder phase appeared after annealing at temperatures above 500°C for both types of crystals. This powder phase had the Bi-2212 composition, which indicated that the integrity of the Bi-2212 single crystals was significantly degraded by the annealing process.