Physical Chemistry , 1st ed.

can be treated statistically, why not apply statistical thermodynamics to crys-

tals, whose atomic contributions to the properties of the solid as a whole can

probably also be treated statistically?

The first person to make serious headway with this approach was Albert

Einstein. In 1907, Einstein proposed to understand the motions of the atoms

in the crystal using Planck’s idea of quantized energy. A crystal is composed of

Natoms, say. These Natoms can vibrate within their crystal lattice in the x,

the y, or the zdirection, giving a total of 3Npossible vibrational motions.

Einstein assumed that the frequencies of the vibrations were the same, some

frequency labeled (^) E, or the Einstein frequency.If this were the case, and we are
only considering vibration-type motions of the atoms in the crystal, then the
heat capacity of the crystal can be determined by applying the vibrational part
onlyof the heat capacity from the vibrational partition function:
CV
3 N
i 1
k
h
k
T
E

2

(1
e
e
h

(^) E
h
/
k
E
T
/kT) 2
where we are taking the vibrational component of the heat capacity from equa-
tion 18.54 and using the Einstein frequency as the frequency of vibration. Since
the Einstein frequency is constant for all 3Nterms in the sum, the heat capac-
ity becomes
CV 3 Nk
h
k
T
E

2

(1
e
e
h

(^) E
h
/
k
E
T
/kT) 2
As is our habit, we define a temperature
E 
h
k
(^) E
 (18.65)
where Eis the Einstein temperatureof the crystal. Einstein’s expression for the
heat capacity of a crystal is therefore
CV 3 Nk



T

E



2



(1

e

e





E/



T

E/T) 2 (18.66)

Notice how the Einstein temperature and the absolute temperature of the

crystal always appear together as the fraction E/T. Notice, too, that there is

nothing in equation 18.66 that is sample-dependent other than the Einstein

temperature E. This means that if the heat capacity of any crystal were plot-

ted versus E/T, all of the graphs would look exactly the same. This is one ex-

ample of what is called a law of corresponding states.Einstein’s derivation of a

low-temperature heat capacity of crystals was the first to predict such a rela-

tionship for all crystals.

How do we determine the Einstein temperature Ewithout knowing the

characteristic “vibrational frequency” of the atoms in the crystal? Typically, ex-

perimental data is fitted to the mathematical expression in equation 18.66 and

a value of the Einstein temperature is used to allow for the best possible fit to

experimental results. For example, a plot of experimental measurements of the

heat capacity versus Tdivided by E(which is proportional to T, whereas E/T

is inversely proportional to Tand less easy to graph as T→0 K) is shown in

Figure 18.4. Notice that there is reasonable agreement between experiment and

theory, suggesting that Einstein’s statistical thermodynamic basis of the heat

capacity of crystals has merit. Table 18.6 lists a few experimentally determined

Einstein temperatures for crystals.

However, the Einstein equation deviates from experimental values at very

18.10 Crystals 645

Table 18.6 Einstein temperatures of
various crystals
Material E(K)
Al 240
C (dia) 1220
Pb 67