Question b26f0

Nov 1, 2016

${C}_{\text{rms":C_"av":C_"mp}} = 1 : 0.9213 : 0.8165$

Explanation:

These are the root mean square, average, and most probable velocities of a gas molecule.

According to Kinetic Molecular Theory,

${C}_{\text{rms}} = \sqrt{\frac{3 R T}{M}}$

C_"av" = sqrt((8RT)/(πM))

${C}_{\text{mp}} = \sqrt{\frac{2 R T}{M}}$

The ratio with ${C}_{\text{rms}} = 1$

C_"rms":C_"av":C_"mp" = sqrt((3RT)/M):sqrt((8RT)/(πM)):sqrt((2RT)/M)

Multiply all values by $\sqrt{\frac{M}{3 R T}}$

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

${C}_{\text{rms":C_"av":C_"mp}} = 1 : 0.9213 : 0.8165$

The ratio with ${C}_{\text{av}} = 1$

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 ${C}_{\text{mp}} = 1$

Multiply all values by $\sqrt{\frac{M}{2 R T}}$

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#