These are the root mean square, average, and most probable velocities of a gas molecule.
According to Kinetic Molecular Theory,
#C_"rms" = sqrt((3RT)/M)#
#C_"av" = sqrt((8RT)/(πM))#
#C_"mp" = sqrt((2RT)/M)#
The ratio with #C_"rms" = 1#
#C_"rms":C_"av":C_"mp" = sqrt((3RT)/M):sqrt((8RT)/(πM)):sqrt((2RT)/M)#
Multiply all values by #sqrt(M/(3RT))#
#C_"rms":C_"av":C_"mp" = sqrt((3RT)/M)×sqrt(M/(3RT)):sqrt((8RT)/(πM))×sqrt(M/(3RT)):sqrt((2RT)/M)×sqrt(M/(3RT)) = 1: sqrt(8/(3π)): sqrt(2/3#
#C_"rms":C_"av":C_"mp" = 1: 0.9213: 0.8165#
The ratio with #C_"av" = 1#
Multiply all values by #sqrt((πM)/(8RT))#
#C_"rms":C_"av":C_"mp" = sqrt((3RT)/M)×sqrt((πM)/(8RT)):sqrt((8RT)/(πM))×sqrt((πM)/(8RT)):sqrt((2RT)/M)×sqrt((πM)/(8RT)) = sqrt((3π)/8):1 : sqrt(π/4) = 1.009 :1: 0.8862#
The ratio with #C_"mp" =1#
Multiply all values by #sqrt(M/(2RT))#
#C_"rms":C_"av":C_"mp" = sqrt((3RT)/M)×sqrt(M/(2RT)):sqrt((8RT)/(πM))×sqrt(M/(2RT)):sqrt((2RT)/M)×sqrt(M/(2RT)) = sqrt(3/2):sqrt(4/π) :1 = 1.225 :1.128:1#
