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total radiation law



= const, x T 4 , where. S = 3 x io 10 E = 3 x lo 1



THE QUANTUM THEORY 43.

both of which have been experimentally verified. Planck's expression-
is thus a very comprehensive one. 1

Some Numerical Values.

At this point it is of interest to calculate the values of the two-
fundamental constants h and k which occur in Planck's equation, h
is the universal proportionality-factor connecting the energy e of a
quantum with the frequency v ; k, as will be shown later, is the gas
constant R, reckoned not for a gram-mole but for a single molecule. .
The calculation may be carried out as follows.

Kurlbaum (Wied. Ann., 65, 759, 1898) has found by experiment
that the total energy emitted from i square centimetre of a "black
body" in i second, the temperature of the body being 100 C. and
that of the air being taken as o C., amounts to 0*0731 watt/cm. 2 , that
is

S = S 100 o C .- S o C .= 4'2 x 7-31 x io 5 .ergs/cm. 2 -sec.
By applying Stefan's Law we obtain

S = o-( 373 4 ~ 2 73 4 ).
_ 4-2 x 7-31 x io 5 ergs

(373* - 273 4 ) ' cm. 2 -sec.-degrees 4 .



The physical significance of a- is evidently the total radiation
emitted from a black body per second, when the temperature differ-
ence between the black body and the surroundings is i degree, the
temperature of the black body being i absolute, the surroundings
being at o absolute. Corresponding to this emission at i absolute,

1 To emphasise further the fact that Planck's formula is at variance with the
principle of ^wjpartition of energy among various degrees of freedom, it is inter-
esting to calculate the energy of an electron vibrating with a frequency identical
with that of the ultra-violet region when light is emitted, and compare this energy
with the energy of an atom or of a gaseous molecule possessing the mean kinetic
energy characteristic of ordinary temperature (say 300 absolute), (cf. J. Stark,
Zeitsch. physik. Chem., 86, 53, 1913). The frequency v for the ultra-violet region
will be about 5 x io 14 per second. Planck's expression for the mean energy of a
single resonator is, as we have already seen

' U

For ordinary (low) temperatures T = 300 this expression reduces to

hv

D = hve &T, since v is large and T small. The mean energy of an electron
resonator is therefore U, where U = i x io~ 16 ergs. The mean kinetic energy of

a gas molecule at the ordinary temperature = ~T = 6 x io~ 14 ergs, so that instead

of equipartition of energy we find that the energy of the electron vibrating in the
atom or molecule is only one five-hundredth part of the mean kinetic energy of
translation of the molecule itself.



44 A SYSTEM OF PHYSICAL CHEMISTR Y



we see that E, i.e. the total energy density, is ^, that is

4-2 x 7-31 x io 5 ergs

E l0abs = - ' 4 rr = 7-061 X I0~ 15 - .

3 x io 10 (373 4 - 273 4 ) c.c.

pOO pOO

Now E = EA*/X, or u.4v and E\ or u v is given by Planck's

Jo Jo

expression. Using Planck's expression in the form in which u v occurs,
we get



-

the term T being omitted in the final expression, since it is simply
unity. The integration may be effected by series, and we obtain
.finally

Ei'abs. = ^r x 1-0823.

Setting this equal to the "observed" value of E P a b s ., viz.
7-061 x io~ 16 , we obtain

~= 1-1682 x io" .... (3)

Further, Lummer and Pringsheim ( Verh. d. Deutsch. physik. GeselL,
2, 176, 1900) have determined the value of X max . T, where Xmax. is the
wave-length corresponding to the maximum value of E x from a black
body radiating at a given temperature. The expression X max . x T is
a constant independent of temperature as Wien has shown, the numeri-
cal value found by Lummer and Pringsheim being 0-294 cm. degiees.
Now, by differentiating Planck's formula (equation (2)) with respect
to X, and putting the differential equal to zero, when X = X max . we
obtain



whence X max . x T =



4-9651^

h 4-0651 x 0-204 f ^

or -1F^ 2 = 4'866 x lo" 11 . . (4)

R 3 x io

By combining equations (3) and (4) we obtain finally

h = 6-55 x io~ 27 erg/sec.
k = 1-346 x io~ 16 erg/degree.

Millikan (Proc. Nat. Acad. Sa'., 3, 3 T 4 (1917)) gives as the
most accurate value of the charge e on an electron, the quantity
4-774 0-005 x io~ 10 electrostatic units. Knowing this quantity and
the value of e/m, viz. 1-767 x io 7 , and also knowing Rydberg's con-



THE QUANTUM THEORY 45

stant for the Balmer series in the hydrogen spectrum, 1 viz. 3-290 x io 15
according to Millikan, the value of Planck's constant h may be calcu-
lated with a high degree of precision. The value given by Millikan is

h = 6'547 o'oii x io~ 27

which is in close agreement with that "observed" by Millikan (6-56)
and by Webster (6-53).

