# 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

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

Solution: According to kinetic gas equation:

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

Related posts:

1. Kinetic Theory of Gases Kinetic theory of gases D. Bernaulli (1738) forwarded this theory...
2. Linear Differential Equations Linear Differential Equation of nth Order Linear differential equation is...
3. Pressure Temperature Law Pressure Temperature Law (Amonton’s Law): According to this Law at...
4. Instantaneous velocity and speed. Instantaneous velocity: Beside instantaneous velocity , average velocity is a...
5. Bernoulli’s Equation The Bernoulli’s differential equation is written in the form  :...