Topic 8.8.18 min read
Carnot Efficiency and Carnot Theorem
Carnot efficiency is the maximum possible engine efficiency between two reservoirs and depends only on their absolute temperatures, not on the working substance.
Choose Your Level
Beginner Level
An ideal engine becomes more efficient when its hot reservoir is hotter or its cold reservoir is colder. It cannot reach 100% efficiency under real conditions.
Recommended for Class 11
Carnot Efficiency and Carnot Theorem
Overview
Carnot efficiency is the maximum possible engine efficiency between two reservoirs and depends only on their absolute temperatures, not on the working substance.
Learning Outcomes
- Carnot efficiency is η = WQ₁.
- Since W = Q₁ − Q₂, η = 1 − Q₂Q₁.
- For a reversible Carnot engine, Q₂Q₁ = T₂T₁.
- Therefore ηC = 1 − T₂T₁, with temperatures in kelvin.
- Efficiency depends only on reservoir temperatures.
- No engine between the same temperatures can exceed Carnot efficiency.
Intermediate Level
Efficiency Derivation
Efficiency is useful work output divided by heat input. For one Carnot cycle, W = Q₁ − Q₂.
Temperature Form
For a reversible Carnot cycle, Q₂Q₁ = T₂T₁. Both temperatures must be absolute temperatures.
What Controls Efficiency?
| Change | Effect | Reason |
|---|---|---|
| increase T₁ | ηC increases | T₂T₁ falls |
| decrease T₂ | ηC increases | less heat is rejected |
| T₁ = T₂ | ηC = 0 | no temperature difference |
| T₂ → 0 K | ηC approaches 100% | unattainable limit |
Carnot Theorem
All reversible engines between the same reservoirs have the same efficiency. Real engines suffer friction, finite-temperature heat transfer and other losses, so their efficiencies are lower.
Textbook Example 8.4
Find the efficiency of an engine operating between 400 °C and 30 °C.
T₁ = 400 + 273 = 673 K
T₂ = 30 + 273 = 303 K
ηC = (1 − 303673) × 100%
Final answer: ηC ≈ 55%
Maximum Work
A Carnot engine works between 600 K and 300 K and absorbs 2.0 kJ per cycle. Find efficiency and work.
ηC = 1 − 300600 = 0.50
W = ηCQ₁
W = (0.50)(2.0 kJ)
Final answer: ηC = 50% and W = 1.0 kJ
Common Mistakes
- Convert Celsius to kelvin.
- T₁ must be greater than T₂.
- Carnot efficiency is a theoretical maximum.
- Do not invert T₂T₁.
Short-Question Preparation
Q. Derive Carnot efficiency.
A. η = WQ₁ = 1 − Q₂Q₁ = 1 − T₂T₁.
Q. What determines Carnot efficiency?
A. Only the absolute temperatures of the two reservoirs.
Q. State Carnot’s theorem.
A. No engine between the same temperatures can be more efficient than a Carnot engine.
Q. Why is 100% efficiency impossible?
A. It would require T₂ = 0 K or an infinite T₁, neither attainable.
Sticky Notes
- Carnot efficiency is η = WQ₁.
- Since W = Q₁ − Q₂, η = 1 − Q₂Q₁.
- For a reversible Carnot engine, Q₂Q₁ = T₂T₁.
- Therefore ηC = 1 − T₂T₁, with temperatures in kelvin.
- Efficiency depends only on reservoir temperatures.
- No engine between the same temperatures can exceed Carnot efficiency.
60-Second Revision
ηC = WQ₁ = 1 − Q₂Q₁ = 1 − T₂T₁. Use kelvin. Raising T₁ or lowering T₂ increases the limit. No engine can exceed it.
Summary
Carnot efficiency is the maximum possible engine efficiency between two reservoirs and depends only on their absolute temperatures, not on the working substance.
Ask-the-Tutor Modes
- Explain it simply.
- Walk through the cycle.
- Derive the efficiency.
- Solve a numerical.
- Test me with an MCQ.
Exam Focus
Exam Focus
Important points
- Carnot efficiency is η = WQ₁.
- Since W = Q₁ − Q₂, η = 1 − Q₂Q₁.
- For a reversible Carnot engine, Q₂Q₁ = T₂T₁.
- Therefore ηC = 1 − T₂T₁, with temperatures in kelvin.
- Efficiency depends only on reservoir temperatures.
- No engine between the same temperatures can exceed Carnot efficiency.
Important formulas
- W = Q₁ − Q₂
- η = WQ₁
- ηC = 1 − Q₂Q₁
- ηC = 1 − T₂T₁
Common exam traps
- Convert Celsius to kelvin.
- T₁ must be greater than T₂.
- Carnot efficiency is a theoretical maximum.
- Do not invert T₂T₁.
Long question structure
Definition → assumptions → stages or derivation → graph → numerical.
Common Mistakes
Common Mistakes
- Mistake
Convert Celsius to kelvin.
- Mistake
T₁ must be greater than T₂.
- Mistake
Carnot efficiency is a theoretical maximum.
- Mistake
Do not invert T₂T₁.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
Carnot efficiency is η = WQ₁.
Since W = Q₁ − Q₂, η = 1 − Q₂Q₁.
For a reversible Carnot engine, Q₂Q₁ = T₂T₁.
Therefore ηC = 1 − T₂T₁, with temperatures in kelvin.
Carnot efficiency is the maximum possible engine efficiency between two reservoirs and depends only on their absolute temperatures, not on the working substance.
- Definition
- Carnot efficiency is the maximum possible engine efficiency between two reservoirs and depends only on their absolute temperatures, not on the working substance.
Did you understand this topic?
Your answer is saved with your learning record. It does not mark the topic as mastered.
Ready to test yourself?
You've completed the learning section.
Exam Practice
Important Question
Q1. Derive Carnot efficiency.
Important Question
Q2. What determines Carnot efficiency?
Important Question
Q3. State Carnot’s theorem.
Important Question
Q4. Why is 100% efficiency impossible?