The Significance of the Constant k, and a Determination of the Number
of Molecules in one gram-molecule.

Consider once more the Planck expression for the energy of vibra-
tion of a large number of similar resonating particles emitting mono-
chromatic radiation, viz.

2 U =

The expression e/ * T - i may be expanded thus
~ + ~( &*) + higher powers.

K 1 2 \K 1 /

If now we are dealing with a system vibrating at very high tempera-
ture it will be seen that the above expression becomes 7^. That is,.

at high temperatures

^ Ne



Exactly the same result is obtained at less high temperatures if the
system is vibrating very slowly, for in this case v is small (relatively),
and since = hv the quantity c is likewise small. In both cases the
quantity e vanishes from the expression for the sum of the energies of
the vibrating particles, this energy being simply proportional to the
absolute temperature. Under these conditions we reobtain the rt suits
of the ordinary kinetic theory, i.e. the principle of equipartition of
energy, it being no longer necessary to consider the energy as other
than continuous. The principle of equipartition of energy is therefore
true as a limiting case for large values of T, and for small values of v.
Suppose that we are dealing with a solid radiating energy at a tempera-
ture sufficiently high that the energy of vibration of the resonators could
be represented by NT. The resonators, as employed by Planck, are
linear, i.e. they possess i degree of freedom, to which one would ascribe
(if the equipartition principle applied, i.e. if T is sufficiently high) ^RT
kinetic and ^RT potential, in all RT units of energy per gram- mole or

1 Cf. Bohr's theory of the atom (Chap. V.), according to which Rydberg's con-
stant =



46 A SYSTEM OF PHYSICAL CHEMISTRY

gram-atom, if a monatomic solid be considered, and if the atoms be
identified with the resonators. 1

Hence, the term 2U reckoned for N " linear " atoms where N is
now regarded as the number of atoms in a gram-atom (or molecules in
a gram-mole) should be identical with RT. That is, RT = NT, or
R = N. The constant k is therefore simply the gas constant reckoned
for a single molecule.

We may see in another way that k has this significance. As has
been pointed out already, Jeans 2 has shown that for very long waves
the law of the partition of energy, or of energy-density between waves
of different lengths, as expressed in a formula, must contain the wave-
length term to the inverse fourth power. A similar formula was de-
duced by Lord Rayleigh on the classical theory, and holds well for
the very long wave- length region. Jeans' equation is

Ex =



where Ry is here the gas constant reckoned for a single molecule. Now
for this very long wave region we have seen that Planck's expression

may be simplified, the term hv _ becoming - or



i b hv ' ' he

That is, Planck's equation becomes



Ex = ^~ - - = S^TX- 4 .
A nc

On comparing this with Jeans' equation, it is at once evident that
the two become identical and they must be identical if they are
equally to reproduce experimental results in the long-wave region if
k is identical with Ry, i.e. k is the gas constant per single molecule.
This may be tested at once by calculating N from the known value R
(per mole), and from the value of k calculated from radiation data,
using Planck's formula. It this way it is found that N = 6-175 x io 23 ,
a number which agrees very well indeed with the values obtained by
Perrin and Millikan (cf. Vol. I., Chap. I.). This "radiation" method
may therefore be regarded as a new and independent method of cal-
culating the Avogadro constant. The value of N just given leads to
a value for the charge on an electron, viz. 4-69 x io~ 10 electrostatic
units, which agrees well with Millikan's value, 4*774 x io~ 10 . 3

1 The actual atoms in solids vibrate with three degrees of freedom. For the
present we are considering "linear" atoms or atoms possessing i degree of
freedom.

2 Jeans, Phil. Mag., [6], 17, 229, 1909.

3 In the above calculation the value of k was obtained from radiation data.
The most accurate because the most direct method of obtaining k is simply to divide
Rby N.

According to Millikan the most accurate value for N is -062 x io 23 .






THE QUANTUM THEORY



47



Numerical Values of the Size of the Quantum e in different Spectral

Regions.

In the following table are collected some values of (hv) extending
over a wide range. The longest wave measured by Rubens, in 1910,
was A, = QO/X, though still longer waves have been measured more
recently. The shortest wave-length measured is of the order O-I/A, or
IOO/A/X (Schumann).



Wave-length A.


