Problem 16 - Pearson Physics

Question

Calculate A heat engine takes in \(1220 \mathrm{~J}\) of heat from the hot reservoir and exhausts \(680 \mathrm{~J}\) of heat to the cold reservoir. (a) How much work is done by the engine? (b) What is the efficiency of the heat engine?

Step-by-Step Solution

Answer

a) 540 J; b) 44.26% efficiency.

Step 1: Understand the Problem

We have a heat engine that absorbs 1220 J of energy from a hot reservoir and expels 680 J of energy to a cold reservoir. We need to calculate the work done by the engine and its efficiency.

Step 2: Apply the First Law of Thermodynamics

According to the first law of thermodynamics, the work done by the engine (W) is the difference between the heat absorbed from the hot reservoir (Q_hot) and the heat expelled to the cold reservoir (Q_cold). Mathematically, we express this as: \[ W = Q_{hot} - Q_{cold} \] Substitute the given values: \[ W = 1220 ext{ J} - 680 ext{ J} = 540 ext{ J} \] So, the work done by the engine is 540 J.

Step 3: Calculate the Efficiency

The efficiency (e) of a heat engine is defined as the ratio of the work done by the engine to the heat absorbed from the hot reservoir: \[ e = \frac{W}{Q_{hot}} \] Substitute the known values: \[ e = \frac{540 ext{ J}}{1220 ext{ J}} \approx 0.4426 \] Convert the efficiency to percentage by multiplying by 100: \[ e \approx 44.26\% \] Therefore, the efficiency of the heat engine is approximately 44.26%.

Key Concepts

Heat Engine

A heat engine is a fascinating device that allows us to convert thermal energy into mechanical work. It forms a crucial part of many modern machines, including automobiles and power plants. The operation of a heat engine is based on the flow of heat between a hot reservoir and a cold reservoir. By exploiting this temperature difference, the engine extracts energy from the hot reservoir. It then converts part of this energy into work, while the remaining energy is expelled to the cold reservoir.

• The **hot reservoir** is the source of thermal energy for the engine.
• The **cold reservoir** is where the unused energy is released.
• The work done by the engine is the energy that has been harnessed for practical use.

Understanding the relationship between these components is fundamental for grasping how heat engines function in relation to thermodynamics.

First Law of Thermodynamics

The first law of thermodynamics, also known as the law of energy conservation, plays a key role in understanding heat engines. This principle states that energy cannot be created or destroyed; it can only be transferred or changed from one form to another.For a heat engine, this means:

• The total energy absorbed from the hot reservoir equals the sum of the work done by the engine and the energy expelled to the cold reservoir.
• Energy input, losses, and useful work are all balanced, ensuring nothing is unaccounted for.

Mathematically, it's expressed as: \[ Q_{ ext{hot}} = W + Q_{ ext{cold}} \]Where:

• \(Q_{ ext{hot}}\) is the energy absorbed from the hot reservoir.
• \(Q_{ ext{cold}}\) is the energy expelled to the cold reservoir.
• \(W\) is the work done by the engine.

The first law ensures that any calculation involving work or energy transfer in thermodynamic systems is grounded in the conservation principle.

Work Done

In the context of heat engines, calculating the work done is essential. Work represents the output of the heat engine, illustrating the energy converted from thermal to mechanical form. In our example, to find the work done by the engine, we subtract the heat expelled to the cold reservoir from the heat absorbed from the hot reservoir.Let's look at the formula:\[ W = Q_{ ext{hot}} - Q_{ ext{cold}} \]By substituting the given values:

• \(Q_{ ext{hot}} = 1220\, \text{J} \)
• \(Q_{ ext{cold}} = 680\, \text{J} \)

Thus, we find:\[ W = 1220\, \text{J} - 680\, \text{J} = 540\, \text{J} \]So, the work done by the engine is 540 Joules. This outcome demonstrates how effectively the engine converts absorbed heat into useful work, a critical aspect for assessing its performance.

Efficiency Calculation

The efficiency of a heat engine is a measure of how well it converts heat from the hot reservoir into work. A higher efficiency indicates a more effective engine. Calculating efficiency involves comparing the work output to the heat input, providing insight into energy utilization.The formula for efficiency \(e\) is:\[ e = \frac{W}{Q_{ ext{hot}}} \]Substitute the known values from our problem;

• \(W = 540\, \text{J} \)
• \(Q_{ ext{hot}} = 1220\, \text{J} \)

Hence,\[ e = \frac{540\, \text{J}}{1220\, \text{J}} \approx 0.4426 \]To express this value as a percentage, multiply by 100, resulting in an efficiency of approximately 44.26%.This percentage tells us that about 44.26% of the input heat energy is successfully converted into work, while the rest is lost to the cold reservoir. It's a vital aspect to consider when evaluating energy systems.