THE UNIVERSITY OF BRITISH COLUMBIA
 
Physics 108 Assignment # 2:
 
KINETIC THEORY OF GASES
 
Wed. 12 Jan. 2005 - finish by Wed. 19 Jan.

1. Quantum Tension in a String
   A single electron is confined to a single-walled carbon nanotube (SWNT) of length $L = 1$ $\mu$m but is free to move up and down the length of the SWNT. (Since the SWNT is only about 1.2 nm in diameter, you may think of it as a long string). If this system is cooled to nearly 0 K so that the electron is in its lowest possible energy state (the "ground state"), what is the tension in the SWNT "string" due to the electron's confinement?
   Hint: Use de Broglie's hypothesis ($\lambda = h/p$) and think in terms of standing waves. Then use a classical picture of a particle of momentum $p = mv$ bouncing back and forth off the ends of the string . . . .

2. 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, a SWNT or a "nanowire" made from GaAs/AlGaAs structures.)

3. One-Dimensional Maxwellian Speed Distribution
   1. 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: Again, use de Broglie's hypothesis and think of standing waves.

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

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