Anomous
Last Activity: 3 Years ago
The efficiency of the engine is defined by ε=W/Q
1
and is shown in the text to be
ε=
T
1
T
1
−T
2
⇒
Q
1
W
=
T
1
T
1
−T
2
The coefficient of performance of the refrigerator is defined by K=Q
4
/W and is shown in the text to be
K=
T
3
−T
4
T
4
⇒
W
Q
4
=
T
3
−T
4
T
4
Now Q
4
=Q
3
−W, so
(Q
3
−W)/W=T
4
/(T
3
−T
4
)
The work done by the engine is used to drive the refrigerator, so W is the same for the two. Solve the engine equation for W and substitute the resulting expression into the refrigerator equation.
The engine equation yields W=(T
1
−T
2
)Q
1
/T
1
and the substitution yields
T
3
−T
4
T
4
=
W
Q
3
−1=
Q
1
(T
1
−T
2
)
Q
3
T
1
−1
Solving for Q
3
/Q
1
,
we obtain
Q
1
Q
3
=(
T
3
−T
4
T
4
+1)(
T
1
T
1
−T
2
)=(
T
3
−T
4
T
3
)(
T
1
T
1
−T
2
)=
1−(T
4
/T
3
)
1−(T
2
/T
1
)
With T
1
=400K,T
2
=150K,T
3
=325K, and T
4
=225K, the ratio becomes
Q
3
/Q
1
=2.03The efficiency of the engine is defined by ε=W/Q
1
and is shown in the text to be
ε=
T
1
T
1
−T
2
⇒
Q
1
W
=
T
1
T
1
−T
2
The coefficient of performance of the refrigerator is defined by K=Q
4
/W and is shown in the text to be
K=
T
3
−T
4
T
4
⇒
W
Q
4
=
T
3
−T
4
T
4
Now Q
4
=Q
3
−W, so
(Q
3
−W)/W=T
4
/(T
3
−T
4
)
The work done by the engine is used to drive the refrigerator, so W is the same for the two. Solve the engine equation for W and substitute the resulting expression into the refrigerator equation.
The engine equation yields W=(T
1
−T
2
)Q
1
/T
1
and the substitution yields
T
3
−T
4
T
4
=
W
Q
3
−1=
Q
1
(T
1
−T
2
)
Q
3
T
1
−1
Solving for Q
3
/Q
1
,
we obtain
Q
1
Q
3
=(
T
3
−T
4
T
4
+1)(
T
1
T
1
−T
2
)=(
T
3
−T
4
T
3
)(
T
1
T
1
−T
2
)=
1−(T
4
/T
3
)
1−(T
2
/T
1
)
With T
1
=400K,T
2
=150K,T
3
=325K, and T
4
=225K, the ratio becomes
Q
3
/Q
1
=2.03The efficiency of the engine is defined by ε=W/Q
1
and is shown in the text to be
ε=
T
1
T
1
−T
2
⇒
Q
1
W
=
T
1
T
1
−T
2
The coefficient of performance of the refrigerator is defined by K=Q
4
/W and is shown in the text to be
K=
T
3
−T
4
T
4
⇒
W
Q
4
=
T
3
−T
4
T
4
Now Q
4
=Q
3
−W, so
(Q
3
−W)/W=T
4
/(T
3
−T
4
)
The work done by the engine is used to drive the refrigerator, so W is the same for the two. Solve the engine equation for W and substitute the resulting expression into the refrigerator equation.
The engine equation yields W=(T
1
−T
2
)Q
1
/T
1
and the substitution yields
T
3
−T
4
T
4
=
W
Q
3
−1=
Q
1
(T
1
−T
2
)
Q
3
T
1
−1
Solving for Q
3
/Q
1
,
we obtain
Q
1
Q
3
=(
T
3
−T
4
T
4
+1)(
T
1
T
1
−T
2
)=(
T
3
−T
4
T
3
)(
T
1
T
1
−T
2
)=
1−(T
4
/T
3
)
1−(T
2
/T
1
)
With T
1
=400K,T
2
=150K,T
3
=325K, and T
4
=225K, the ratio becomes
Q
3
/Q
1
=2.03
