THE UNIVERSITY OF BRITISH COLUMBIA
 
Physics 122 Assignment # 2:
 
KINETIC THEORY OF GASES
 
Wed. 09 Jan. 2002 - finish by Wed. 16 Jan.

1.

One-Dimensional Ideal Gas
Making use of the EQUIPARTITION THEOREM, derive an equation analogous to the familiar 3D IDEAL GAS LAW ( $pV = N \tau$) for an ideal gas confined to a one-dimensional ``box'' of length L. (Some examples would be N electrons moving freely along a single DNA molecule, a trans-polyacetylene chain or a ``nanowire'' made from GaAs/AlGaAs structures.)

2.

One-Dimensional Maxwellian Speed Distribution

(a)

What is the thermal speed distribution ${\cal D}(v)$ [in the textbook's notation, N(v)/N, as in Eq. (22-14) on p. 503] for an ideal gas confined to a one-dimensional ``box''? It would be nice if you could find the right leading factors (involving temperature and various constants) to normalize the distribution so that

$$\int_0^\infty {\cal D}(v) \; dv \; = \; 1 \; ,$$

but I am mainly looking for its dependence on the speed v.
Hint: Use de Broglie's hypothesis ( $\lambda = h/p$) and think in terms of standing waves.

(b)

Sketch this distribution for a given temperature and compare its shape with that shown in the Figures on p. 503 of the textbook.

(c)

What can you say about the most probable speed $v_{\rm p}$ in the two different cases?

3.

Heat Capacity of Nitrogen Gas
Sketch (including axis labels and scales) the heat capacity per molecule, C₁, of an ideal gas of diatomic nitrogen (N₂) molecules in an isolated closed vessel of fixed volume, as a function of temperature $\tau$ from room temperature¹ up to a temperature high enough to dissociate the molecules into separate atoms. You may have to make some estimates based on simple mechanics and common sense.