An ideal gas undergoes a polytropic process described by the equation PV^x = constant . It expands from an initial state with pressure P₀ and volume V₀ to a final volume 2V₀ . If the work done by the gas during this expansion is P₀V₀ 2 , the value of the exponent x is :

Step-by-step solution

For a polytropic process PV^x = constant , the final pressure is given by P₂ = P₀ ( V₀ 2V₀ )^x = P₀ 2^ -x . The work done by the gas is given by: W = P₁V₁ - P₂V₂ x-1 Substituting the known values: W = P₀V₀ - (P₀ 2^ -x )(2V₀) x-1 = P₀V₀(1 - 2^ 1-x ) x-1 We are given that W = P₀V₀ 2 . Equating the two expressions: 1 - 2^ 1-x x-1 = 1 2 Testing the given options or solving by inspection, for x = 2 : LHS = 1 - 2¹⁻² 2-1 = 1 - 2⁻¹ 1 = 1 - 0.5 = 0.5 = 1 2 Thus, x = 2 perfectly satisfies the condition. Answer: 2