Question

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

Answer 1

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.