GATE 2023 PH – Question 65
Two identical bodies kept at temperatures 800 K and 200 K act as the hot and the cold reservoirs of an ideal heat engine, respectively. Assume that their heat capacity (C) in Joules/K is independent of temperature and that they do not undergo any phase change. Then, the maximum work that can be obtained from the heat engine is n× C Joules. What is the value of n (in integer)?
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 200
Explanation
The two identical bodies have heat capacity $C$ (independent of temperature) and start at 800 K and 200 K. An ideal (reversible) engine runs between them, and both reservoirs change temperature until they reach a common final temperature $T_f$.
**Reversible process: total entropy change is zero.**
$$\Delta S=C\ln\frac{T_f}{800}+C\ln\frac{T_f}{200}=0\;\Rightarrow\;T_f^2=800\times200\;\Rightarrow\;T_f=400\text{ K}.$$
**Maximum work** = heat taken from the hot body minus heat delivered to the cold body:
$$W=C\,(800-T_f)-C\,(T_f-200)=C\,(800+200-2T_f)=C\,(1000-800)=200\,C .$$
So $n=\mathbf{200}$.