Clausius inequality entropy change

So, we can assign temperatures consistently with our experience of hotness. Now it s time to move on and prove both the existence of entropy as a state function and the Clausius inequality. This requires employing the concept of the reversible process. A reversible process is one each step of which may be exactly reversed by an infinitesimal change in the external conditions prevailing at the time of that step. The two requirements needed for a real process to approximate reversibility are (1) the process proceeds slowly compared to all internal... 

This is known as the Clausius inequality and has important applications in irreversible processes. For example, dS > (dQ/T) for an irreversible chemical reaction or material exchange in a closed heterogeneous system, because of the extra disorder created in the system. In summary, when we consider a closed system and its surroundings together, if the process is reversible and if any entropy decrease takes place in either the system or in its surroundings, this decrease in entropy should be compensated by an entropy increase in the other part, and the total entropy change is thus zero. However, if the process is irreversible and thus spontaneous, we should apply Clausius inequality and can state that there is a net increase in total entropy. Total entropy change approaches zero when the process approaches reversibility. 

Clausius inequality and the change of entropy for nonequilibrium processes... 

Let us change this inequality by incorporating the entropy notion. Use a cycle like that presented in Figure 3.21. Let us consider also that part of the cycle (la2) is reversible and the other (lb2) is irreversible. From the Clausius inequality it then follows that... 

The foundation of rational thermodynamics is the Clausius-Planck inequality defining the change of entropy between two equilibrium states, 1 and 2... 

At the third stage the equilibrium thermodynamics was created by Clausius, Helmholtz, Boltzmann and Gibbs. Since that time the equilibrium principles started to develop as applied to macroscopic systems of any physical nature. The main, second law of thermodynamics was discovered by Clausius (Clausius, 2008). He found out the existence of the state function, entropy (S), that can change in the isolated systems exclusively towards increase. The inequality that shows such monotonicity of change... 

---