Problem 18 - Master Resource Book in JEE Main Physics

Question

The efficiency of a Carnot engine is \(50 \%\) and temperature of sink is \(500 \mathrm{~K}\). If temperature of source is kept constant and its efficiency raised to \(60 \%\), then the required temperature of sink will be (a) \(100 \mathrm{~K}\) (b) \(600 \mathrm{~K}\) (c) \(400 \mathrm{~K}\) (d) \(500 \mathrm{~K}\)

Step-by-Step Solution

Answer

The required temperature of the sink is 400 K.

Step 1: Understand Carnot Engine Efficiency

The efficiency of a Carnot engine is given by the formula \( \eta = 1 - \frac{T_C}{T_H} \), where \( \eta \) is the efficiency, \( T_C \) is the absolute temperature of the sink, and \( T_H \) is the absolute temperature of the source.

Step 2: Calculate Temperature of Source (Initial Scenario)

Given the initial efficiency of the Carnot engine is \( 50\% \), which is \( 0.5 \) in decimal form. The temperature of the sink \( T_C = 500 \) K. We substitute these into the efficiency formula: \( 0.5 = 1 - \frac{500}{T_H} \).\Solving for \( T_H \), we get \( \frac{500}{T_H} = 0.5 \Rightarrow T_H = \frac{500}{0.5} = 1000 \) K.

Step 3: Set Up New Efficiency Calculation

Now, the efficiency is raised to \( 60\% \), which is \( 0.6 \) in decimal form. We keep \( T_H = 1000 \) K constant and solve the new efficiency equation: \( 0.6 = 1 - \frac{T_C'}{1000} \), where \( T_C' \) is the new temperature of the sink.

Step 4: Solve for New Sink Temperature

We rearrange the equation \( 0.6 = 1 - \frac{T_C'}{1000} \) to \( \frac{T_C'}{1000} = 1 - 0.6 = 0.4 \). Then, solving for \( T_C' \), we find \( T_C' = 0.4 \times 1000 = 400 \) K.

Key Concepts

Introduction to Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat, work, temperature, and energy. This field provides fundamental insights into how energy is transferred and transformed.
It is crucial for understanding processes like refrigeration, engine operation, and even the human body. At its core, thermodynamics revolves around four laws that describe these concepts, with the first and second laws being especially pertinent for understanding Carnot engines.

• The **First Law of Thermodynamics**, also known as the conservation of energy principle, states that energy cannot be created or destroyed, only transferred or converted from one form to another.
• The **Second Law of Thermodynamics** introduces the concept of entropy, stating that energy transformations are not 100% efficient due to the inevitable increase in disorder or entropy, particularly in closed systems.

These laws lay the foundation for analyzing the efficiency of engines, including the Carnot engine.

Understanding Temperature of Sink

In the context of thermodynamics and Carnot engines, the **temperature of the sink** refers to the temperature of the cold reservoir that absorbs heat expelled by the engine. This is crucial because it dictates the efficiency of a thermodynamic cycle.
In a Carnot engine, the efficiency is calculated using the formula \[\eta = 1 - \frac{T_C}{T_H}\]where \(T_C\) is the temperature of the sink and \(T_H\) is the temperature of the source. The lower the temperature of the sink relative to the source, the higher the potential efficiency of the engine.

• A larger difference between \(T_H\) and \(T_C\) indicates a more efficient energy conversion process, as less energy is wasted in the form of heat.
• Adjusting \(T_C\) helps manage efficiency outcomes, such as shifting \(T_C\) from 500 K to 400 K to improve the Carnot engine's efficiency from 50% to 60%.

Understanding how to manipulate the temperature of the sink allows for optimizing engine performance and efficiency.

Exploring the Carnot Cycle

The Carnot cycle is a theoretical model that describes the most efficient series of processes an engine can undergo. Developed by Sadi Carnot, this cycle provides the foundation for understanding maximum efficiency of heat engines in thermodynamics.
It consists of four reversible processes:

• **Isothermal Expansion** - The gas expands while absorbing heat from the hot reservoir at a constant temperature.
• **Adiabatic Expansion** - The gas continues to expand without heat exchange, causing its temperature to drop.
• **Isothermal Compression** - Heat is expelled to the cold reservoir while the gas is compressed at a constant temperature.
• **Adiabatic Compression** - The gas is compressed further, rising back to its initial temperature without heat exchange.

The Carnot cycle serves as an ideal benchmark for the efficiency of real-world engines, showing how perfectly coordinated energy transformations can achieve maximum performance. Actual devices must deal with irreversibilities and non-idealities, making Carnot's assumptions a useful goal rather than an absolute reality.