Answer:
option c is the correct answer
Explanation:
The efficiency (\(η\)) of a Carnot engine operating between two heat reservoirs at temperatures [tex]\(T_1\)[/tex][tex]T_2\)[/tex]is [tex](\(T_1 > T_2\))[/tex]the Carnot efficiency formula:
[tex]\[ η_{\text{Carnot}} = 1 - \frac{T_2}{T_1} \][/tex]
Now, the coefficient of performance [tex](\(COP\))[/tex]of a heat pump working between the same temperatures is related to the Carnot efficiency. The COP of a heat pump is defined as:
[tex]\[ COP_{\text{heat pump}} = \frac{1}{η_{\text{Carnot}}} \][/tex]
Substitute the expression [tex]\(η_{\text{Carnot}}\)[/tex]into the equation for COP:
[tex]\[ COP_{\text{heat pump}} = \frac{1}{1 - \frac{T_2}{T_1}} \][/tex]
Now, compare this with the provided answer choices:
[tex]\[ \text{A. } 1η \]\[ \text{B. } \frac{(1−η)}{η} \]\[ \text{C. } \frac{1}{η} \]\[ \text{D. } 1 + \frac{1}{η} \][/tex]
[tex]\[ \text{C. } \frac{1}{η} \][/tex]
Therefore, option c is the correct answer
C. 1/η
The coefficient of performance (COP) of a heat pump working between the same temperatures as a Carnot engine with efficiency (η) is given by COPhp = 1/(η).
The coefficient of performance (COP) of a heat pump working between the same temperatures as a Carnot engine with efficiency (η) can be calculated as COPhp = 1/(η). This is an important and interesting fact because the COP of a heat pump is always greater than 1, indicating that it has more heat transfer (Qh) than work put into it.
The efficiency of a perfect, or Carnot, engine is given by η = 1 - (Tc/Th), where Tc is the temperature of the cold reservoir and Th is the temperature of the hot reservoir. Therefore, the smaller the temperature difference, the smaller the efficiency and the greater the COPhp.
In summary, for a heat pump working between the same temperatures as a Carnot engine, the coefficient of performance is given by COPhp = 1/(η).
