A monoatomic gas at pressure P having volume V expands isothermally to a volume 3V and then adiabatically to a volume 81 V. The final pressure of the gas is ( = 5/3 ).

Step-by-step solution

For the isothermal expansion from volume V to 3V , the temperature remains constant. P₁ V₁ = P₂ V₂ P V = P₂ (3V) P₂ = P 3 For the subsequent adiabatic expansion from volume 3V to 81V , the relation between pressure and volume is given by P₂ V₂^ = P₃ V₃^ . Given that the gas is monoatomic, = 5 3 . P₃ = P₂ ( V₂ V₃ )^ P₃ = P 3 ( 3V 81V )^ 5 3 P₃ = P 3 ( 1 27 )^ 5 3 P₃ = P 3 ( 1 3^3 )^ 5 3 P₃ = P 3 1 3^5 P₃ = P 3^6 Answer: P (3)^6