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)Crms=√3RTM
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
