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**

Q_{1} = 300 kJ, T_{1} = 500K, Q2 = 210 kJ, T_{2} = 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.**