A gaseous mixture consisting of 1 mole of an ideal monoatomic gas and 1 mole of an ideal rigid diatomic gas at an initial temperature T is compressed adiabatically to one-fourth of its initial volume. The work done on the gas mixture during this process is:

Step-by-step solution

For the mixture, the total number of moles is n = n₁ + n₂ = 1 + 1 = 2 . The molar heat capacity at constant volume for the mixture is: C_ v, mix = n₁ C_ v1 + n₂ C_ v2 n₁ + n₂ = 1 ( 3 2 R ) + 1 ( 5 2 R ) 2 = 4R 2 = 2R The effective adiabatic exponent _ mix is: _ mix = 1 + R C_ v, mix = 1 + R 2R = 3 2 For an adiabatic process, T_i V_i^ - 1 = T_f V_f^ - 1 . Given V_f = V_i 4 and T_i = T : T V_i^ 1 2 = T_f ( V_i 4 )^ 1 2 T = T_f ( 1 2 ) T_f = 2T The work done ON the gas in an adiabatic process is equal to the change in