91Ó°ÊÓ

The number of molecules of the gas is \(N\).

The speed of the molecules is \(v\).

The expression to calculate the number of the molecules with the speed is given as follows.

\(\Delta N = N\int_v^{\Delta v + v} {f\left( v \right)dv} \) …… (i)

Here,\(f\left( v \right)\)is the Maxwell-Boltzmann distribution.

Let’s assume \(f\left( v \right)\) is very small.

Rewrite the equation (i).

\(\Delta N = Nf\left( v \right)\Delta v\)

Substitute \(1\) for \(\Delta v\) into above equation.

\(\begin{array}{l}\Delta N = Nf\left( v \right)\left( 1 \right)\\\Delta N = Nf\left( v \right)\end{array}\)

From the above discussion, \(Nf\left( v \right)\) is the number of the molecules and the \(Nf\left( v \right)\) is the number of the molecules with the speed interval \(v\) to \(v + \Delta v\) and \(f\left( v \right)\) represents the probability per unit speed interval. Therefore, \(Nf\left( v \right)\), is not the number of the molecules with the exactly speed \(v\).

Hence the \(Nf\left( v \right)\), is not the number of the molecules with the exactly speed \(v\). But the number of the molecules is given by \(Nf\left( v \right)\).