Problem 17 - 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: Understanding Carnot Efficiency Formula

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 temperature of the cold reservoir (sink), and \( T_h \) is the temperature of the hot reservoir (source). The temperatures should be in Kelvin.

Step 2: Calculate Source Temperature for Initial Efficiency

The initial efficiency is given as 50%, or \( \eta = 0.50 \). The temperature of the sink (\( T_c \)) is 500 K. Plug these values into the efficiency equation: \[0.50 = 1 - \frac{500}{T_h}\]Solving for \( T_h \), we get \( T_h = 1000 \text{ K} \).

Step 3: Set Up New Efficiency Equation

With the efficiency raised to 60%, the new efficiency \( \eta = 0.60 \). The temperature of the source remains the same, \( T_h = 1000 \text{ K} \). Substitute these into the efficiency equation:\[0.60 = 1 - \frac{T_c}{1000}\]

Step 4: Solve for New Temperature of Sink

Solve for the new temperature of the sink (\( T_c \)):\[\frac{T_c}{1000} = 0.40\]Multiplying both sides by 1000 gives:\[T_c = 400 \text{ K}\]

Step 5: Final Step: Compare Solution with Options

The calculated value for the temperature of the sink necessary to achieve 60% efficiency is 400 K. Check with the provided options: (a) 100 K, (b) 600 K, (c) 400 K, (d) 500 K. The correct option is (c) 400 K.

Key Concepts

Carnot Efficiency

Carnot efficiency is a measure of the ideal efficiency of a heat engine, operating between two temperature reservoirs. In a Carnot cycle, the efficiency formula is given by the equation: \[ \eta = 1 - \frac{T_c}{T_h} \] where:

• \( \eta \) stands for efficiency, expressed as a fraction or percentage.
• \( T_c \) is the temperature of the colder reservoir, or the sink, in Kelvin.
• \( T_h \) is the temperature of the hotter reservoir, or the source, also in Kelvin.

Carnot efficiency represents the maximum possible efficiency a heat engine can achieve between two given temperatures. No real engine can surpass this efficiency.
Understanding the limits set by Carnot efficiency helps in designing more efficient engines and evaluating how close real-world engines get to this ideal.

Thermodynamics

Thermodynamics is the branch of physics that deals with heat, work, and energy. It explains how energy transforms from one form to another and obeys fundamental laws.
In the context of a Carnot cycle, several thermodynamic principles come into play. The key principles include:

• The First Law of Thermodynamics (Conservation of Energy): It states that energy cannot be created or destroyed, only transferred or converted. In a heat engine, heat energy is converted into work.
• The Second Law of Thermodynamics: It states that entropy, or disorder, tends to increase in the universe. In practical terms, it means no engine can be 100% efficient as some energy is always lost to the surroundings as waste heat.

Understanding thermodynamics allows us to harness energy more effectively, and it provides the foundational background needed to study systems like heat engines and their limitations.

Heat Engine Principles

Heat engines operate based on fundamental principles of converting heat energy into mechanical work. A heat engine typically involves cycling between two thermal reservoirs at different temperatures.
The primary principles include:

• Absorbing heat energy from a high-temperature source.
• Converting part of this absorbed heat into work output.
• Rejecting the remaining heat to a low-temperature sink.

A Carnot engine is a theoretical model of a heat engine that operates in an idealized reversible cycle, called the Carnot cycle. It consists of two isothermal processes, where the engine exchanges heat at constant temperature, and two adiabatic processes, where the engine undergoes work without any heat exchange.
The Carnot cycle provides a benchmark to measure real engine efficiencies against, reminding us of where improvements can be made and the unattainable nature of perfect efficiency.