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

[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.

 

[paranormal]

I can provide a mathematical and physical explanation for the question. The question is asking for the fraction of molecules in an ideal gas that have velocities within a certain range of angles, specifically between 29.5° and 30.5° for φ and between 44.5° and 45.5° for θ. This can be represented as a volume in phase space, where each point represents a unique combination of position and velocity for a molecule. The volume within this range of angles would represent the fraction of molecules with velocities in that range.

To calculate this fraction, we can use the Maxwell-Boltzmann distribution, which describes the distribution of velocities for molecules in an ideal gas. This distribution is given by the equation:

f(v) = 4π (m/2πkT)^3/2 * v^2 * e^(-mv^2/2kT)

Where:
f(v) is the fraction of molecules with velocity v
m is the mass of the molecule
k is the Boltzmann constant
T is the temperature in Kelvin

To calculate the fraction of molecules within the given range of angles, we can integrate this distribution over the range of velocities that correspond to those angles. This can be done using the integral:

∫f(v)dv = ∫4π (m/2πkT)^3/2 * v^2 * e^(-mv^2/2kT) dv

Where the limits of integration are the velocities corresponding to the given angles. This integral will give us the fraction of molecules with velocities within the given range of angles.

In conclusion, the fraction of molecules in an ideal gas with velocities between 29.5° and 30.5° for φ and between 44.5° and 45.5° for θ can be calculated using the Maxwell-Boltzmann distribution and integrating over the corresponding velocities. This fraction represents the volume of phase space occupied by molecules with velocities in that range.