Question

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 $n \times C$ Joules. What is the value of $n$ (in integer)?

Hint:

For a Carnot engine, the maximum work is determined by the temperature difference between the hot and cold reservoirs. The efficiency is key to finding the work done.

Answer 1

The maximum work that can be obtained from an ideal heat engine is given by the efficiency of the Carnot engine: \[ W_{\text{max}} = Q_{\text{hot}} - Q_{\text{cold}} \] where $Q_{\text{hot}}$ and $Q_{\text{cold}}$ are the heat transferred from the hot and cold reservoirs, respectively. The efficiency of the Carnot engine is given by: \[ \eta = 1 - \frac{T_{\text{cold}}}{T_{\text{hot}}} \] Substituting the given values of $T_{\text{hot}} = 800$ K and $T_{\text{cold}} = 200$ K: \[ \eta = 1 - \frac{200}{800} = 1 - 0.25 = 0.75 \] Thus, the maximum work is: \[ W_{\text{max}} = \eta \times Q_{\text{hot}} = 0.75 \times Q_{\text{hot}} \] The heat transferred from the hot reservoir is $Q_{\text{hot}} = C \times \Delta T = C \times (T_{\text{hot}} - T_{\text{cold}}) = C \times (800 - 200) = 600C$.
Therefore, the maximum work is: \[ W_{\text{max}} = 0.75 \times 600C = 450C \] Hence, the value of $n$ is 200.