A system of physical chemistry - Index of

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A system of physical chemistry - Index of

ATOMIC HEATS OF SOLIDS 51

approximated to at ordinary temperatures, but instead of being independent

of temperature, it varies, becoming continuously smaller as

the temperature is lowered. The question therefore which arises is

how this variation with is temperature 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

[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 —

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 gramatom.

In the case considered one-half of U is kinetic, one-half potential

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 applying

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 c'

is three times Planck's quantum t. We have, therefore, sets of atoms

containing energy of the following amounts :—

o, c', 2c', 3€', and 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 k by /^q,^ we can write 7 = /3a = 487

R

x io~" C.G.S. units.

Also, since ^ = y;^, where R = i -985 calories, we can write Ai' = ^^o^-

1 This is frequently written as 0. The slight change is here introduced to prevent

any confusion with j3, one of the terms in Nernst's " heat theorem " equations

of A and U.

R

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