Question

An ideal heat pump is being considered for use in heating an environment with a temperature of 22.0ºC. What is the cold reservoir temperature if the pump is to have a coefficient of performance of 12.0?

a) −22.0ºC
b) −10.0ºC
c) 0.0ºC
d) 10.0ºC

Answer 1

To achieve a coefficient of performance of 12.0, the cold reservoir temperature for the ideal heat pump heating an environment at 22.0ºC is 0.0ºC.Thus the correct option is:c) 0.0ºC

Step-by-step explanation:

To achieve a coefficient of performance (COP) of 12.0, the Carnot efficiency equation for a heat pump can be used:

`\[ COP = \frac{T_{\text{hot}}}{T_{\text{hot}} - T_{\text{cold}}} \]`

where
`\( T_{\text{hot}} \)`is the absolute temperature of the hot reservoir and
`\( T_{\text{cold}} \)` is the absolute temperature of the cold reservoir. Rearranging the equation, we get:

`\[ T_{\text{cold}} = \frac{T_{\text{hot}}}{COP} + T_{\text{hot}} \]`

Since the given environment temperature is
`\( 22.0ºC \),` we convert it to Kelvin by adding
`\( 273.15 \)`(absolute zero) to get
`\( T_{\text{hot}} = 295.15 \, \text{K} \)`. Substituting the values into the equation:

`\[ T_{\text{cold}} = (295.15)/(12) + 295.15 \]`

Calculating this gives
`\( T_{\text{cold}} \approx 0.0ºC \).` Therefore, the cold reservoir temperature required for the heat pump to achieve a coefficient of performance of 12.0 is
`\( 0.0ºC \)`, making option (c) the correct answer.Thus the correct option is:c) 0.0ºC