We know that, at constant volume, Heat supplicd, ΔQ1=ΔU=CyΔT At constant pressure, heat supplied, ΔQ=C.ΔT=ΔU+ΔW Ratio of heat capacity at constant pressure to constant volume is γ=
Cr
Cy
Now,
ΔQ
ΔU
=
CpΔT
CVΔT
=
Cp
CV
=γ Now,
ΔQ
ΔW
=
ΔQ
ΔQ−ΔU
=
CpΔT
CpΔT−CVΔT
=
Cr
Cp−Cv
=
Cp
Cv(
Cp
Cv
−1)
=
γ
γ−1
For refrigerator, coefficient of performance, β=
Q2
W
=
Heat‌absorbed‌from‌food‌stuff
Electrical‌work‌done
∴ β=
Q2
Q1−Q2
Coefficient of heat pump =
Heat‌released
Electrical‌work
i.e.α=
Q1
W
=
Q4
Q1−Q2
We know that,
Q1
Q2
=
T1
T2
So, we get, β=
Q2
Q1−Q2
=
Q2
Q4
1−
Q2
Q1
=
T2
T1
1−
T2
T1
=
T2
T1−T2
and α=
Q
Q1−Q2
=
Q1
Q2
Q1
Q2
−1
=
T1
T2
T1
T2
−1
=
T1
T1−T2
Hence, we get the best match as given by A→4,B→3,C→2 and D→1.