Question

Show that the root means the square speed of the molecules of a gas is directly proportional to the square root of the absolute temperature of the gas.  

Answer 1

1. The average value of the pressure of the gas is,
   P = `1/3 "N"/"V" "m"overline("v"^2)`
2. Thus, the mean square velocity of the molecule will be,
   `overline("v"^2) = (3"PV")/("Nm")`
3. Using the ideal gas equation,
   PV = nRT
   `overline("v"^2) = (3"nRT")/("Nm")`
4. But, n = `"N"/"N"_"A"`
   ∴ `overline("v"^2) = (3"NRT")/("N"_"A""Nm")`
5. Also, mN_A = M₀ (Molar mass of the gas)
   ∴ `overline("v"^2) = (3"RT")/("M"_0)`
   ∴ `sqrt(overline("v"^2)) = "V"_"rms" = sqrt((3"RT")/("M"_0))`
6. As R and M₀ in the above equation are constant,
   ∴ V_rms ∝ `sqrt"T"`