Physical Chemistry Third Edition

1182 28 The Structure of Solids, Liquids, and Polymers

The ratio of the drift speed to the average speed is

Ratio

1. 06 × 10 −^4 ms−^1

1. 154 × 105 ms−^1

 9. 2 × 10 −^10

The expression for the conductivity can also be expressed in terms of the mean free

path between collisions. The mean free path is approximately given by

λ〈v〉τ (28.4-10)

where〈v〉is an average speed of the electrons. This is not the average drift speed, which

is a small speed superimposed on a large speed. It is an average of the actual speeds,

which is nearly the same as the equilibrium value, because of the smallness of the drift

velocity. If we use the root-mean-square speed of Example 28.10, the conductivity is

σ

Ne^2 λ

( 3 mkBT)^1 /^2

(28.4-11)

EXAMPLE28.11

Find the mean free path for electrons in gold at 20◦C.

Solution

λ

( 3 mkBT)^1 /^2 σ

Ne^2



[3(9. 11 × 10 −^31 kg)(1. 38 × 10 −^23 JK−^1 )(293 K)]^1 /^2 (4. 46 × 107 ohm−^1 m−^1 )

(5. 90 × 1028 m−^3 )(1. 60 × 10 −^19 C)

 3. 1 × 10 −^9 m

This value seems reasonable, as it is roughly 10 lattice distances, and an electron might pass

a number of cores before it collides with one.

The Drude model is a crude model, but it contains the accepted mechanism for

electrical resistance in solids, which is the effect of collisions with the cores of the

crystal. There are a number of more sophisticated theories than the Drude theory.

However, the results of these theories are similar in their general form to Eq. (28.4-9).

The major differences are in the interpretation of the quantitiesN,τ, andm.^13 One

problem with the Drude theory is that the conductivities of most common metals are

found experimentally to be approximately inversely proportional to the temperature,

instead of being inversely proportional to the square root of the temperature, as in

Eq. (28.4-11). One can rationalize this by arguing that the mean free path should

decrease as the temperature rises, because of the increased vibrational amplitude

of the cores, making them into targets with larger effective sizes at higher

temperature.

(^13) J. S. Blakemore,op. cit., p. 162ff (note 4).