'-*r.


,-kv.


A = 90/t .


3-3 X IO 12


2*1 x io 14 ergs


\ = 6fJL


5 x io 13


3-3 x io- 13 ,,


A = 2/u. (This is the region of maxi"|
mum intensity when the body is at j-


1*5 x io 14


9-9 x io- 13 ,,


1646 absolute) . . . . I






A = O'8w. (This is approximately the\
limit of the visible red) . . ./


375 x io 14


2'47 x io 12


A = o'4/*. (This is approximately the\
limit of the visible violet) . . /


7-5 x io 14


4-95 x io- 1 * ,,


A = o'2)ii. (This is the limit obtainable \
with a quartz prism) . . . /


i '5 x io is


9-9 x io 12


A = o'i/x. Schumann rays


3'o x io 15


1-9 x io u



The value of can thus vary from io" 11 to io~ 14 ergs, according to
the vibration frequency of the resonator, i.e. the vibration frequency of
the radiation. As already pointed out, the resonators emitting short
waves are probably electrons, those emitting long waves are probably
the atoms. As regards the total energy emitted from a heated body,
we can neglect the portion due to wave-lengths shorter than o'8/x, i.e.
we can neglect the visible and ultra-violet regions. This is true, of
course, only for bodies the temperature of which is not higher than
1600 C. If the temperature of the radiator be raised much higher
than this, the contribution from the visible may no longer be negligible,
since in accordance with Wien's displacement law the position of maxi-
mum energy emission shifts towards the shorter wave region the higher
the temperature. A body with the temperature of the sun (say
6000 C.), radiating purely thermally (without chemical effects in ad-
dition, as is the case with the sun), would have its maximum emission
of energy at X = o'5/x approximately, i.e. in the blue-green region. It
is of course quite impossible to realise a temperature so high as this
under any experimental conditions.

In connection with Einstein's view of the quantum hypothesis,
namely, the view that radiation itself consists of small units travelling
through space, it is necessary to point out one important phenomenon
which it is very difficult to reconcile with this view. The phenomenon
is that of interference of light, when two beams starting in the same
phase have traversed paths which differ in length. The difficulty of



48 A SYSTEM OF PHYSICAL CHEMISTRY

expecting interference on the Einstein view is very considerable, espe-
cially when it is recalled that interference can be produced as in
certain experiments of Michelson with a difference of path of 40 cms.,
a distance which is enormously great compared with one wave-length.

PLANCK'S LATER QUANTUM HYPOTHESIS.

Planck's Modified Equation for the Energy of an Oscillator.

(Cf. Max Planck, Ber. d. Deut. physik. GeselL, 13, 138, 1911.)

The essential modification here introduced is that while emission
of radiant energy takes place in quanta, and therefore discontinuously,
absorption can take place continuously. Planck was led to consider
this as more probable than the older view, one important reason
having to do with the question of time. Supposing absorption took
place in quanta only, then an oscillator might be exposed to radiation
so weak as not to yield even one quantum, under which condition the
oscillator would absorb no energy at all. This would be especially true
for large values of v (large units). Further, even in those cases in
which absorption could occur, it is conceivable that in weak radiation
there might be a time interval.

On the former view the energy U of an oscillator could be expressed
by U = e, where n is an integer. If absorption is continuous, the
total energy can no longer be regarded as an exact multiple of e, but
could in general be regarded as consisting of nt units plus a quantity p
over. That is

U = e + p.

Now the limits for the value of p are zero and one e. If we are
considering a large number of oscillators p will have an average value

hv

- or , so that we can write

2 2

TT kv

\J = ?l( + .

2

On the older view Planck's equation for U is

u = ^_ i

which evidently correspondsjo e.

The new expression for tJ can therefore be written

hv hv hv

U = -T-71- h =






Planck points out several consequences of this new expression.



PLANCK'S LATER QUANTUM HYPOTHESIS 49

When T = o, U is not = o, but = - This " residual energy " re-
mains with the oscillator even at o absolute. It cannot lose it since
it cannot emit anything less than one hv. " At high temperatures and
for long waves in the region where the Jeans- Rayleigh Law holds the
new formula passes into the old."

For short waves (visible and ultra-violet region) e hvlkr becomes
large compared with unity, and we have

U = -".

2

As regards the question of the specific heats of solids considered in
the following chapter, it is pointed out by Planck that measurements of
specific heat cannot be used to compare the older and newer forms of
U since C v is </U/*/T, and differentiation removes the additional term

. Planck doubts whether the' new expression can be tested experi-
mentally by any direct means. 1

Up to this point the most important experimental evidence which
we have considered in support of the quantum theory is the fact that
Planck's radiation formula quantitatively reproduces experimental values
over the entire range of the spectrum investigated. In the next chapter
we shall deal with further experimental evidence in connection with the
atomic heats of solids which likewise is strongly in favour of the" con-
cept of quanta.

