Question

An inventor claims that he has created a water-driven engine with an efficiency of 0.200 that operates between thermal reservoirs at \(4^{\circ} \mathrm{C}\) and \(20 .{ }^{\circ} \mathrm{C}\). Is this claim valid?

Short answer

Answer: No, the inventor's claim is invalid as the engine's efficiency of 0.200 is higher than the maximum possible efficiency (Carnot efficiency) of 0.0546 for an engine operating between the given temperatures.

Step-by-step solution

Conversion of temperatures to Kelvin

Convert the given temperatures in Celsius to Kelvin. To do so, add 273.15 to the Celsius temperature value. We'll call the lower temperature T1 and the higher temperature T2. T1 = 4°C + 273.15 = 277.15 K T2 = 20°C + 273.15 = 293.15 K

Calculate the Carnot efficiency

The Carnot efficiency is the maximum possible efficiency of any heat engine operating between two temperatures T1 and T2. It is determined by the following formula: Carnot efficiency = 1 - (T1 / T2) Now, let's plug in the values for T1 and T2: Carnot efficiency = 1 - (277.15 / 293.15) = 1 - 0.9454 ≈ 0.0546

Compare the efficiencies

Now that we've calculated the Carnot efficiency (0.0546), let's compare it to the efficiency claimed by the inventor (0.200). If the inventor's engine efficiency is greater than the Carnot efficiency, his claim is invalid: Inventor's efficiency: 0.200 Carnot efficiency: 0.0546 Since 0.200 > 0.0546, the inventor's claim of an engine with an efficiency of 0.200 is invalid, as it is higher than the maximum possible efficiency (Carnot efficiency) for an engine operating between the given temperatures.

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, radiation, and physical properties of matter. The key idea of thermodynamics is the conversion of heat energy into mechanical energy, which is precisely what heat engines do.

There are several laws of thermodynamics that govern how energy is transferred and transformed. The first law, also known as the law of energy conservation, states that energy cannot be created or destroyed in an isolated system. The second law introduces the concept of entropy, indicating that thermodynamic processes are irreversible and that energy tends to disperse or spread out unless acted upon by work. This plays a fundamental role in determining the efficiency of engines, as some energy is always lost as waste heat, which is why no engine can be 100% efficient.

In examining the inventor's claim, the first step is to analyze the thermodynamic process involved, understand the energy transfers, and use relevant equations to calculate the efficiency of the claimed process—a task that involves the use of the Carnot efficiency.

Heat Engines

Heat engines are devices that convert thermal energy into mechanical work. The basic operation of a heat engine involves transferring heat from a higher temperature source (the hot reservoir) to a lower temperature sink (the cold reservoir) and, in the process, doing work. The efficiency of a heat engine is a measure of how well it can convert the input heat into useful work.

One of the most important concepts when discussing heat engines is the Carnot cycle, which is a theoretical model that defines the maximum possible efficiency a heat engine can achieve between two thermal reservoirs. Although real engines cannot reach this idealized efficiency due to various practical limitations such as friction and material imperfections, the Carnot cycle sets the benchmark for the best possible performance. The efficiency of a Carnot engine is dependent solely on the temperatures of the hot and cold reservoirs and not on the specific details of the engine itself. This ties into the inventor's claim, which can be assessed by comparing the claimed efficiency with the theoretical limit set by the Carnot cycle.

Temperature Conversion

Temperature conversion is a fundamental practice in science and engineering as different processes operate more naturally within different temperature scales. The most commonly used temperature scales are Celsius (°C) and Kelvin (K). In thermodynamics, the Kelvin scale is particularly important because it is an absolute temperature scale, starting at absolute zero, where all molecular motion ceases.

The conversion from Celsius to Kelvin is straightforward: simply add 273.15 to the Celsius temperature. This is crucial when computing the Carnot efficiency because the formula requires temperatures to be in Kelvin. It's important to perform this step accurately to ensure correct calculation of thermal efficiency.

The step-by-step solution to the exercise showed how to convert the temperatures of the thermal reservoirs from Celsius to Kelvin. Using these Kelvin temperatures, we were then able to calculate the Carnot efficiency and thereby evaluate the validity of the inventor's claim. In doing so, we illustrated not only the practical application of temperature conversion but also its critical role in thermodynamic calculations and the assessment of engine efficiencies.