Flux of molecules striking a wall

There are many derivations of elementary processes in kinetic theory. We present one more here since the result comes up often. We want to know the number of molecules striking a wall (perpendicular to the x-axis) per unit time and area. This is simply the number density times the x component of velocity averaged over the velocity distribution. We proceed by finding n0 vx using the Maxwell-Boltzmann distribution for the x component (the other factors for the y and z components integrate to unity). We consider only the positive component of vx:

flux/(± area) = noVx = no J Avx exp I - ^ ) dvx (2.27)

(m0/(2nkBT))1/2 and a2 = kBT/m0. The integral can be evaluated with A to give

And finally:

[flux of molecules hitting a wall].

If we apply this to leaks through a small hole in the wall the process is called effusion. The formula holds when the hole is smaller than the mean free path of the molecules so that they flow through the hole without collisions; otherwise the gas acts like a fluid when passing through the opening and one must use fluid mechanics methods rather than kinetic theory.

Table 2.5 Gas notation

p

pressure (N m-2 = Pa; 100 Pa = 1 hPa = 1 mb)

V

volume (m3)

p

mass density (kgm-3)

a

specific volume (m3 kg-1), a = p-1

mo

mass of an individual molecule (kg); for H, m0 = 1.67 x 10-27 kg

no

number density (molecules m-3)

kB

Boltzmann's constant: 1.381x 10-23 J K-1 molecule-1

Mg

gram molecular weight; for hydrogen, MH =1 g mol-1

MMG

the gram molecular weight divided by 1000

Md

dry air effective molecular weight, Md = 28.97 g mol-1

Me

effective gram molecular weight of a mixture of gases

Mi

bulk mass of constituent i (kg)

Na

Avogadro's number: 6.022x1023 molecules mol-1

V

number of moles of a gas

R*

universal gas constant: 8.3143 JK-1 mol-1

Rd

gas constant for dry air: 287JK-1 kg-1

R, Rg

gas constant for a particular gas, G (J K-1 kg-1)

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