The Clausius theorem is a fundamental principle in thermodynamics. It states that for any cyclic process, the cyclic integral of heat transfer divided by absolute temperature is less than or equal to zero. For reversible cycles, this integral equals zero, while for irreversible cycles, it is strictly less than zero.
For a reversible cycle, we analyze the process step by step. We divide the cycle into infinitesimal steps, where each step involves heat transfer delta Q at temperature T. The ratio of heat to temperature for each step is summed over the entire cycle. For reversible processes, this integral equals zero, which leads to the definition of entropy as a state function.