Fraction of gas molecules with velocity angles between 29.5°–30.5° and 44.5°–45.5°

[Phyrrus]

Homework Statement

Approximately what fraction of molecules of a gas (assumed ideal) have velocities for
which the angle φ lies between 29.5° and 30.5°, while θ lies between 44.5° and 45.5°?

Homework Equations

The Attempt at a Solution

What does the question even mean geomtrically?

 

[francesco85]

Since the Maxwell Boltzmann distribution for a free diluite gas depend only on the modulus of the velocity, the fraction of the particles that have velocity between the angles defined by $\phi_1$ and $\phi_2$, $\theta_1$ and $\theta_2$ is simply the fraction of the area of part of the spherical surface determined by those parameters and the surface of the sphere; namely

fraction= $\frac{1}{4\pi}(\phi_2-\phi_1)($cos$(\theta_1)-$cos$(\theta_2))$

all the angles are expressed in radiants.

 

[Phyrrus]

Thanks mate, is there somewhere I could get a detail analysis and derivation of this, because it wasn't in any lectures and isn't in my text at all.

 

[francesco85]

You are welcome; the best place where one can study all this stuff at the introductory level is Huang's textbook "Statistical Mechanics".
f.