SME
Thermal & Statistical Physics · Problem type
The Otto cycle
Use this for “A gasoline engine has a compression ratio of 9.5 and burns fuel to add 2400 J of heat each cycle. With γ = 1.4 for air, find its ideal efficiency, the work per cycle and the heat thrown out.”
Diagram
QL = QH − W = 1,000 − 565
QL = 435 J
Efficiency = W/QH = 565/1,000 = 56.5%
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.
e
=
1
−
r
1
−
=
%
QH
Heat in
example · J
W
Work out
calculated · J
QL
Heat out
calculated · J
Formulas
e
=
100
×
(1
−
r
1 −
γ
)
56.47
=
100
×
(1
−
8
1 − 1.4
)
W
=
e
100
×
QH
564.7
=
56.47
100
×
1,000
QH
=
W
+
QL
1,000 = 564.7 + 435.3
Assumptions
•
The ideal (air-standard) Otto cycle: two slow adiabats and two constant-volume heatings and coolings of an ideal gas.
•
The compression ratio r = Vₘₐₓ ÷ Vₘᵢₙ; γ is about 1.4 for air.
•
The first law over one cycle: ΔU = 0, so the heat in QH = W + QL.
Step-by-step
WE KNOW
r = 8, γ = 1.4, QH = 1,000 J
FIND
efficiency (e), work out (W), heat out (QL)
Step 1 · Find efficiency (e)
Use e = 100 × (1 − r¹⁻ᵞ)
Along
each
adiabat
T
V
γ
− 1
stays
the
same,
so
every
temperature
drops
by
r
1 −
γ
on
the
way
out;
that
share
of
the
heat
is
thrown
out.
e
=
100
×
(1
−
8
1 − 1.4
)
e
=
100
×
(1
−
8
−0.4
)
e
=
100
×
(1
−
0.4353)
e
=
100
×
0.5647
e
=
56.47%
Step 2 · Find work out (W)
Use W = e ÷ 100 × QH
The efficiency is the share of the heat in that comes out as work.
W
=
56.47
100
×
1,000
W
=
0.5647
×
1,000
W
=
564.7
J
Step 3 · Find heat out (QL)
Use QH = W + QL
The gas ends each cycle as it began (ΔU = 0), so the heat not turned into work leaves at the exhaust.
QL
=
QH
−
W
QL = 1,000 − 564.7
QL = 435.3 J
Check
Put the numbers back in. Both sides should match.
56.47
=
100
×
(1
−
8
1 − 1.4
)
✓
564.7
=
56.47
100
×
1,000
✓
1,000 = 564.7 + 435.3 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
Laws of thermodynamics
Main lesson
Adiabatic expansion
Use this for “2.00 mol of a diatomic ideal gas at 400 K expands slowly with no heat in or out, from 4 L to 12 L. Find its final temperature and the work it does.”
Heat capacities of an ideal gas
Use this for “3.00 mol of helium warms by 40 K. How much heat does it take at constant volume, how much at constant pressure, and how much work does it do at constant pressure?”