Question

For a thermodynamic cycle to be reversible, it is necessary that

• A. $\oint \frac{dQ}{T} = 0$
• B. $\oint \frac{dQ}{T} \lt 0$
• C. $\oint \frac{dQ}{T} \gt 0$
• D. $\oint \frac{dQ}{T} = \infty$

Hint:

Think of "=" as "Perfect" (Reversible) and "<" as "Real" (Irreversible). Since a reversible cycle is a perfect theoretical concept, it must balance out exactly to Zero.

Answer 1

The Correct Option is A

The Clausius Inequality is a fundamental theorem in thermodynamics that provides a criterion for the reversibility of a cycle based on the Second Law of Thermodynamics. 1. The Clausius Inequality: The theorem states that for any thermodynamic cycle, the cyclic integral of $\frac{dQ}{T}$ must satisfy: $$\oint \frac{dQ}{T} \leq 0$$

2. Interpreting the Integral: The value of this cyclic integral tells us about the nature of the cycle:

• Case 1 (Reversible Cycle): If the cycle is perfectly reversible (no friction, no rapid expansions, and heat transfer only across infinitesimal temperature differences), then: $$\oint \frac{dQ}{T} = 0$$

• Case 2 (Irreversible Cycle): If there are any internal or external irreversibilities (which occur in all real-world processes), then: $$\oint \frac{dQ}{T} \lt 0$$

• Case 3 (Impossible Cycle): If the integral is greater than zero, the cycle violates the Second Law of Thermodynamics.

3. Conclusion: For a cycle to be truly reversible, there must be no net change in the entropy of the universe. This condition is mathematically satisfied only when the cyclic integral of the entropy transfer is exactly zero.