## Efficiency of Carnot Engine

### Task number: 1802

Carnot engine operates with efficiency of 40 %. How much must the temperature of the hot reservoir increase, so that the efficiency increases to 50 %? The temperature of the cold reservoir remains at 9 °C.

#### Hint

Carnot efficiency is given by the relation

\[\eta=\frac{T_1-T_2}{T_1},\]where

*T*_{1}is the temperature of hot reservoir and*T*_{2}is the temperature of cold reservoir.#### Analysis

Write down the relationships for the initial efficiency of the Carnot engine and for the efficiency after changing the temperature of the hot reservoir. Using these equations, evaluate the initial and the final temperature of the hot reservoir. The temperature increase of the hot reservoir can be determined as the difference of these two temperatures.

#### Given values

*η*_{1}= 40 % = 0.4initial efficiency of Carnot engine *η*_{2}= 50 % = 0.5efficiency of Carnot engine after increasing the temperature of hot reservoir *t*_{2}= 9 °C =>*T*_{2}= 282 Ktemperature of cold reservoir Δ *T*_{1}= ?temperature increase of hot reservoir #### Solution

First we write down the relationships for the initial efficiency

\[\eta_1=\frac{T_1-T_2}{T_1}, \qquad \qquad \qquad \eta_2=\frac{T_1^*-T_2}{T_1^*},\]*η*_{1}of Carnot engine and for the efficiency*η*_{2}after changing the temperature of the hot reservoir:where

*T*_{1}is the initial temperature of the hot reservoir,*T*_{1}^{*}is the new temperature of the hot reservoir, and*T*_{2}is the temperature of the cold reservoir.Now we evaluate the unknown temperatures

\[T_1=\frac{T_2}{1-\eta_1}, \qquad \qquad \qquad T_1^*=\frac{T_2}{1-\eta_2}.\]*T*_{1}and*T*_{1}^{*}from these relations:The unknown temperature difference Δ

\[\Delta T_1=T_1^*-T_1\] and by substituting the above evaluated temperatures we obtain \[\Delta T_1=\frac{T_2}{1-\eta_2}-\frac{T_2}{1-\eta_1}.\]*T*_{1}of the hot reservoir is given by:#### Numerical solution

\[\Delta T_1=\frac{T_2}{1-\eta_2}-\frac{T_2}{1-\eta_1}=\left(\frac{282}{1-0.5}-\frac{282}{1-0.4}\right)\,\mathrm{K}\,\dot{=}\, 94\,\mathrm{K}\]#### Answer

The temperature of the hot reservoir must increase by 94 K, i.e. by 94 °C.