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Unit 2•Lesson 1 2 min read

Energy Efficiency and Entropy

7/18

Learning Objectives

Define energy efficiency.
Explain how the Second Law of Thermodynamics (entropy) places a fundamental limit on the efficiency of heat engines.
Describe the Carnot cycle and its significance as an ideal thermodynamic cycle.

Efficiency: Getting the Most Out of Energy

Energy efficiency is the goal to reduce the amount of energy required to provide products and services. In a physical context, it is the ratio of useful energy output to total energy input.

Efficiency (η) = (Useful Energy Output) / (Total Energy Input)

Efficiency is usually expressed as a percentage. No process can be 100% efficient due to the Second Law of Thermodynamics.

The Second Law of Thermodynamics and Efficiency

The Second Law of Thermodynamics can be stated in several ways, but a key implication for energy is:

It is impossible to construct a device that operates in a cycle and produces no other effect than the transfer of heat from a cooler body to a hotter body. (Clausius statement)

It is impossible to convert heat completely into work in a thermodynamic cycle. (Kelvin-Planck statement)

This means that any heat engine (a device that converts thermal energy into mechanical work, like a car engine or a power plant turbine) must reject some waste heat into a cold reservoir (the environment). This is a consequence of the fact that all real-world processes increase the total entropy (disorder) of the universe. The waste heat is the 'disorder' produced by the engine.

The Carnot Cycle: The Theoretical Maximum Efficiency

The Carnot cycle is a theoretical, ideal thermodynamic cycle that gives the maximum possible efficiency that a heat engine can have when operating between two temperatures. No real engine can be more efficient than a Carnot engine.

The efficiency of a Carnot engine depends only on the absolute temperatures of the hot reservoir (Tₕ) from which it takes heat, and the cold reservoir (Tₑ) to which it rejects waste heat.

Carnot Efficiency (η_carnot) = 1 - (Tₑ / Tₕ)

Temperatures must be in Kelvin.
This formula shows that to increase efficiency, one should make the hot reservoir as hot as possible and the cold reservoir as cold as possible.
It also shows that 100% efficiency (η=1) is impossible, as it would require the cold reservoir to be at absolute zero (Tₑ = 0 K), which is forbidden by the Third Law of Thermodynamics.

Key Terms

Energy Efficiency
The ratio of useful energy output to total energy input for a device or process.
Second Law of Thermodynamics
The law of physics stating that the total entropy of an isolated system can only increase over time. It implies that heat cannot be completely converted into work in a cycle.
Heat Engine
A device that converts thermal energy into mechanical work.
Carnot Cycle
A theoretical, ideal, reversible thermodynamic cycle that represents the upper limit of efficiency for any heat engine operating between two given temperatures.

Check Your Understanding

1

A heat engine takes in 1000 J of heat from a hot reservoir and performs 400 J of work. What is its efficiency?

2

According to the Second Law of Thermodynamics, why can no heat engine be 100% efficient?

3

A Carnot engine operates between a hot reservoir at 600 K and a cold reservoir at 300 K. What is its maximum theoretical efficiency?

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