Problem 18 - University Physics with Modern Physics

Question

A Carnot heat engine uses a hot reservoir consisting of a large amount of boiling water and a cold reservoir consisting of a large tub of ice and water. In 5 minutes of operation, the heat rejected by the engine melts 0.0400 kg of ice. During this time, how much work \(W\) is performed by the engine?

Step-by-Step Solution

Answer

The work performed by the Carnot engine is approximately 4,895 J.

Step 1: Identify the heat absorbed by the cold reservoir

The amount of heat absorbed by the cold reservoir is related to the melting of ice. The heat required to melt ice is given by the formula: \( Q_c = mL_f \), where \( m = 0.0400 \, \text{kg} \) is the mass of the ice and \( L_f = 334,000 \, \text{J/kg} \) is the latent heat of fusion for ice.

Step 2: Calculate the heat absorbed by the cold reservoir

Using the values identified, the heat absorbed by the cold reservoir (\( Q_c \)) is: \[ Q_c = 0.0400 \, \text{kg} \times 334,000 \, \text{J/kg} = 13,360 \, \text{J} \]

Step 3: Understand Carnot engine efficiency

A Carnot engine operates between two temperatures: the boiling water temperature \( T_h \) and the ice-water mixture temperature \( T_c \). The efficiency (\( \eta \)) of a Carnot engine is given by \( \eta = 1 - \frac{T_c}{T_h} \).

Step 4: Use efficiency to find work done

The efficiency \( \eta \) is also the ratio of work done \( W \) to heat absorbed by the hot reservoir \( Q_h \), so \( \eta = \frac{W}{Q_h} \). We don't directly have \( Q_h \), but know \( Q_h = Q_c + W \). Rearranging gives \( W = Q_h - Q_c \).

Step 5: Estimate the hot and cold reservoir temperatures

For water boiling, \( T_h = 373 \, \text{K} \) and for the ice-water mixture, \( T_c = 273 \, \text{K} \). Substituting these into the efficiency formula: \( \eta = 1 - \frac{273}{373} = 0.268 \).

Step 6: Calculate work done by the engine

Substituting \( Q_c \) and \( \eta \) in the work formula: \[ \eta = \frac{W}{Q_h} = \frac{W}{W + Q_c} \] Solving \( 0.268 = \frac{W}{W + 13,360} \), obtain \( W = 4,895.45 \, \text{J} \).

Key Concepts

Understanding Heat Transfer in Carnot Engines

In a Carnot heat engine, heat transfer occurs between a hot reservoir and a cold reservoir. The engine draws heat from the hot reservoir and transfers some of it to perform work, while the remaining heat is released to the cold reservoir. This process is critical as it involves energy conversion and balance.
Heat transfer is quantified using the formula:

• The hot reservoir supplies heat energy referred to as \( Q_h \).
• The cold reservoir absorbs the heat energy \( Q_c \), calculated as shown in the problem solution using the formula \( Q_c = mL_f \), where \( m \) is the mass of the substance (ice in this case), and \( L_f \) is the latent heat of fusion.

These components are essential in calculating the work done by the engine. If you understand how energy flows in these transformations, you'll be better equipped to comprehend many other thermodynamic processes.

The Role of Latent Heat of Fusion

The latent heat of fusion is the energy required to change a substance from a solid to a liquid phase without changing its temperature. In this exercise, latent heat is important because it illustrates why the ice melts even when remaining at 0°C.
When the Carnot engine expels heat to the ice-water mixture, the energy absorbed by the ice causes it to melt without increasing in temperature. This is calculated by:

• Using the latent heat of fusion formula \( Q_c = mL_f \).
• For ice, \( L_f \) is approximately \( 334,000 \) J/kg, which means a significant amount of energy is absorbed just to alter the physical state, not to change temperature.

This concept explains how substantial energy amounts can be transferred without temperature rise. It’s a key aspect in various applications of physics and engineering ranging from refrigeration to climate sciences.

Carnot Efficiency Explained

Carnot efficiency is a measure of the ideal efficiency of a heat engine operating between two temperature limits. It represents the maximum ratio of work output to heat input for engines working between two reservoirs. The formula to determine this is:\[ \eta = 1 - \frac{T_c}{T_h} \]Where:

• \( T_h \) is the absolute temperature of the hot reservoir.
• \( T_c \) is the absolute temperature of the cold reservoir.

In our problem, substituting in the boiling water and ice-water temperatures gives an efficiency of 0.268. This means only 26.8% of the heat from the hot reservoir is converted into work, highlighting a key feature of thermodynamic systems: not all energy input can be turned into useful work. Understanding this gives a glimpse into the limitations and design considerations of real-world engines and how engineers strive to maximise efficiency within these constraints.

Approaching Thermodynamics Problem Solving

Solving thermodynamics problems, such as calculating work in a Carnot heat engine, requires a clear understanding of the underlying principles. Following a systematic approach ensures accuracy and comprehension.Firstly, identify known variables and equations related to the problem. Here, understanding the relation between heat absorbed, temperatures, and work is crucial. Then, use this information to methodically apply formulas:

• Calculate heat absorbed by the cold reservoir using specific formulas like \( Q_c = mL_f \).
• Find the efficiency using the temperature-dependent efficiency formula \( \eta = 1 - \frac{T_c}{T_h} \).
• Relate these to find work through approximation and solving equations, such as \( W = Q_h - Q_c \).

By taking a structured approach, each component of the problem becomes more manageable, helping you understand each part's role in the complete system, which is essential for advanced thermodynamics studies.