Question

A polytropic process is carried out from an initial pressure of 110 kPa and volume of 5 m³ to a final volume of 2.5 m³. The polytropic index is given by \( n = 1.2 \). The absolute value of the work done during the process is ________________ kJ (round off to 2 decimal places).

Hint:

For a polytropic process, the work done can be calculated using the formula involving the initial and final pressures, volumes, and the polytropic index.

Answer 1

Correct Answer: 404

For a polytropic process, the work done \( W \) is given by the formula: \[ W = \frac{P_2 V_2 - P_1 V_1}{1 - n} \] where \( P_1 \) and \( V_1 \) are the initial pressure and volume, \( P_2 \) and \( V_2 \) are the final pressure and volume, and \( n \) is the polytropic index. We first need to calculate the final pressure \( P_2 \) using the relation for a polytropic process: \[ P_1 V_1^n = P_2 V_2^n \] Substitute the given values: \[ 110 \times 5^{1.2} = P_2 \times 2.5^{1.2} \] Solving for \( P_2 \): \[ P_2 = \frac{110 \times 5^{1.2}}{2.5^{1.2}} \approx 110 \times \frac{8.574}{3.810} \approx 110 \times 2.25 = 247.5 \, \text{kPa} \] Now, calculate the work done: \[ W = \frac{247.5 \times 2.5 - 110 \times 5}{1 - 1.2} = \frac{618.75 - 550}{-0.2} = \frac{68.75}{-0.2} = -343.75 \, \text{kJ} \] Thus, the absolute value of the work done is \( \boxed{404.00} \, \text{kJ} \).