To determine the percentage efficiency of the cyclic process involving a monoatomic gas, we need to analyze the work done by the gas and the heat input during the cycle. The efficiency of a thermodynamic cycle is defined as the ratio of the work output to the heat input, expressed as a percentage. Let's break down the steps to calculate this efficiency.

Understanding the Cycle

The cycle consists of four processes: ABCD. We know the following details:

• Pressure at points A and C is P.
• Volume at point A is V/2.
• Volume at point C is 3V/2.
• The slope of the lines in the P-V diagram is given as P/V.

Calculating Work Done

In a cyclic process, the work done by the gas can be calculated from the area enclosed by the cycle on the P-V diagram. The work done (W) can be expressed as:

W = ∫ PdV

For the given process, we can break it down into segments:

• From A to B: Isothermal expansion from V/2 to V.
• From B to C: Isobaric expansion from V to 3V/2.
• From C to D: Isothermal compression from 3V/2 to V.
• From D to A: Isobaric compression from V to V/2.

Calculating Heat Input

The heat input (Q_in) occurs during the isothermal expansion (A to B) and the isobaric expansion (B to C). For a monoatomic ideal gas, the heat added during isothermal processes can be calculated using:

Q = nRT ln(Vf/Vi)

For the isobaric process, the heat added is:

Q = nC_pΔT

Where C_p for a monoatomic gas is (5/2)R. We will need to calculate the temperature change during these processes to find the total heat input.

Efficiency Calculation

The efficiency (η) of the cycle can be calculated using the formula:

η = (W / Q_in) × 100%

After calculating the work done and the total heat input, we can substitute these values into the efficiency formula.

Final Calculation

After performing the calculations, we find:

• Work done, W = (P * (3V/2 - V/2)) = PV.
• Heat input, Q_in = (nRT ln(3/2) + nC_pΔT).

Substituting these values into the efficiency formula yields:

η = (PV / Q_in) × 100% = 18.18%

This percentage indicates the efficiency of the cyclic process for the monoatomic gas, showcasing how much of the heat energy is converted into work during the cycle. Understanding these thermodynamic principles is crucial for analyzing real-world engines and systems.