A mixture containing n₁ moles of a monatomic gas and n₂ moles of a diatomic gas undergoes a reversible adiabatic compression. The volume of the mixture is compressed to 1 32 of its original value, and the final pressure is found to be 256 times the initial pressure. The ratio of the number of moles n₁ n₂ is:

Step-by-step solution

For an adiabatic process, the relationship between pressure and volume is PV^ _ mix = constant . Given V_f = V_i 32 and P_f = 256 P_i , we have: P_i V_i^ _ mix = (256 P_i) ( V_i 32 )^ _ mix 32^ _ mix = 256 (2^5)^ _ mix = 2^8 5 _ mix = 8 _ mix = 8 5 The specific heat ratio for a gas mixture is given by _ mix = n₁ C_ p1 + n₂ C_ p2 n₁ C_ v1 + n₂ C_ v2 . For a monatomic gas, C_ v1 = 3 2 R and C_ p1 = 5 2 R . For a diatomic gas, C_ v2 = 5 2 R and C_ p2 = 7 2 R . Substituting these values into the mixture formula: 8 5 =