Question

Does a heat engine that has a thermal efficiency of 100 percent necessarily violate \((a)\) the first law and \((b)\) the second law of thermodynamics? Explain.

Short answer

If so, which law(s) are violated? Answer: A heat engine with 100 percent thermal efficiency does not violate the first law of thermodynamics, as it conserves energy. However, it does violate the second law of thermodynamics, as it defies the natural increase of entropy and exceeds the limitations set by the Carnot efficiency.

Step-by-step solution

Understanding Thermal Efficiency

Thermal efficiency of a heat engine is the ratio of the work output to the heat input. In other words, it's a measure of how much of the heat energy input is converted to useful work output. A thermal efficiency of 100 percent means that all the heat energy input is converted to useful work, and there is no waste heat generated.

Analyzing the First Law of Thermodynamics

The first law of thermodynamics states that energy cannot be created or destroyed, only converted from one form to another. A heat engine with 100 percent efficiency converts all the heat energy input to useful work output, with no waste heat. This means that energy is conserved in the process, which is consistent with the first law of thermodynamics. Therefore, a heat engine with 100 percent efficiency does not violate the first law of thermodynamics.

Analyzing the Second Law of Thermodynamics

The second law of thermodynamics states that natural processes tend to increase the total entropy of the universe. The efficiency of a heat engine is limited by the Carnot efficiency, which is given by \(1 - \frac{T_c}{T_h}\), where \(T_c\) is the temperature of the cold reservoir, and \(T_h\) is the temperature of the hot reservoir. To achieve 100 percent efficiency, the engine would need to have either \(T_c = 0\) or \(T_h\) approaching infinity, both of which are physically impossible. Achieving 100 percent thermal efficiency would result in the extraction of all heat energy from the hot reservoir and none being transferred to the cold reservoir, effectively reversing the natural increase of entropy. Thus, a heat engine with 100 percent efficiency will violate the second law of thermodynamics. In conclusion: \((a)\) A heat engine with 100 percent thermal efficiency does not violate the first law of thermodynamics, as it conserves energy by converting all heat energy input to useful work output. \((b)\) A heat engine with 100 percent thermal efficiency does violate the second law of thermodynamics, as it defies the natural increase of entropy and exceeds the limitations set by the Carnot efficiency.