1 Planck refers to a paper by Stark (Phys. Zeitsch., g, 767 (1908)), on the
application of the quantum theory to canal rays. As regards an attempt to dis-
tinguish between the old and new form of the quantum theory, see A. Einstein and
O. Stern (Annalen der Physik, [4], 40, 551, 1913).



VOL. III.



CHAPTER III.

(Physical equilibrium in solids) Theory of atomic heats of solids Equations
of Einstein, Nernst-Lindemann, and Debye.

Einstein s Extension of Planck's Quantum Theory to the Calculation of
Specific Heats of Solids (Crystalline Substances) and Supercooled
Liquids ( * ' Amorphous Solids "). ( Cf. Einstein, Annalen der Physik,
[4], 22, 180, 1907.)

THE specific heat of a substance at constant volume is defined as the
increase in total- energy when the substance rises i in temperature.
Let us take as our unit of mass the gram- mole or gram-atom (in the
case of monatomic substances), and we then can write

CD = .

NOTE. -U here stands for total energy possessed by the substance at
a given temperature. It is not to be confused with the significance
attached to U in the " elementary thermodynamical treatment " (Chap. I.,
Vol. II.), in which " U " stood for decrease in total energy due to chemical
^reaction.

Let us now restrict our attention to solids (or supercooled liquids),
taking as particular instances the metallic elements. These, as we have
seen, are regarded as monatomic, so that the gram-atom and gram-
molecule are identical terms in these cases. Now we want to find out
to what the total or internal energy of a metal is due. It is usual to
regard the solid state as characterised by vibrations of the atoms about
their respective centres of gravity. Such vibrations can, of course, take
place in the three dimensions of space, i.e. each atom possesses three
degrees of freedom. As we have seen, each vibration represents energy,
one-half of which is kinetic, one-half potential, as long as the amplitude
of the vibration is not too great. This vibrational energy is regarded
as representing all the internal energy possessed by the atom, at least
at low temperatures (at high temperatures the energy of vibration of
the electrons inside each atom would have to be considered, but at
ordinary temperature and at lower temperatures the total energy of
the solid may be ascribed to the vibration of the atoms). We have
already discussed this, and we have seen that on applying the principle
of equipartition of energy the atomic heat of metals should be
^R = 5 '9 5 5 cals. per degree, and that this should be independent of
temperature. As already pointed out, this numerical value is certainly

50



A TO MIC HE A TS Of SOLIDS 5 1

approximated to at ordinary temperatures, but instead of being in-
dependent of temperature, it varies, becoming continuously smaller as
the temperature is lowered. The question therefore which arises is
how this variation with temperature is to be accounted for. Einstein
in 1907 made the first successful attempt at the solution of this problem
by suggesting that Planck's quantum theory which Planck himself had
applied with so much success to the problem of the emission of radiant
energy could also be applied to the vibrational energy of the atoms,
i.e. to the total internal energy of the solid, the temperature coefficient
of which is identical with the specific or atomic heat of the substance
in question : Planck's expression for the average energy of vibration of
a linear resonator (i.e. an atom vibrating along one of the dimensions of
space) is, as we seen

_ hv

U = ghvlkt _ r '

[It will be observed that we are employing Planck's earlier hypothesis.]
The energy of vibration of an atom capable of vibrating along the
three dimensions of space will be three times this quantity, and if we
denote this average vibrational energy per gram-atom by U, we get

U, 3Nfr .

gtolkr _ !

The significance of U is identical with that which has been ascribed to
it in Chap. II., Vol. II., namely, the total energy per mole or gram-
atom. It% the case considered one-half of U is kinetic, one-half po-
tential energy. N denotes the number of atoms in one gram-atom.
On Planck's earlier view the vibrational energy possessed by each atom
must be an even multiple of one quantum. On the " classical "
view (" structureless energy," so to speak) we should say that all
possible differences in energy content would manifest themselves in
a system made up of a large number of vibrating particles. On apply-
ing the unitary theory of energy we must recognise that a number of
atoms have no vibrational energy at all, i.e. are at rest. Of those
vibrating the energy content cannot fall below the quantum e', where '
is three times Planck's quantum e. We have, therefore, sets of atoms
containing energy of the following amounts :

so on -



In order to bring the expression for U given above into the form
used by Einstein, Nernst, and others, we shall make a slight change in
the symbols. If we denote the ratio of Planck's two fundamental con-

stants k and h by /Jo, 1 we can write -7 = /3 = 4-87 x io~ n C.G.S. units.

