f a reversed Carnot cycle operates between the temperature limits of 27°C and –3°C, then the ratio of the COP of a refrigerator to that of a heat pump (COP of refrigerator/ COP of heat pump) based on the cycle is __________ (round off to 2 decimal places).

The correct answer is 0.9 (or 0.90 rounded to two decimal places).

Step-by-Step Explanation:

1. Convert Temperatures to Kelvin:
In thermodynamic cycle analysis, temperature values must always be converted from Celsius to Kelvin (absolute temperature scale). We add 273.15 (typically approximated as 273 for simplicity) to the Celsius values:
• Lower temperature limit (refrigerator evaporator / cold reservoir), $T_L$:
$T_L=-3°\text{C}=273-3=270\text{K}$
• Higher temperature limit (condenser / hot reservoir), $T_H$:
$T_H=27°\text{C}=273+27=300\text{K}$

2. Calculate the Coefficient of Performance (COP) of the Refrigerator:
The COP of a reversed Carnot cycle operating as a refrigerator represents the ratio of the heat removed from the cold reservoir to the net work input. In terms of absolute temperatures, it is expressed as:
${\left(\text{COP}\right)}_R=\frac{T_L}{T_H-T_L}$
Substituting the Kelvin values:
${\left(\text{COP}\right)}_R=\frac{270}{300-270}=\frac{270}{30}=9$

3. Calculate the Coefficient of Performance (COP) of the Heat Pump:
The COP of a reversed Carnot cycle operating as a heat pump represents the ratio of the heat delivered to the hot space to the net work input. In terms of absolute temperatures, it is expressed as:
${\left(\text{COP}\right)}_{\text{HP}}=\frac{T_H}{T_H-T_L}$
Substituting the Kelvin values:
${\left(\text{COP}\right)}_{\text{HP}}=\frac{300}{300-270}=\frac{300}{30}=10$

4. Determine the Ratio of the COPs:
Now, we find the ratio of the COP of the refrigerator to that of the heat pump:
$\text{Ratio}=\frac{{\left(\text{COP}\right)}_R}{{\left(\text{COP}\right)}_{\text{HP}}}=\frac{9}{10}=0.9$

Note: Alternatively, using the standard thermodynamic identity linking the two cycles operating between the same temperature limits:
${\left(\text{COP}\right)}_{\text{HP}}={\left(\text{COP}\right)}_R+1$
With ${\left(\text{COP}\right)}_R=9$, we confirm that ${\left(\text{COP}\right)}_{\text{HP}}=9+1=10$, giving the ratio:
$\frac{9}{10}=0.90$