Molecular Velocity





Molecular Velocities

In kinetic theory of gas the velocity of molecules is expressed in the following three terms

(i)   Root-mean square velocity or R.M.S. Velocity:

It may be defined as, “The square root of the mean value of the squares of the velocities of all molecules”. It is denoted by u. If V_1, V_2, V_3, \cdots V_n are the velocities for n molecules, then u = \sqrt{\dfrac{V_1^2 + V_2^2 + V_3^2 + \cdots + V_n^2}{n}}

(ii)    The average velocity is the arithmetic mean of the velocities of all molecules. It is denoted by v and is given by the following equation:

v = \dfrac{(V_1 + V_2 + V_3 + \cdots V_n)}{n}[

Average velocity (v) = 0.9213 x R. M. S. velocity (u)

(iii)    The most probable velocity is the velocity possessed by maximum number of molecules of the gas at a given temperature. It is denoted by \alpha and is given by the following equation:

\alpha = \dfrac{2RT}{M} = \dfrac{2RT}{mN} (since M= mN)

Or,

\alpha = \sqrt{\dfrac{2u}{3}}

These three velocities are related to each other as:

Y : v : \alpha = 1.0 : 0.9213 : 0.8177

\alpha = 1.0 : 1.128 : 1.224 so \alpha < v < u

Maxwell’s distribution of velocities

maxwell

Example 1: Calculate the root mean square velocity of nitrogen molecule at N.T.P.

Solution: According to kinetic gas equation:

u = \sqrt{\dfrac{3PV}{M}} \\[3mm] = \sqrt{\dfrac{(3)(76 \times 13.6 \times 981 dyn cm^{-2})(22400 cm^3 mol^{-1})}{28 mol^{-1}}} \\[3mm] 4.93 \times 10^4 cms^{-1}

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