A rigid cubical vessel of side holds nitrogen

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

A rigid cubical vessel of side

a

holds nitrogen, treated as an ideal gas of point molecules of mass

m = 4.65\times10^{-26}\,\mathrm{kg}

that bounce elastically off the rigid walls. Gravity is ignored and the velocity distribution is isotropic and unchanging. The number density is

n = N/V = 2.0\times10^{25}\,\mathrm{m^{-3}}

and the temperature is

350\,\mathrm{K}

. ((a)) Starting from a single molecule bouncing between two opposite walls, derive

p = \tfrac13 nm\langle v^2\rangle

, naming the one assumption you use about the components of velocity. ((b)) Taking the speed distribution

f(v) = 4\pi\left(\dfrac{m}{2\pi k_BT}\right)^{3/2}v^2e^{-mv^2/2k_BT}

as given, use the Gaussian integral

\int_{-\infty}^{\infty}e^{-\alpha x^2}\,\mathrm{d}x = \sqrt{\pi/\alpha}

and differentiation with respect to

\alpha

to show that

\langle v^2\rangle = 3k_BT/m

, and hence that the wall-collision result becomes

pV = Nk_BT

. ((c)) Evaluate

v_{\rm rms}

and

p

for this sample.

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