Question

Somebody claims to have developed a new reversible heat-engine cycle that has a higher theoretical efficiency than the Carnot cycle operating between the same temperature limits. How do you evaluate this claim?

Short answer

Answer: To evaluate this claim, we must compare the efficiency of the new heat-engine cycle (\(\eta_{new}\)) with the efficiency of the Carnot cycle (\(\eta_{Carnot}\)). If \(\eta_{new} > \eta_{Carnot}\), then it is possible for the new heat-engine cycle to have a higher theoretical efficiency. However, it is essential to consider that the Carnot cycle represents the best possible achievable efficiency, and surpassing it in practice might be unlikely.

Step-by-step solution

Understand the Carnot cycle and efficiency

The Carnot cycle is a theoretically perfect thermodynamic cycle that serves as an ideal benchmark for heat engines. It consists of two isothermal processes and two adiabatic processes. The efficiency of a Carnot cycle depends only on the temperatures of the hot and cold reservoirs, which are denoted as \(T_H\) and \(T_C\), respectively. The theoretical maximum efficiency of a heat engine operating between these two temperature limits can be calculated using the Carnot efficiency formula: Carnot efficiency (\(\eta_{Carnot}\)) = \(1 - \frac{T_C}{T_H}\)

Obtain the efficiency of the new heat-engine cycle

Get the claimed efficiency of the new heat-engine cycle. Let's denote the efficiency of the new heat-engine cycle as \(\eta_{new}\).

Compare the new cycle efficiency with the Carnot efficiency

To evaluate the claim that the new heat-engine cycle has a higher theoretical efficiency than the Carnot cycle operating between the same temperature limits, we need to compare \(\eta_{new}\) with \(\eta_{Carnot}\). If \(\eta_{new} > \eta_{Carnot}\), then the claim is true, and the new heat-engine cycle has a higher theoretical efficiency than the Carnot cycle. If \(\eta_{new} \leq \eta_{Carnot}\), then the claim is false, as the new heat-engine cycle has a lower or equal theoretical efficiency when compared to the Carnot cycle operating between the same temperature limits.

Conclusion

By comparing the efficiencies of both the new heat-engine cycle and the Carnot cycle, we can determine if this claim is true or false. A higher theoretical efficiency for the new heat-engine cycle would indicate the possibility of increased performance compared to the Carnot cycle; however, it is essential to keep in mind that the Carnot cycle represents the best possible achievable efficiency, and it might be unlikely to surpass it in practice.