added as heat (𝛿Q) and as work (𝛿W), which may vary depending on the path. In a cyclic process, therefore, the entropy of the system at the beginning of the cycle must equal to the entropy at the end of the cycle (because the entropy is a state function), , regardless of whether the process is reversible or irreversible. In irreversible cases, the net entropy is added to the system reservoirs per thermodynamic cycle while in reversible cases, no entropy is created or added to the reservoirs. If the amount of energy added by heating can be measured during the process, and the temperature can be measured during the process, then the Clausius inequality can be used to determine whether the process is reversible or irreversible by carrying out the integration in the Clausius inequality. If integral result is equal to zero then it is a reversible process, while if greater than zero then an irreversible process (less than zero cannot be possible). Proof The temperature that enters in the denominator of the integrand in the Clausius inequality is the temperature of the external thermal reservoir with which the system exchanges heat. At each instant of the process, the system is in contact with an external reservoir. Because of the Second Law of Thermodynamics, in each infinitesimal heat exchange process between the system and the reservoirs, the net change in entropy of the "universe", so to say, is , where Sys and Res stand for System and Reservoir, respectively. In the proof of the Clausius theorem or inequality, a sign convention of heat is used; in the perspective of an object under consideration, when heat is absorbed by the object then the heat is positive, while when heat leaves from the object then the heat is negative. When the system takes heat from a hotter (hot) reservoir by an infinitesimal amount (), for the net change in entropy to be positive or zero (i.e., non-negative) in this step (called the step 1 here) to fulfill the Second Law of Thermodynamics, the temperature of the hot reservoir needs to be equal to or greater than the temperature of the system at that instant; if the temperature of the system is given by at that instant, then as the entropy change in the system at the instant, and forces us to have: This means the magnitude of the entropy "loss" from the