Chapter 24, Problem 52A

(a)

To determine

The actual maximum efficiency of the heat engine.

Answer to Problem 52A

The actual maximum efficiency of the heat engine is $\text{48\%}$ .

Explanation of Solution

Given:

The temperature of the hot source

  $T_{\text{hot}}=300°\text{C}$

Temperature of the sink

  $T_{\text{cold}}=\text{25}°\text{C}$

Formula used:

The maximum possible efficiency of any heat engine is equal to the ideal efficiency of the engine.

The ideal efficiency of a heat engine is given by the expression,

  $\text{Ideal efficiency}=\frac{T_{\text{hot}}-T_{\text{cold}}}{T_{\text{hot}}}$

Calculation:

Express the temperatures of the hot reservoir and the sink in Kelvin.

  $\begin{array}{l}{T}_{\text{hot}}=300°\text{C}+273=573\text{ K}\\ T_{\text{cold}}=\text{25}°\text{C}+273=298\text{ K}\end{array}$

Substitute the values of the variables in the formula and calculate the ideal efficiency.

  $\begin{array}{c}\text{Ideal efficiency}=\frac{T_{\text{hot}}-T_{\text{cold}}}{T_{\text{hot}}}\\ =\frac{\left(573\text{ K}\right)-\left(298\text{ K}\right)}{\left(573\text{ K}\right)}\\ =0.48\end{array}$

Express the efficiency of the engine in percentage.

  $\begin{array}{c}\text{Ideal efficiency}=0.22\times 100\%\\ =22\%\end{array}$

Conclusion:

Thus, the actual maximum efficiency of the heat engine is $\text{48\%}$ .

(b)

To determine

The error made in choosing temperature scales.

Answer to Problem 52A

The scientist had failed to convert the temperature in Celsius to kelvin.

Explanation of Solution

Introduction:

The efficiency of a heat engine is dependent on the temperatures of the hot and cold reservoirs as is shown by the following expression:

  $\text{Ideal efficiency}=\frac{T_{\text{hot}}-T_{\text{cold}}}{T_{\text{hot}}}$

The above equation would give the ideal efficiency when the temperatures are expressed in Kelvin as is shown in part (a).

However, if the same expression is solved using the Celsius scale, this would give a false value for the efficiency.

  $\begin{array}{c}\text{Ideal efficiency}=\frac{T_{\text{hot}}-T_{\text{cold}}}{T_{\text{hot}}}\\ =\frac{\left(300°\text{C }\right)-\left(\text{25}°\text{C}\right)}{\left(300°\text{C }\right)}\\ =0.92\end{array}$

Or, the efficiency of the engine would be $\text{92\%}$ , as claimed by the scientist.

The correct efficiency is, however $\text{48\%}$ and hence his claims will not be substantiated.

Conclusion:

Thus, the person failed to convert the temperatures of the hot and the cold reservoirs into Kelvin, which led to claim an efficiency of $\text{92\%}$ , while the real efficiency of the engine is $\text{48\%}$ .