ANS:A - 1

The work done during the compression of one kilogram of air is minimized when the value of the polytropic index (n) is equal to 1. Explanation:

1. Polytropic Process:
   • The compression or expansion of a gas can be described by a polytropic process, which is a reversible process during which the pressure (P) and volume (V) of the gas are related by the equation =constantPVn=constant, where �n is the polytropic index.
   • The polytropic index (n) characterizes the specific type of process:
     • If n=1, the process is isothermal (constant temperature).
     • If n>1, the process is polytropic compression (adiabatic, but not isothermal).
     • If n<1, the process is polytropic expansion (adiabatic, but not isothermal).
2. Work Done in Compression:
   • The work done (W) during the compression of one kilogram of air can be calculated using the formula: W=1−nP1​V1​−P2​V2​​
   • Where P1​ and V1​ are the initial pressure and volume, P2​ and V2​ are the final pressure and volume, and n is the polytropic index.
3. Minimization of Work Done:
   • To minimize the work done during compression, we want to minimize the expression P1​V1​−P2​V2​​.
   • This expression is minimized when n is equal to 1. When n=1, the polytropic process becomes isothermal, and the work done during compression reduces to the minimum value for the given initial and final states.
4. Conclusion:
   • Therefore, the work done during the compression of one kilogram of air is minimized when the value of the polytropic index (n) is equal to 1.