Question

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?

Short answer

1. The temperature of the lower-temperature reservoir is$266°\mathrm{C}$
2. The temperature of the higher-temperature reservoir is$341°\mathrm{C}$

Step-by-step solution

The given data

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

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

Understanding the concept of the Carnot engine

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)

Calculation of temperature of the low-temperature reservoir

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$266°\mathrm{C}$

b) Calculation of the temperature of the higher-temperature reservoir

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$341°\mathrm{C}$