An ideal gas with an unknown number of degrees of freedom is adiabatically compressed to 1 8 of its original volume. It is then isothermally expanded back to its original volume. If the final pressure of the gas is found to be 4 times its initial pressure, the degrees of freedom of the gas is

Step-by-step solution

Let the initial state of the gas be (P₀, V₀) . During the adiabatic compression to V₁ = V₀ 8 , the new pressure P₁ is given by: P₀ V₀^ = P₁ ( V₀ 8 )^ P₁ = P₀ 8^ Next, the gas undergoes isothermal expansion back to its original volume V₀ . The final pressure P₂ is given by Boyle's law: P₁ V₁ = P₂ V₀ (P₀ 8^ ) ( V₀ 8 ) = P₂ V₀ P₂ = P₀ 8^ - 1 We are given that the final pressure is 4 times the initial pressure, so P₂ = 4 P₀ . P₀ 8^ - 1 = 4 P₀ 8^ - 1 = 4 2^ 3( - 1) = 2^2 3( - 1) = 2 = 5 3 The relationship between and th