Question

An inventor claims that he has created a water-driven engine with an efficiency of 0.200 that operates between thermal reservoirs at \(4.0^{\circ} \mathrm{C}\) and \(20.0^{\circ} \mathrm{C}\). Is this claim valid?

Short answer

Answer: No, the inventor's claim is not valid as the calculated maximum efficiency for a heat engine operating between these temperatures is approximately 0.0545, which is significantly lower than the claimed efficiency of 0.200.

Step-by-step solution

Convert temperatures to Kelvin

To convert the temperatures from Celsius to Kelvin, we add 273.15 to each temperature: \(T_1 = 4.0^{\circ}C + 273.15 = 277.15K\) \(T_2 = 20.0^{\circ}C + 273.15 = 293.15K\)

Calculate the maximum efficiency

The maximum efficiency of a heat engine operating between two temperatures is given by the efficiency of a Carnot engine: \(Efficiency_{max} = 1 - \frac{T_1}{T_2}\) Substitute the values for \(T_1\) and \(T_2\) from Step 1 to calculate the maximum efficiency: \(Efficiency_{max} = 1 - \frac{277.15}{293.15} \approx 0.0545\)

Compare the claimed efficiency to the maximum efficiency

The inventor claims an efficiency of 0.200, while the calculated maximum efficiency is approximately 0.0545 for a heat engine operating between the given thermal reservoirs. Since the claimed efficiency is significantly higher than the maximum efficiency, the inventor's claim is not valid.

Key concepts

Heat Engine

Imagine a heat engine as a sort of magical box that can do work for us, like turning a wheel or moving a car. But it's not magic—it follows the rules of physics. At its core, a heat engine takes in heat energy from a high-temperature source, converts part of this energy into work, and releases the remaining energy to a lower temperature sink. This process happens in a cycle, similar to how a carousel goes round and round.

To understand our inventor's claim about their water-driven engine, we picture it as a heat engine sitting between a hot pot of water at 20 degrees Celsius, and a cooler pot at 4 degrees Celsius. According to the exercise, the engine boasts an efficiency of 20% (0.200). In simple words, efficiency is a measure of how good the engine is at turning heat into useful work. If the engine is 100% efficient, it would mean all the heat is transformed into work with no waste, but that's not possible because of the second law of thermodynamics, which we'll touch on shortly.

Thermodynamic Temperature Conversion

When dealing with heat engines and thermodynamics, scientists like to use a temperature scale that starts at absolute zero, called the Kelvin scale. Absolute zero is the theoretical point where there's no heat and all particles stop moving. To convert from Celsius, a more everyday temperature scale, to Kelvin, we do a simple dance step: just add 273.15 to the Celsius value. This dance is critical because the formulas we use only work with Kelvin.

For this thought experiment, the warm pot of water was 20.0 degrees Celsius, which converts to 293.15K, and the cooler pot was at 4.0 degrees Celsius, bumping it up to 277.15K after we do the temperature tango. This step is essential because it sets the stage for us to calculate the best possible efficiency our engine can have—and that's where the Carnot engine comes into play.

Second Law of Thermodynamics

The second law of thermodynamics is like the party pooper of the physics world—it tells us that we can't have all that we want. This rule says that some energy will always be wasted when we convert heat to work. It's a fundamental principle that ensures a heat engine can never be 100% efficient.

One of the ways this law manifests is in the form of a limit on how efficient heat engines can be. This maximum efficiency is what the Carnot engine represents—it's the dream engine that works with perfect theoretical efficiency. It helps us understand the best we can do. When we calculate that a real engine, like our inventor's, is more efficient than a Carnot engine, that sets off alarm bells. It's like claiming you've found a way to make a perpetual motion machine—cool, but impossible according to the laws of physics as we know them.

Thermal Reservoirs

Thermal reservoirs are like giant bathtubs for heat energy. They can absorb or give off heat without changing their own temperature. In the context of our heat engine, these reservoirs are the warm and cool pots of water. A real thermal reservoir might be something like the ocean, which can absorb tons of heat from the atmosphere without heating up very much.

The efficiency of heat engines is tied to the temperatures of these reservoirs. According to our understanding of the second law of thermodynamics, the bigger the temperature gap between the hot and cold reservoirs, the higher the potential efficiency. In our inventor's case, the small gap between the two temperatures means there isn't much room for efficiency, and the low calculated maximum efficiency reflects that reality. By comparing the given temperatures, we can tell if an efficiency claim is reasonable or if it's too good to be true, like the claim of the water-driven engine being 20% efficient between just a 16-degree Celsius difference.