SME
Grade 11 Science · Problem type
A heat engine and the Carnot limit
Use this for “An engine takes in 2000 J from a 600 K source and does 500 J of work. How much heat goes to the 300 K sink, and how efficient is it compared with the most it could be?”
Diagram
Qc = Qₕ − W = 2,000 − 500
Qc = 1,500 J
Efficiency = W/Qₕ = 500/2,000 = 25%
Carnot limit = 1 − Tc/Tₕ = 1 − 300/600 = 50%
Some heat always goes to the cold reservoir: no engine turns all its heat into work.
This is an example. Type your own values to start a new problem.
Qₕ
Heat in from the hot reservoir
example · J
W
Work
example · J
Qc
Heat out to the cold reservoir
calculated · J
e
Efficiency
calculated · %
Tₕ
Hot reservoir temperature
example · K
Tc
Cold reservoir temperature
example · K
eC
Carnot limit
calculated · %
r
Share of the Carnot limit
calculated · %
Formulas
Qc
=
Qₕ
−
W
1,500 = 2,000 − 500
e
=
100
×
W
/Qₕ
25
=
100
×
500
2,000
eC = 100 × (1 − Tc/Tₕ)
50
=
100
×
(1
−
300
600
)
r
=
100
×
e
/eC
50
=
100
×
25
50
Assumptions
•
First law: the heat taken in becomes work plus the heat given out, Qₕ = W + Qc.
•
Second law: some heat always goes to the cold reservoir.
•
The best possible efficiency is the Carnot limit, 1 − Tc/Tₕ, with temperatures in kelvins.
Step-by-step
WE KNOW
Qₕ = 2,000 J, W = 500 J, Tₕ = 600 K, Tc = 300 K
FIND
heat out to the cold reservoir (Qc), efficiency (e), Carnot limit (eC), share of the Carnot limit (r)
Step 1 · Find heat out to the cold reservoir (Qc)
Use Qc = Qₕ − W
The heat not turned into work goes to the cold reservoir.
Qc = 2,000 − 500
Qc = 1,500 J
Step 2 · Find efficiency (e)
Use e = 100 × W/Qₕ
The share of the heat that became work, as a percent.
e
=
100
×
500
2,000
e
=
25%
Step 3 · Find Carnot limit (eC)
Use eC = 100 × (1 − Tc/Tₕ)
No engine between these temperatures can beat 1 − Tc/Tₕ (in kelvins).
eC
=
100
×
(1
−
300
600
)
eC = 100 × 0.5
eC = 50%
Step 4 · Find share of the Carnot limit (r)
Use r = 100 × e/eC
How
close
the
engine
comes
to
the
best
possible:
e
over
the
Carnot
limit.
r
=
100
×
25
50
r
=
50%
Check
Put the numbers back in. Both sides should match.
1,500 = 2,000 − 500 ✓
25
=
100
×
500
2,000
✓
50
=
100
×
(1
−
300
600
)
✓
50
=
100
×
25
50
✓
Topic check
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