# What fraction of molecules in an ideal gas have velocites between φ1 & φ2 and θ1 & θ2

1. Mar 25, 2012

### Phyrrus

1. The problem statement, all variables and given/known data

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°?

2. Relevant equations

3. The attempt at a solution

What does the question even mean geomtrically?

2. Mar 25, 2012

### francesco85

Re: what fraction of molecules in an ideal gas have velocites between φ1 & φ2 and θ1

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.

3. Mar 26, 2012

### Phyrrus

Re: what fraction of molecules in an ideal gas have velocites between φ1 & φ2 and θ1

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.

4. Mar 27, 2012

### francesco85

Re: what fraction of molecules in an ideal gas have velocites between φ1 & φ2 and θ1

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