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.
Key equation
W=Q1-Q2
Key equation
η=WQ1
Key equation
η=Q1-Q2Q1=1-Q2Q1
Temperature Form
For a reversible Carnot cycle, Q₂Q₁ = T₂T₁. Both temperatures must be absolute temperatures.
Key equation
ηC=1-T2T1
Key equation
ηC(\%)=(1-T2T1)×100
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.
1
Working
T₁ = 400 + 273 = 673 K
2
Step 2
T₂ = 30 + 273 = 303 K
3
Step 3
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.
1
Working
ηC = 1 − 300600 = 0.50
2
Step 2
W = ηCQ₁
3
Step 3
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.
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Exam preparation
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.
Common mistakes
Convert Celsius to kelvin.
T₁ must be greater than T₂.
Carnot efficiency is a theoretical maximum.
Do not invert T₂T₁.
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.
Carnot Efficiency and Carnot Theorem
Sticky Notes · Quick Revision
1.
Key Point 1
Carnot efficiency is η = WQ₁.
2.
Key Point 2
Since W = Q₁ − Q₂, η = 1 − Q₂Q₁.
3.
Key Point 3
For a reversible Carnot engine, Q₂Q₁ = T₂T₁.
4.
Key Point 4
Therefore ηC = 1 − T₂T₁, with temperatures in kelvin.
