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Work and Heat, Second Law of Thermodynamics, Carnot Engine
Typology: Lecture notes
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Q
Hot Body Body Body Body
Cold (^) Hot Cold
R Q 2
Q 2 + Q 1 Q 2 ≡ Q 2
Q 1 E
E+R W =
Q
1
Hot Reservoir
Cold QH QL Reservoir Cyclic Engine ∆U = 0
W^ =^ Q
H^ −
QL
η = work output heat input
⇒ η = 1 −
Work
1
2
3
4
Adiabatic
Adiabatic
P 05
QL
QH
V 2 V 1
T 2
CV dT = CV (T 3 − T 2 ) and QL = −
T 4
CV dT = CV (T 4 − T 1 )
T 1 V 1 γ −^1 = T 2 V 2 γ−^1 T 4 V 1 γ −^1 = T 3 V 2 γ−^1
η = 1 −
1
2 3
4 P 0 5 V 2 V 3 V 1
QH
QL
Adiabatic Adiabatic
Work
T 2
CP dT = CP (T 3 − T 2 ) and QL = −
T 4
CV dT = CV (T 4 − T 1 )
T 1 V 1 γ −^1 = T 2 V 2 γ−^1 T 4 V 1 γ −^1 = T 3 V 3 γ−^1
)γ− 1
η = 1 −
γ
r (^) Cγ − 1 rC − 1
, where rC =
Hot reservoir
Cold reservoir
E′^ C
TH
TL
Q′ H
Q′ L
QH
QL
W ′ W
Hot reservoir
Cold reservoir
E′^ C
TH
TL
Q′ H
Q′ L
QH
QL
W ′ W
W ′^ = Q H′ − Q L′ and W = QH − QL
η′^ > ηC →
implying
η′^ ≤ ηC
Qtriple point
Carnot
P
V
1
3
4
2
θH
θL
pV (^) = (^) nRθ H pV = nRθ L
pV γ = k
pV (^) γ = k
2 → 3 : pV = nRθH , 4 → 1 : pV = nRθL
∫ (^) b
Q 23 = nRθH ln V V^32 Q 41 = nRθL ln V V^41
= θH^ ln(V^3 /V^2 ) θL ln(V 4 /V 1 )
−CV dθ = p dV = nRθ dV V
θ
dθ θ = nR ln Vf Vi
Cold reservoir T 0
Hot reservoir Hot reservoir
RA RB
E
Q 1 Q 2
Q 0 A Q 0 Q 0 B
Q 1 A Q 2 B
W WA WB
T 1 T 2
W ≤ WA + WB and Q 0 A + Q 0 B − Q 0 ≤ 0
and similarly Q 0 B = Q 2 T^0 T 2
p
V
A B
C
2
1
1
δQ T
A
2
δQ T
B
= 0 and
1
δQ T
A
2
δQ T
C
1
δQ T
B
2
δQ T
C
1
δQ T
B
2
δQ T
C
1
δQ T
δQ = T dS
1
δQir T
2
δQr T
1
δQir T
1
δQr T