In Section 17.6, we derived the expression for the flux JN...

In Section $17.6,$ we derived the expression for the flux $J_{N}$ of gas molecules through a surface in terms of velocities $v_{x}$ of molecules in the direction perpendicular to the surface. The derivation can be made in a more general way by considering that the molecules can approach the surface with velocity $v$ at angles $\theta$ and $\phi$ in the system of spherical coordinates. In this case, the differential of the flux is given by \[\mathrm{d} J_{N}=\frac{\rho}{4 \pi} v F(v) \mathrm{d} v \cos \theta \sin \theta \mathrm{d} \theta \mathrm{d} \phi\] Integrate this equation to obtain the flux $J_{N}$

In Section $17.6,$ we derived the expression for the flux $J_{N}$ of gas molecules through a surface in terms of velocities $v_{x}$ of molecules in the direction perpendicular to the surface. The derivation can be made in a more general way by considering that the molecules can approach the surface with velocity $v$ at angles $\theta$ and $\phi$ in the system of spherical coordinates. In this case, the differential of the flux is given by
\[\mathrm{d} J_{N}=\frac{\rho}{4 \pi} v F(v) \mathrm{d} v \cos \theta \sin \theta \mathrm{d} \theta \mathrm{d} \phi\]
Integrate this equation to obtain the flux $J_{N}$

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Key Concepts

Spherical Coordinates Integration

Using spherical coordinates to perform integration is crucial when accounting for all possible directions of molecular motion. In this framework, angles ? and ? naturally appear, with the integration measure including factors like sin? to account for the surface area element on a sphere, and factors like cos? to project the velocity component onto the surface normal.

Molecular Flux

Molecular flux represents the number of gas molecules passing through a unit area per unit time. It is computed by summing or integrating the contributions from all molecules, each characterized by its velocity and the direction relative to the surface. This concept is central in kinetic theory and transport phenomena.

Velocity Distribution Function

The velocity distribution function, often denoted as F(v), gives the probability density for the molecules to have a specific speed v. Integrating over the speeds with the appropriate weighting factor from F(v) allows calculation of macroscopic properties such as flux by considering contributions from molecules at all speeds.

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