Question

1. (a) what is the hot reservoir temperature of a carnot engine that has an efficiency of 42.0% and a cold reservoir temperature of 27.0°c? (b) what must the hot reservoir temperature be for a real heat engine that achieves 0.700 of the maximum efficiency, but still has an efficiency of 42.0% (and a cold reservoir at 27.0°c)? (c) does your answer imply practical limits to the efficiency of car gasoline engines?

Explanation:

Step1: Convert cold reservoir temperature to Kelvin

The formula to convert Celsius to Kelvin is \(T = t + 273.15\).
For \(t=- 27.0^{\circ}C\), \(T_{c}=27.0 + 273.15=300.15\space K\)

Step2: Use Carnot efficiency formula for part (a)

The Carnot efficiency formula is \(\eta_{C}=1-\frac{T_{c}}{T_{h}}\).
We know \(\eta_{C} = 0.420\), and \(T_{c}=300.15\space K\).
Rearrange the formula for \(T_{h}\): \(\frac{T_{c}}{T_{h}}=1 - \eta_{C}\), then \(T_{h}=\frac{T_{c}}{1-\eta_{C}}\)
Substitute the values: \(T_{h}=\frac{300.15}{1 - 0.420}=\frac{300.15}{0.580}\approx517.5\space K\)

Step3: For part (b)

Let the maximum (Carnot) efficiency be \(\eta_{C}\). We know \(\eta = 0.700\eta_{C}\) and \(\eta=0.420\)
So, \(\eta_{C}=\frac{\eta}{0.700}=\frac{0.420}{0.700} = 0.600\)
Again, using \(\eta_{C}=1-\frac{T_{c}}{T_{h}}\), and \(T_{c} = 300.15\space K\)
Rearrange for \(T_{h}\): \(T_{h}=\frac{T_{c}}{1-\eta_{C}}\)
Substitute \(\eta_{C}=0.600\) and \(T_{c}=300.15\space K\)
\(T_{h}=\frac{300.15}{1 - 0.600}=\frac{300.15}{0.400}=750.4\space K\)

Step4: For part (c)

Car gasoline engines are real - world heat engines. The second - law of thermodynamics and the concept of Carnot efficiency imply that there are practical limits. Real engines cannot reach Carnot efficiency, and even getting a significant fraction (like in part (b) where we had a factor of \(0.7\) of Carnot efficiency) requires high \(T_{h}\). High \(T_{h}\) has material - limitation (e.g., melting of engine components) and other practical constraints (e.g., fuel combustion limits)

Answer:

(a) \(T_{h}\approx518\space K\)
(b) \(T_{h} = 750\space K\)
(c) Yes, the answer implies practical limits. Real - world engines (like car gasoline engines) are subject to the second - law of thermodynamics. Achieving high efficiencies (even a fraction of Carnot efficiency) requires high \(T_{h}\), which is limited by material properties (e.g., melting point of engine parts), fuel combustion characteristics, and other engineering constraints. So, there are practical upper - bounds on the efficiency of car gasoline engines.