For a reversible change in a thermodynamic system, the second law of thermodynamics can be expressed in terms of entropy. For an irreversible process, the entropy change of the system and surroundings is greater than zero. The mathematical statement that combines these ideas is known as the Clausius inequality. The Clausius inequality is expressed mathematically as [ \oint \frac{\delta q}{T} \leq 0 ] where (\delta q) represents an infinitesimal amount of heat supplied to or removed from the system, (T) is the absolute temperature at which the heat transfer occurs, and the symbol (\oint) indicates integration over a complete thermodynamic cycle. For a reversible cycle, [ \oint_{\text{rev}} \frac{\delta q}{T}=0 ] whereas for an irreversible cycle, [ \oint_{\text{irrev}} \frac{\delta q}{T}<0 ] Thus, the Clausius inequality may be written in the general form [ \boxed{\oint \frac{\delta q}{T}\leq 0} ] This relation was developed from the second law of thermodynamics and provides an important mathematical criterion for distinguishing reversible and irreversible processes. To understand its significance, consider a thermodynamic system undergoing a cyclic process. At the end of a complete cycle, the system returns to its original state. Therefore, all state functions, such as internal energy, enthalpy, and entropy, return to their initial values. Heat and work, however, are path functions and depend on the manner in which the cycle is carried out. For a reversible cycle, the heat absorbed by the system at each stage divided by the corresponding absolute temperature, when integrated around the entire cycle, gives zero: [ \oint \frac{\delta q_{\text{rev}}}{T}=0 ] This result provides the basis for defining entropy. For a reversible process between states 1 and 2, [ dS=\frac{\delta q_{\text{rev}}}{T} ] and hence [ \Delta S=S_2-S_1 =\int_1^2\frac{\delta q_{\text{rev}}}{T} ] Since entropy is a state function, its change depends only on the initial and final states and not on the path followed. For an irreversible process, the corresponding relation is [ dS>\frac{\delta q}{T} ] for the system considered together with the appropriate surroundings, and more generally, [ \Delta S\geq\int_1^2\frac{\delta q}{T} ] Equality applies to a reversible process, while the greater-than sign applies to an irreversible process. This is another important form of the Clausius inequality. The Clausius inequality is significant because it provides a quantitative expression of the second law of thermodynamics. The first law states that energy is