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Challenge: Problem 3

Consider the . According to investigations of thermal conductivity, the free electrons in an alkali metal must have very high velocities that are relatively independent of the temperature, up to a thousand degrees centigrade or so. These electrons can be treated as composing a "gas", since they are kinetic particles confined in a definite volume by the atomic lattice of the metal. However, electrons differ from molecules, such as those of hydrogen and helium, in many important respects. Modern statistics gives a representation of the distribution of electrons with respect to energy such as that shown in fig.8.9. The line S indicates the "energy surface" of the electrons at room temperature. The vertical dotted line represents the energy surface at absolute zero. The area is shaded in order to indicate that the energy space indicated is : there is no room for more electrons at the lower energies.

Obviously, a different sort of statistics is involved with different rules. In this case, we consider a "momentum space" (see fig. 8.10) divided into small regions or cells-such as . The sphere drawn in the figure represents the "surface of the maximum energy" at absolute zero. As the temperature increases, this surface becomes increasingly "fuzzy".

It has been calculated, using modern theory, that the electrons in a metallic conductor will not make their full (ideal gas) contribution the specific heat before a temperature of 30,000° K is reached, before which, of course, the metal will have completely vaporized.

Review this account of the electron gas in the light of what you have learnt and the items on the RESPONSE ARRAY. Choose items which indicate concepts you would use in order to understand what has been said about the electron gas.

1
break up of material particles into smaller particles
2
momentum or velocity space
3
energy cells that accommodate only one particle at a time
4
number of arrangements corresponding to the distribution of energy
5
mass of particles
6
translational kinetic energy
7
average thermal energy of a particle is an integral multiple of 1/2 kT
8
number of particles possible in a particular energy cell
9
internal energy states of particles
10
at absolute zero, entropy is zero
11
state of greatest probability
12
number of degrees of freedom for thermal energy
13
rotational energy
14
intrinsic structure of energy cells, such as their size and the "separation" betweeen them
15
= ( 1 / kT )
16
zero point energy of a system
17
vibrational energy
18
effective continuum of energy states
19
internal energy is the sum of the energies of the particles
20
thermal equilibrium entails maximum entropy