Physical Chemistry Third Edition

1180 28 The Structure of Solids, Liquids, and Polymers

We divide this equation byN, multiply bydt, and perform an integration from

t0tott′:

1

N

dN

dt

dt

1

N

dN −

1

τ

dt

∫N(t′)

N(0)

1

N

dN −

1

τ

∫t′

0

dt

ln

(

N(τ)

N(0)

)

−

t

τ

N(t)N(0)e−t/τ (28.4-2)

where we replacet′bytin the equation forN(t).

In an external electric fieldEEE the force on an electron iseEEE. The Drude model

assumes that we can apply classical mechanics to the electrons. Classical mechanics

cannot successfully be applied to individual electrons, but it can sometimes be an

adequate approximation for some average properties. From Newton’s second law the

electric field produces a constant acceleration equal to−eEEE/m. If the electron does

not undergo a collision, the change in velocity of an electron in timetis

∆v(t)−

eEEEt

m

(28.4-3)

This change in velocity is imposed on whatever velocity the electron originally had.

Before the field was imposed no current was flowing, so that the average velocity of

the conduction electrons vanished. After the change in velocity is imposed on every

electron, Eq. (28.4-3) represents the average final velocity. The average distance trav-

eled during the time from 0 to timet′is obtained by integrating Eq. (28.4-3) from time

0 to timet′:

r(t′)−

eEEEt′^2

2 m

(28.4-4)

We assume that the average velocity returns to zero value after the collision.

The rate at which electrons are undergoing collisions at timetis

dN

dt

−

N(t)

τ

−

N(0)

τ

e−t/τ (28.4-5)

The contribution of all electrons to the “electron transport” (equivalent to electrons per

unit area per second times the time of transport) is

∫∞

0

r(t)

dN

dt

dt

−eEEEn(0)

2 m

∫∞

0

t^2 e−t/τdt−

eEEEN(0)^2

m

(28.4-6)

This formula for the electron transport is the same as ifN(0) electrons per unit volume

all had the “drift velocity” given by Eq. (28.4-3) for an acceleration time equal toτ,

vdrift−

eEEEτ

m

(28.4-7)

and traveled for a time equal toτ. The electric current per unit area,j, is equal to the

charge on one electron,−e, times the electron flux

j(−e)

(

−

eEEEN(0)τ

m

)



Ne^2 τ

m

EEE (28.4-8)