A scientist says that the efficiency of his heat engine which work at source temperature $127^{\circ } C $ and sink temperature $27^{\circ} C $ is $26\% ,$ then

Question

A real gas within a closed chamber at $ 27^\circ \text{C} $ undergoes the cyclic process as shown in the figure. The gas obeys the equation $ PV^3 = RT $ for the path A to B. The net work done in the complete cycle is (assuming $ R = 8 \, \text{J/molK} $):

Answer 1

To determine the possibility of the efficiency claimed by the scientist, we need to understand the concept of efficiency of a heat engine and compare it with the theoretical maximum efficiency given by the Carnot efficiency formula.

The efficiency (\eta) of a heat engine depends on the temperatures of the heat source T_1 and heat sink T_2, both of which must be in Kelvin. The Carnot efficiency is given by the formula:

\eta_{Carnot} = 1 - \frac{T_2}{T_1}

Let's first convert the temperatures from Celsius to Kelvin.

Source temperature in Kelvin: T_1 = 127^\circ C + 273 = 400 K
Sink temperature in Kelvin: T_2 = 27^\circ C + 273 = 300 K

Next, apply these values to the Carnot efficiency formula:

\eta_{Carnot} = 1 - \frac{300}{400} = 1 - 0.75 = 0.25 \text{ or } 25\%

The actual efficiency of the heat engine is claimed to be 26%, which exceeds the Carnot efficiency of 25%. The Carnot efficiency represents the maximum possible efficiency for a heat engine operating between the given temperature limits. It is a theoretical limit based on the second law of thermodynamics, which states that no heat engine can be more efficient than the Carnot cycle. Therefore, an efficiency of 26% is impossible under these conditions.

Hence, the correct answer is:

it is impossible

Answer 2