In the derivation of the Clausius inequality, [itex]T[/itex] is the temperature of the reservoir at that point in the cycle, but in the definition of entropy it becomes the temperature of the system. This seems to work for a Carnot cycle, where the two are the same, but for other processes, such as an object with constant heat capacity [itex]C[/itex] at temperature [itex]T_0[/itex] cooling due to heat exchange with a resevoir at temperature [itex]T < T_0 [/itex], is where I start to get confused. In that case we calculate the system's entropy change to be [itex] C \ln (T/T_0)[/itex] and the reservoir's as [itex]C(T_0-T)/T[/itex]. In fact, I guess we have to come up with a reversible process connecting the two states. What could that be?