300 kJ of heat is supplied at constant temperature of 500 K, to a heat engine. The heat rejection takes place at 300 K. The following result are obtai

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300 kJ of heat is supplied at constant temperature of 500 K, to a heat engine. The heat rejection takes place at 300 K. The following result are obtained (a) 210 kJ (b) 180 kJ (c) 150 kJ. Identify whether result is a reversible cycle. Irreversible cycle or impossible cycle respectively

1. reversible / irreversible / impossible respectively
2. irreversible / reversible / impossible respectively
3. irreversible / impossible / reversible respectively.
4. None of the above

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Correct Answer - Option 2 : irreversible / reversible / impossible respectively

Concept:

Clausius inequality states that $\oint \frac{{{\rm{dQ}}}}{{\rm{T}}} \le 0$

It provides the criteria for the reversibility of a cycle.

If $\oint \frac{{{\rm{dQ}}}}{{\rm{T}}} = 0$, the cycle is reversible,

$\oint \frac{{{\rm{dQ}}}}{{\rm{T}}} < 0$, the cycle is irreversible and possible

$\oint \frac{{{\rm{dQ}}}}{{\rm{T}}} > 0,$​​ The cycle is impossible

$\oint \frac{{{\rm{dQ}}}}{{\rm{T}}} = \frac{Q_1}{T_1} - \frac{Q_2}{T_2}$

Calculation:

Given:

Case 1

Q1 = 300 kJ, T1 = 500K, Q2 = 210 kJ, T2 = 300K,

$\oint \frac{{{\rm{dQ}}}}{{\rm{T}}} = \frac{Q_1}{T_1} - \frac{Q_2}{T_2}$

$\oint \frac{{{\rm{dQ}}}}{{\rm{T}}} = \frac{300}{500} - \frac{210}{300}$

$\oint \frac{{{\rm{dQ}}}}{{\rm{T}}} =- 0.1$

$\oint \frac{{{\rm{dQ}}}}{{\rm{T}}} < 0$ the cycle is irreversible and possible

Case 2

Q1 = 300 kJ, T1 = 500K, Q2 = 1800 kJ, T2 = 300K,

$\oint \frac{{{\rm{dQ}}}}{{\rm{T}}} = \frac{Q_1}{T_1} - \frac{Q_2}{T_2}$

$\oint \frac{{{\rm{dQ}}}}{{\rm{T}}} = \frac{300}{500} - \frac{180}{300}$

$\oint \frac{{{\rm{dQ}}}}{{\rm{T}}} =0$

$\oint \frac{{{\rm{dQ}}}}{{\rm{T}}} = 0$, the cycle is reversible,

Case 3

Q1 = 300 kJ, T1 = 500K, Q2 = 150 kJ, T2 = 300K,

$\oint \frac{{{\rm{dQ}}}}{{\rm{T}}} = \frac{Q_1}{T_1} - \frac{Q_2}{T_2}$

$\oint \frac{{{\rm{dQ}}}}{{\rm{T}}} = \frac{300}{500} - \frac{150}{300}$

$\oint \frac{{{\rm{dQ}}}}{{\rm{T}}} = 0.1$

$\oint \frac{{{\rm{dQ}}}}{{\rm{T}}} > 0,$​ The cycle is impossible.