Problem 46 - Pearson Physics

Question

Engine 1 takes in \(100 \mathrm{~J}\) of heat from a hot reservoir and does \(20 \mathrm{~J}\) of work. Engine 2 takes in the same amount of heat from the hot reservoir and does \(25 \mathrm{~J}\) of work. Is the efficiency of engine 1 greater than, less than, or equal to the efficiency of engine 2? Explain.

Step-by-Step Solution

Answer

Engine 1 is less efficient than Engine 2.

Step 1: Understand Efficiency Formula

Efficiency (\eta) is defined as the ratio of work output to heat input for an engine. Mathematically, the formula is given by \( \eta = \frac{W}{Q_{in}} \), where \( W \) is the work done by the engine and \( Q_{in} \) is the heat absorbed from the hot reservoir.

Step 2: Calculate Efficiency of Engine 1

For Engine 1, the work done \( W_1 = 20 \mathrm{~J} \) and the heat absorbed \( Q_{in,1} = 100 \mathrm{~J} \). Plug these values into the efficiency formula: \( \eta_1 = \frac{20}{100} = 0.20 \) or \( 20\% \).

Step 3: Calculate Efficiency of Engine 2

For Engine 2, the work done \( W_2 = 25 \mathrm{~J} \) and the heat absorbed \( Q_{in,2} = 100 \mathrm{~J} \). Using the efficiency formula: \( \eta_2 = \frac{25}{100} = 0.25 \) or \( 25\% \).

Step 4: Compare Efficiencies of the Engines

Compare the efficiencies calculated: Engine 1 has an efficiency of \( 20\% \), while Engine 2 has an efficiency of \( 25\% \). Therefore, the efficiency of Engine 1 is less than the efficiency of Engine 2.

Key Concepts

Heat Engine

A heat engine is a fascinating device that converts thermal energy into mechanical work. It operates by taking in heat from a source, like a hot reservoir, and uses that energy to perform work. The process is cyclic, meaning the engine goes through a series of steps repeatedly to produce work. During each cycle, part of the thermal energy is converted to work, while the rest is expelled as waste heat into a cold reservoir.
Heat engines are the backbone of many machines we encounter daily, such as car engines and power plants. They play a crucial role in our technology-driven world by efficiently converting energy from one form to another. Understanding how heat engines operate is key to improving their efficiency and performance.

Work Output

Work output is the amount of energy that a heat engine converts into useful mechanical work during its cycle. It is one of the most critical parameters used to measure how well an engine performs. In simple terms, it’s the energy that leaves the engine in a useful form.
The work output depends on various factors such as the design of the engine and the efficiency of the energy transformation process. For instance, in our original exercise, Engine 1 had a work output of 20 J, while Engine 2 had a work output of 25 J. This variation affects the engine’s efficiency and is crucial for assessing the engine's performance.
High work output relative to the heat input often indicates a more efficient engine, as it means a larger portion of the absorbed heat is being used to perform work rather than being wasted.

Heat Input

Heat input refers to the thermal energy absorbed by a heat engine from a hot reservoir. It is the starting point of the engine's energy conversion process and is essential for the engine's ability to produce work.
In the given exercise, both Engine 1 and Engine 2 had the same heat input of 100 J. This uniform heat input allows for a direct comparison of their efficiencies based solely on their work outputs. The heat input is crucial because it sets the potential maximum work output if the engine were perfectly efficient.
Real-world engines are never perfectly efficient due to unavoidable energy losses, such as friction and sound, making the concept of heat input vital for understanding the energy balance in the engine cycle.

Efficiency Calculation

The efficiency of a heat engine is a measure of how well it converts heat input into useful work. It is expressed as a percentage and is calculated using the formula:\[\eta = \frac{W}{Q_{in}} \times 100\%\]where \( W \) represents the work output and \( Q_{in} \) represents the heat input. This equation allows us to compare the performance of different engines. For example, Engine 1 in our exercise has an efficiency of 20%, calculated as \( \frac{20}{100} \times 100\% \). In contrast, Engine 2 has an efficiency of 25%, or \( \frac{25}{100} \times 100\% \).
These results show that Engine 2 is more efficient than Engine 1, as it transforms a greater fraction of its heat input into work. Efficiency calculations are essential not only for determining engine performance but also for identifying areas where improvements can be made to reduce waste and optimize energy usage.