Number of molecules moving in a given direction at a given speed

[Pushoam]

Homework Statement

I didn't get it.
Consider velocity space.
The no. of velocity vectors for a given speed is proportional to the surface area of a sphere of radius v in the velocity space.
So, a fraction of velocity vectors with speed v in an elemental area dA is##\frac{ dA}A##.
A = ##\Omega v^2##
## dA = d \Omega v^2##
##\frac {dA}A = \frac{d \Omega v^2}{4\pi v^2}
\\ \frac {dA}A =\frac{d \Omega}{4\pi }##How to relate it to eq.6.7?

I didn't get this,too.

Please help me.

Homework Equations

The Attempt at a Solution

 

[Pushoam]

I have to calculate fraction of molecules traveling in a certain direction at a certain speed .
Let's define the certain direction by a direction which is at an angle ##\theta## from a standard direction, let's say z- direction ( as I am going to use spherical coordinates).
Let's denote the certain speed by "v".
Let's denote the required quantity by H(##\theta,v##).
No. of molecules at a certain speed is given by Boltzmann- Maxwell eqn. which is denoted here by f(v)dv.
In velocity space, fraction of velocity vectors in a small solid angle d##\Omega## corresponding to elemental area dA is given by ##\frac{d\Omega}{4\pi}##
The area dA is the area occupied by the velocity vector( similar to the position vector in position space) , when it is moved from an angle ##\theta ## to ##\theta +d \theta## in velocity space, where ##\theta ## is an angle made by the velocity vector with a standard direction.

In this case,
##d\Omega = 2\pi \sin {\theta} d\theta##
So, H(##\theta,v##) = ##\frac{f(v)dv \sin {\theta} d\theta}2##

Is this correct?