Problem 79 - Pearson Physics

Question

Think \& Calculate A heat engine does 2700 J of work with an efficiency of \(0.18\). What is (a) the heat taken in from the hot reservoir and (b) the heat given off to the cold reservoir? (c) If the efficiency of the engine is increased, does your answer to part (a) increase, decrease, or stay the same? Explain.

Step-by-Step Solution

Answer

(a) 15000 J, (b) 12300 J, (c) Decrease.

Step 1: Understand the Efficiency Formula

The efficiency (\( \eta \)) of a heat engine is given by the formula: \( \eta = \frac{W}{Q_\text{in}} \), where \( W \) is the work done by the engine and \( Q_\text{in} \) is the heat taken in from the hot reservoir. Here, we know \( \eta = 0.18 \) and \( W = 2700 \) J.

Step 2: Calculate Heat Taken in from the Hot Reservoir

Rearrange the efficiency formula to solve for \( Q_\text{in} \):\[ Q_\text{in} = \frac{W}{\eta} = \frac{2700}{0.18} = 15000 \text{ J} \].Thus, the heat taken in from the hot reservoir is 15000 J.

Step 3: Understand the Energy Conservation in Heat Engines

In a heat engine, the heat taken in from the hot reservoir is either used to do work or given off to the cold reservoir. This is expressed as \( Q_\text{in} = Q_\text{out} + W \). Here, \( Q_\text{out} \) is the heat given off to the cold reservoir.

Step 4: Calculate Heat Given off to the Cold Reservoir

Rearrange the energy conservation equation to solve for \( Q_\text{out} \):\[ Q_\text{out} = Q_\text{in} - W = 15000 - 2700 = 12300 \text{ J} \].Thus, the heat given off to the cold reservoir is 12300 J.

Step 5: Effect of Increased Efficiency on Heat Taken In

If the efficiency of the engine increases, more of the heat taken in is converted to work, meaning less heat is required to achieve the same work output. So \( Q_\text{in} \) would decrease with increased efficiency.

Key Concepts

Thermodynamics

Thermodynamics is a branch of physics that studies how heat and energy interact with each other. It explores how heat moves in and out of different systems and how it can be transformed into work. Within thermodynamics, we often refer to three main concepts:

• The study of energy transfers and transformations, how energy is conserved, and the efficiency of these processes.
• Understanding different systems: isolated, open, and closed systems, which describe how a system exchanges energy and matter with its surroundings.
• The laws of thermodynamics, which govern how energy behaves. These include the famous First and Second Laws of Thermodynamics.

Knowledge of thermodynamics allows us to analyze systems like heat engines, refrigerators, and even living organisms. In the context of heat engines, we're particularly interested in how efficiently heat can be converted into useful work.

First Law of Thermodynamics

The First Law of Thermodynamics, also known as the Law of Energy Conservation, states that energy cannot be created or destroyed, only transformed from one form to another. This principle can be summarised by the equation: \(\Delta U = Q - W\)where:

• \(\Delta U\) is the change in internal energy of the system,
• \(Q\) is the heat added to the system, and
• \(W\) is the work done by the system.

In a heat engine, the First Law helps us understand how input heat energy \(Q_{\text{in}}\) is partly used to do work \(W\), with the remainder being released as waste heat \(Q_{\text{out}}\) to the cold reservoir. This is reflected in the equation \( Q_{\text{in}} = Q_{\text{out}} + W \), showing that all energy in the system is accounted for. It's an essential concept for predicting how much work a system can perform based on the amount of heat it absorbs.

Energy Conservation

Energy conservation in thermodynamics implies that the total energy of an isolated system remains constant. This principle enables us to predict and calculate energy transformation efficiently.For instance, in the context of a heat engine, energy conservation aligns with the calculated energy flows:

• The input heat \(Q_{\text{in}}\) is distributed between the work output \(W\) and the waste heat \(Q_{\text{out}}\).
• When efficiency increases, a greater portion of the input heat is converted to work, reducing the heat required for the same work output.

Thus, energy conservation ensures that all energy sources and their subsequent outputs can be traced and calculated confidently, as reflected in the laws of thermodynamics. This understanding allows engineers to improve heat engines, making them more efficient by optimizing how they transform and conserve energy.