Problem 22 - University Physics with Modern Physics

Question

20.22. 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 \(\mathrm{kg}\) of ice. During this time, how much work \(W\) is performed by the engine?

Step-by-Step Solution

Answer

Work performed cannot be determined without temperatures of reservoirs.

Step 1: Identify the Given Information

The heat engine operates between a hot reservoir of boiling water and a cold reservoir of ice. The mass of ice melted is given as 0.0400 kg, and the time of operation is 5 minutes.

Step 2: Determine the Latent Heat of Fusion

The latent heat of fusion of ice is a known value, approximately 334,000 J/kg. This is the amount of energy required to melt 1 kg of ice without changing its temperature.

Step 3: Calculate Heat Rejected (Q_c)

The energy required to melt the ice, which is the heat rejected by the engine, can be calculated using the formula \( Q_c = mL \), where \( m = 0.0400 \) kg is the mass of ice, and \( L = 334,000 \) J/kg is the latent heat of fusion. \[ Q_c = 0.0400 \times 334,000 = 13,360 \text{ J} \]

Step 4: Recall Carnot Efficiency and Relationship to Work

Carnot efficiency for a reversible heat engine is given by \( \eta = 1 - \frac{T_c}{T_h} \). The work done by the engine can be calculated as \( W = Q_h - Q_c \), where \( Q_h \) is the heat absorbed from the hot reservoir.

Step 5: Use Efficiency to Find Q_h and W

In an ideal Carnot engine, \( Q_c = Q_h \times \frac{T_c}{T_h} \). Rearrange to find \( Q_h \), and use this to compute work \( W = Q_h - Q_c \). Since only \( Q_c \) is given, more information is needed, such as the temperatures of the reservoirs.

Key Concepts

Latent Heat of Fusion

The latent heat of fusion is a crucial concept when understanding phase changes, specifically from solid to liquid. In simple terms, it denotes the amount of heat needed to convert a solid into a liquid at constant temperature. For ice, the latent heat of fusion is approximately 334,000 J/kg. This value implies that to melt 1 kilogram of ice without raising its temperature, 334,000 joules of energy are required. In our problem, with 0.0400 kg of ice being melted by the engine, we use the heat of fusion to calculate the energy transfer involved.

• For 0.0400 kg, the energy needed can be determined by multiplying the mass with the latent heat of fusion, leading to 13,360 J.
• This energy represents the amount rejected by the Carnot engine.

Latent heat is a key component in thermodynamics, especially in studying how energy moves during phase changes, without altering temperature.

Heat Rejected

In the context of a thermodynamic cycle, heat rejected is the energy discharged or released by an engine to the surrounding environment. For the Carnot heat engine, the ice acts as the cold reservoir where this heat is rejected. By melting the ice, the engine transfers energy, effectively rejecting it. This is quantitatively determined using the latent heat of fusion formula.

• In our example, with a mass of 0.0400 kg of ice, the heat rejected is 13,360 J.

This process highlights the functioning of the heat engine in maintaining its cycle. It captures the unavoidable process of losing some energy to the cold reservoir, pivotal for any real or theoretical engine. Remember, while heat rejected illustrates loss from the engine's perspective, it follows the principle of energy conservation in physical processes.

Carnot Efficiency

Carnot Efficiency is a theoretical metric determining how effectively a heat engine converts absorbed heat into useful work. It's a defining formula given by \( \eta = 1 - \frac{T_c}{T_h} \), where \( T_c \) and \( T_h \) are the absolute temperatures of the cold and hot reservoirs, respectively. This equation reveals that efficiency inversely correlates with the ratio of these temperatures.

• A lower \( T_c \) or a higher \( T_h \) increases efficiency.
• However, reaching 100% efficiency is impossible due to the second law of thermodynamics.

This Carnot cycle serves as the benchmark for real-world engines, which can never exceed this ideal. Understanding efficiency is vital when analyzing how much work an engine can optimally perform given its heat intake and loss with the universal statistics defined by Carnot.

Work Performed

In thermodynamics, the work performed by a heat engine is the net energy output after it has completed a cycle. In a Carnot engine, we can compute the work performed by analyzing the difference between the heat absorbed (\( Q_h \)) and heat rejected (\( Q_c \)). The relationship is defined as \( W = Q_h - Q_c \).

• To solve our problem, we first need the temperatures of both reservoirs to derive \( Q_h \) using the Carnot efficiency.
• Subsequent calculations would then simplify to finding \( W \) after subtracting \( Q_c \) from the obtained \( Q_h \).

The significance of work in this context emphasizes the use of thermodynamic processes to harness usable energy. It's a practical application showing energy transformation, critical in any mechanical system.