A diatomic gas is kept in a perfectly insulated vessel. It is suddenly compressed such that its final pressure becomes 128 times its initial pressure. The ratio of its final volume to its initial volume is:

Step-by-step solution

For a sudden compression in an insulated vessel, the process is adiabatic. The equation for an adiabatic process is: P_i V_i^ = P_f V_f^ Rearranging for the ratio of volumes: ( V_f V_i )^ = P_i P_f V_f V_i = ( P_i P_f )^ 1 For a diatomic gas, the ratio of specific heats is = 7 5 . Given that P_f = 128 P_i , we have P_i P_f = 1 128 . Substituting the values: V_f V_i = ( 1 128 )^ 5 7 Since 128 = 2^7 , we get: V_f V_i = ( 1 2^7 )^ 5 7 = 1 2^5 = 1 32 Answer: 1 32