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Mathematical Statement of the 2nd Law:

Consider a mechanically irreversible adiabatic between two equilibrium states X and Y. In fig. 4.7 this path is reproduced as a broken line between points. Next the fluid is returned to the original state by means two sequential processes: (i) a mechanically reversible adiabatic process Y-Z, and then (ii) a reversible isobaric compression Z-X. Assuming that the mechanically irreversible process X-Y leads to an entropy change, heat transfer must occur during process Z-X (since none occurs on Y-Z). This is because for the reversible return path the same entropy change must occur as between X to Y. Since the return path is reversible we must get Mathematical Statement of the Second Law | Thermodynamics - Mechanical EngineeringThus, integrating this expression for the return path:

Mathematical Statement of the Second Law | Thermodynamics - Mechanical Engineering

4.7 Cycle containing an irreversible adiabatic process A to B
 

Mathematical Statement of the Second Law | Thermodynamics - Mechanical Engineering            .....(4.23)

For the entire cycle we must have ∆Ut= 0. Thus, net total work during the cycle may be written as: Wt = −Qrev . However, since (as depicted by the P-V diagram) a net work would need to be done on the system over the cycle, it follows that Wt > 0; henceQrev< 0. Therefore, using eqn. 4.23, it follows that: that is:  Mathematical Statement of the Second Law | Thermodynamics - Mechanical Engineering that is:Mathematical Statement of the Second Law | Thermodynamics - Mechanical EngineeringThis implies that the entropy change for the original irreversible step A to B, we have:  Mathematical Statement of the Second Law | Thermodynamics - Mechanical Engineering  Therefore, we conclude that the original irreversible step is accompanied by a positive change of entropy. It may be additionally shown that if the original process occurred through irreversible heat transfer process an increase of entropy would result likewise. 

It may be pointed out during the return reversible process, since the heat transfer (during the isobaric path) occurs under reversible conditions it must occur under infinitesimal temperature gradient between the system and surrounding. Thus for the system and the surrounding the net entropy change (gives by: Qrev /T ) is equal and opposite. Consequently the total entropy change of the universe (system + surrounding) is zero during this step. During the initial irreversible path X to Y, while entropy increases for the system, for the surrounding there is no change. Nevertheless, for the universe as such there is a positive change in entropy. 

Since the above conclusion is not premised on any specific internal nature of the system, the above result may be generalized: all irreversible processes are accompanied by a positive change of entropy of the universe. This leads to a mathematical statement of the second law: ∆St≥ 0.

At this point we can revert to the question that was stated at the beginning of this chapter: what change in a system is thermodynamically feasible? The second law provides answer to this, i.e., a process can only proceed in a direction that results in a positive change in the total entropy of the universe, the limiting value of zero being attained only by a fully reversible process. The corollary to this is: no process is possible for which the total entropy of the universe decreases.

The document Mathematical Statement of the Second Law | Thermodynamics - Mechanical Engineering is a part of the Mechanical Engineering Course Thermodynamics.
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FAQs on Mathematical Statement of the Second Law - Thermodynamics - Mechanical Engineering

1. What is the mathematical statement of the Second Law?
Ans. The mathematical statement of the Second Law, also known as the Law of Entropy, is given by the equation ΔS ≥ 0, where ΔS represents the change in entropy in a closed system.
2. How is the Second Law related to entropy?
Ans. The Second Law of Thermodynamics states that the entropy of a closed system will either remain constant or increase over time. Entropy can be thought of as a measure of the disorder or randomness in a system. Therefore, as time progresses, the disorder in a closed system will either stay the same or increase, in accordance with the Second Law.
3. Can the entropy of a closed system decrease?
Ans. No, according to the Second Law of Thermodynamics, the entropy of a closed system cannot decrease. The entropy can either remain constant or increase, but it cannot decrease.
4. How is the Second Law relevant to everyday life?
Ans. The Second Law of Thermodynamics has various applications in everyday life. It explains why hot coffee cools down when left alone, why ice melts when left at room temperature, and why energy tends to disperse and become less useful over time. It helps us understand natural processes and the direction in which they occur.
5. Is the Second Law violated in any natural processes?
Ans. While the Second Law of Thermodynamics holds true for most natural processes, there are certain situations where it may appear to be violated. However, upon closer examination, it is found that these processes are not truly closed systems or that there is an increase in entropy elsewhere to compensate for the apparent decrease. Overall, the Second Law is a fundamental principle that has been consistently observed in numerous physical and chemical systems.
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