A Carnot engine has an efficiency of 22.0 %It operates between constant-temperature reservoirs differing in temperature by $\mathbf{75}\mathbf{.}\mathbf{0}\mathbf{°}$. (a) What is the temperature of the lower temperature, and (b) What is the temperature of a higher-temperature reservoir?

The efficiency of the engine,$\mathrm{\epsilon }=22\%\mathrm{or}0.22$

Temperature difference,${\mathrm{T}}_{\mathrm{H}}-{\mathrm{T}}_{\mathrm{L}}=75°\mathrm{C}$

We use the efficiency of Carnot’s heat engine to calculate the temperature of a low-temperature reservoir and the temperature of a high-temperature reservoir.

Formula:

The efficiency of the Carnot engine,$\mathrm{\epsilon }=1-\frac{{\mathrm{T}}_{\mathrm{L}}}{{\mathrm{T}}_{\mathrm{H}}}$ (i)

First, we have to calculate the temperature of the higher temperature reservoir in (b) and then substitute that value in the above equation to get the temperature of the low-temperature reservoir.

Using equation (1), the temperature of the higher-temperature reservoir of the Carnot’s heat engine can be given as:

$$\begin{gathered} \mathrm{\epsilon }=\frac{{\mathrm{T}}_{\mathrm{H}-}{\mathrm{T}}_{\mathrm{L}}}{{\mathrm{T}}_{\mathrm{H}}} \\ {\mathrm{T}}_{\mathrm{H}}=\frac{{\mathrm{T}}_{\mathrm{H}-}{\mathrm{T}}_{\mathrm{L}}}{\mathrm{\epsilon }} \\ =\frac{75}{0.22} \\ =341°\mathrm{C} \end{gathered}$$

Now, substituting this value in the given data of temperature difference, we can get the temperature of the lower-temperature reservoir as given:

$$\begin{gathered} {\mathrm{T}}_{\mathrm{L}}=\left(341°\mathrm{C}-75°\mathrm{C}\right) \\ =266°\mathrm{C} \end{gathered}$$

Hence, the value of the low temperature is$266°\mathrm{C}$

From the calculation of part (a) for the lower temperature, we get the higher temperature of the reservoir.

Hence, the value of a higher temperature is$341°\mathrm{C}$