Assuming a homogeneous gas in the context of Kinetic Molecular Theory of gases, how does the root-mean-squared velocity relate to the average gas velocity in one dimension?
1 Answer
If you take the squared velocity in a certain direction
#<< v_x^2 >> = 1/Nsum_(i=1)^(N) v_(ix)^2 = (v_(1x)^2 + v_(2x)^2 + . . . + v_(Nx)^2)/N#
For an homogeneous gas, its motion is isotropic, so that
#<< v_x^2 >> = << v_y^2 >> = << v_z^2 >># ,
and thus,
#<< v^2 >> = << v_x^2 >> + << v_y^2 >> + << v_z^2 >>#
#= 3<< v_x^2 >>#
If you then take the square root of
#v_(RMS) = sqrt(<< v^2 >>)#
For gases that follow the Maxwell-Boltzmann Distribution, this is given by:
#v_(RMS) = sqrt((3RT)/M)# where
#R = "8.314472 J/mol"cdot"K"# and#T# is temperature in#"K"# .#M# is the molar mass in#"kg/mol"# .
The function of