Question

A Carnot refrigerator $A$ works between $-{10}^{\circ }\text{C}$ and ${27}^{\circ }\text{C}$, while Carnot refrigerator $B$ works between $-{27}^{\circ }\text{C}$ and ${17}^{\circ }\text{C}$, both removing heat equal to $200\text{J}$ from the freezer. Then which of the following is the correct option regarding the coefficient of performance $\left(\beta \right)$ of both the refrigerators?

• A. ${\beta }_A<{\beta }_B$
• B. ${\beta }_A>{\beta }_B$
• C. ${\beta }_A={\beta }_B=0$
• D. ${\beta }_A={\beta }_B\ne 0$

Answer 1

The correct option is B ${\beta }_A>{\beta }_B$

For refrigerator $A$,
Temperature of sink, $T_2=-10+273=263\text{K}$
Temperature of source, $T_1=27+273=300\text{K}$
Coefficient of performance for a Carnot refrigerator is,
${\beta }_A=\frac{T_2}{T_1-T_2}=\frac{263}{300-263}=7.1$
Simlarly for refrigerator $B$,
Temperature of Sink, $T_2=-27+273=246\text{K}$
Temperature of source, $T_1=17+273=290\text{K}$
Coefficient of performance,
${\beta }_B=\frac{T_2}{T_1-T_2}=\frac{246}{290-246}=5.6$
$\therefore {\beta }_A>{\beta }_B$