An ideal gas ( = 1.5 ) is expanded adiabatically. To reduce the root mean square velocity of molecules two times, the gas should be expanded

Step-by-step solution

The root mean square velocity of gas molecules is given by v_ rms = 3RT M . This implies v_ rms T . To reduce the rms velocity by a factor of 2 , the absolute temperature must be reduced by a factor of 4 . Thus, T_f = T_i 4 . For an adiabatic process, the relation between temperature and volume is TV^ - 1 = constant . Therefore, T_i V_i^ - 1 = T_f V_f^ - 1 . ( V_f V_i )^ - 1 = T_i T_f Given = 1.5 , we have - 1 = 0.5 = 1 2 . ( V_f V_i )^ 1 2 = 4 V_f V_i = 4^2 = 16 The gas should be expanded 16 times. Answer: 16 time