Argon atoms of mass are adsorbed on a flat solid surface

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Kinetic theory and transport in gases · Kinetic theory and transport in gases

Thermal Physics, Ch Temperature, pressure and the kinetic theory of gases

Problem

Argon atoms of mass

m = 6.63\times10^{-26}\,\mathrm{kg}

are adsorbed on a flat solid surface. They slide in the plane without friction and are confined to a square of side

L

by low straight ridges, from which they rebound elastically; motion perpendicular to the surface is frozen out, so each atom has only the two velocity components

v_x

and

v_y

. The layer is dilute (the atoms occupy a negligible fraction of the area) and its velocity distribution is isotropic in the plane. The areal number density is

n_2 = N/L^2

, and the two-dimensional pressure

p_2

is defined as the force on a ridge divided by its length. ((a)) Repeat the wall-collision argument in the plane and derive

p_2 = \tfrac12 n_2 m\langle v^2\rangle

, naming the isotropy statement you use. ((b)) Using equipartition of energy for the two quadratic terms

\tfrac12 mv_x^2

and

\tfrac12 mv_y^2

, show that

p_2 = n_2 k_BT

. ((c)) Assume, as in the three-dimensional derivation, that

v_x

and

v_y

are independent. Starting from the isotropic Gaussian

f(v_x) \propto e^{-mv_x^2/2k_BT}

for each component and using the ring of area

2\pi v\,\mathrm{d}v

in the two-dimensional velocity plane, derive the two-dimensional speed distribution

f(v)

, check that it is normalised, and obtain the most probable speed

v_p

and

\langle v^2\rangle

. ((d)) Evaluate

p_2

and

v_p

for this layer at

300\,\mathrm{K}

with

n_2 = 5.0\times10^{18}\,\mathrm{m^{-2}}

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