41. The efficiency of the engine is defined by ε = W/Q

1

and is shown in the text to be ε = (T

1

− T

2

)/T

1

,

so W/Q

1

= (T

1

− T

2

)/T

1

. The coefficient of performance of the refrigerator is defined by K = Q

4

/W

and is shown in the text to be K = T

4

/(T

3

− T

4

), so Q

4

/W = T

4

/(T

3

− 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

4

T

3

− T

4

=

Q

3

W

− 1 =

Q

3

T

1

Q

1

(T

1

− T

2

)

− 1 .

Solve for Q

3

/Q

1

:

Q

3

Q

1

=

T

4

T

3

− T

4

+ 1



T

1

− T

2

T

1



=

T

3

T

3

− T

4



T

1

− T

2

T

1



=

1

− (T

2

/T

1

)

1

− (T

4

/T

3

)

.