The relation between efficiency (eta) of a Ca | vedclass

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(A) For a Carnot engine working between temperatures $T_1$ (source) and $T_2$ (sink),the efficiency is given by $\eta = 1 - \frac{T_2}{T_1}$.For a ref

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$\eta = \frac{1}{1+\beta}$

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(A) For a Carnot engine working between temperatures $T_1$ (source) and $T_2$ (sink),the efficiency is given by $\eta = 1 - \frac{T_2}{T_1}$.For a refrigerator working between the same temperatures,the coefficient of performance is given by $\beta = \frac{T_2}{T_1 - T_2}$.From the refrigerator equation,we can write $\frac{1}{\beta} = \frac{T_1 - T_2}{T_2} = \frac{T_1}{T_2} - 1$.Therefore,$\frac{T_1}{T_2} = 1 + \frac{1}{\beta} = \frac{\beta + 1}{\beta}$.Taking the reciprocal,$\frac{T_2}{T_1} = \frac{\beta}{1 + \beta}$.Substituting this into the efficiency formula: $\eta = 1 - \frac{T_2}{T_1} = 1 - \frac{\beta}{1 + \beta} = \frac{1 + \beta - \beta}{1 + \beta} = \frac{1}{1 + \beta}$.Thus,the correct relation is $\eta = \frac{1}{1 + \beta}$.

The relation between efficiency $(\eta)$ of a Carnot engine and the coefficient of performance $(\beta)$ of a refrigerator working between the same temperatures is:

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