K

R R

Also, since k = ~, where R = i '985 calories, we can write hv = xr/Sov.

1 This is frequently written as j8. The slight change is here introduced to pre-
vent any confusion with 0, one of the terms in Nernst's " heat theorem " equations
of A and U.



A SYSTEM OF PHYSICAL CHEMISTRY
The expression for U then becomes

U-*R. . ft v



It will be observed that this equation differs from the expression
3RT (obtained by applying the (^/partition principle, energy being

regarded as continuous), by the substitution of the term -jjjj- - in

place of T.

Differentiating U with respect to T we obtain Einstein's equation
for the atomic (or molecular) heat of a solid at constant volume, viz



On the " classical " view, the factor multiplied by 3 R would have been

1;




i-o



0-5

FIG. 5.

unity. The " correction " term will be seen to contain T and is there-
fore a function of temperature. On Einstein's theory one would expect
the atomic heat itself to be a function of temperature, as is experimentally
the case. Qualitatively, therefore, this marks a considerable advance over
the older theory. It still remains to be seen whether the expression
given really reproduces the values of C v at various temperatures quantita-
tively. The shape of the curve for -=, as given by Einstein's formula

is shown in the accompanying figure (Fig. 5), in which the ordinates
represent atomic heats and the abscissae the temperature expressed in

T
terms of -5 . For any given substance the vibration frequency is taken

to be a constant.



It will be seen from the figure that when -= >o-o the term



ATOMIC HEATS OF SOLIDS



53



approximates to unity, so that the atomic heat becomes



equal to 3R. Dulong and Petit's Law is, therefore, a consequence of
Einstein's theory when the temperature is not too low. This region of
temperature is evidently reached at room temperature in the case of
the majority of metallic elements. As in radiation phenomena we see
that the equipartition principle applied in the ordinary way yields results
(in connection with atomic heat) which are in agreement with experi-
ment when the temperature reaches a certain magnitude. Einstein
tested his equation not on the data available in the case of a metal but
on the diamond. Some of the results are given in the subjoined table.
They are also indicated by circles in the figure. Einstein chose the
experimentally determined value for C w at T = 331*3 and hence
calculated v, using the value so obtained to calculate the values of C v at
other temperatures. The experimental data (quoted) refer, as a matter
of fact, to C/,, i.e. the atomic heat at constant pressure. In the case of
the diamond though not so in. other cases the difference between
Cp and Ct, is small. In the following section we shall consider the
question of the independent determination of v and C v respectively.

ATOMIC HEAT OF CARBON (DIAMOND).



T (absolute).


T//V-


C^ calculated from
Einstein's equation.


Cfr observed
(Weber).


222'4


0-1679


0-762


0-76


262*4


0-1980


1-146


I-I 4


2837


0-2141


1-354


1-35


306-4


0*2312


I-582


I- 5 8


331*3


0*2500


1-838


I-8 4


358-5


0-2705


2-118


2-12


413-0


0-3II7


2-661


2-66


479-0


0-3615


3-280


3-28


520-0


0-3924


3'63I


3-63


8797


0*6638


5-290


5-29


1079-7


0-8147


5-387


5 '39


1258-0


0-9493


5-507


5'5i



Measurements of C ? of the diamond at low temperatures down to
abs. have been carried out by Nernst, who discovered the remark-
able fact that the thermal capacity of this substance tends practically to
zero, even at the temperature + 50 abs. Between this temperature

and absolute zero, C v or = o. We have already had occasion to






point this out in connection with Nernst's heat theorem. From the
shape of Einstein's curve (Fig. 5), it will be seen that Cv is tending
towards zero at a temperature higher than o abs. It thus appears that,



54 A SYSTEM OF PHYSICAL CHEMISTRY

as far as the diamond is concerned, Einstein's theory reproduces experi-
mental values with very considerable fidelity. While giving full weight
to such general agreement, it is necessary to point out that this agree-
ment is far from being complete in many other cases. We shall return
to this after having described in outline the experimental methods of
determining C r , and the characteristic vibration frequency v.

Experimental Measurements of the Specific Heats of Solids, especially
at Low Temperature.

(Cf. Nernst, Journ. de Physique, [4], 9, 1910; Nernst, Koref, and
Lindemann, Sitzungsber. Berl. Akad., 1910, vol. i, p, 247 ; ibid., Nernst,


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