# STATISTICAL MECHANICS | Einstein’s theory of specific heat | Debye theory of specific heat

## Einstein’s theory of specific heat

The theory of the specific heat of solids first proposed by Albert Einstein in 1906. In this theory, Einstein assigned the specific heat of solids to the vibrations of the solid and made the simplified assumption that all the vibrations will have the same frequency. Since this was able to derive Dulong and Petit’s law at high temperatures and showed that the specific heat capacity goes to zero as the absolute temperature also goes to zero, this theory was partially successful.

A mono atomic solid consists of a large number of atoms in regular way array, a lattice. Because of thermal agitation each atom vibrates about its mean equilibrium position.

Dulong and Petit  was proved that average energy of each atom (oscillator)  = 3kT

Total energy for N- atoms, E = 3NkT,  where N = Average no.

So, Molar specific heat of solid Hence molar specific heat of all solids is constant and independent of temperature and has value equal to 3R = 5.96 Cal / mole) · This is known as Dulong – Petit’s law.

Though, Dulong- Petit’s law says that molar specific heat capacity of all solids is constant and independent of temperature; it is noticed that, the specific heat of most materials decreases as the temperature is lowered . Classical statistical mechanics fails to explain this variation of the specific heat with temperature mainly lower temperature region. Einstein was the first who was applied quantum concept to explain the theory of specific heat of solids.

According to Einstein –

i) The atoms in a solid are all independent and each atom acts as a simple harmonic oscillator wilt a common frequency ν.

ii) Each atom of a solid has 3 degrees of freedom like mono atomic gas.

iii) Mean energy per degrees of freedom of an oscillator (Planck’s law)

iv) For different solid frequency ν value are different.

Since the Oscillators are considered as boson, each oscillator (atom) has mean energy per degrees of freedom according to Planck’s law.

So, Energy content of a gram-atom of solid consisting of N-atoms is, Gram – atomic specific heat at constant volume, Einstein defined a characteristic temperature as called Einstein characteristic temperature.

Then ………………… 1

Above equation shows that Cv is function of temperature (T)

Case I – At high temperature then  (approximately constant)

This indicates that Dulong and Petit’s law is an an asymptotic law valid only in high temperature range.

Case II – At low temperature  Specific heat, therefore falls exponentially fast rate and Cv → 0 at T → 0 (should approach zero)

## Debye theory of specific heat

### Defects of Einstein theory

Einstein’s theory explains

the temperature variation of specific heat of solids in the very law as well as very high temperature regions.

Although the nature of Einstein’s curve is similar to the experimental curve but there is a deviation in the low temperature region.

when then which indicates that Cv decreases very rapidly. But experimentally, at low temperature specific heat is proportional to T3. This defect of Einstein’s theory was removed by ” Debye”.

To explain the quantum theory of specific heat of solid, Debye make the following assumptions –

i) A crystalline solid is arranged by atoms periodically in 3D Space. They vibrate about their mean equilibrium position due to thermal excitation.

ii) Mean energy of each atom per each degrees of freedom is equal to iii) The atoms are coupled each other . Due to presence of mutual forces their vibration will not be independent to each other. Frequency of each vibration is not same but lies from zero to a certain frequency (νD); called Debye cut off frequency.

When a continuous solid is subjected to elastic vibration two kinds of vibrations are produced

1. Longitudinal vibrations
2. Transverse vibrations

If CL be the velocity of longitudinal wave then the number of modes of longitudinal vibrations per unit volume with frequencies between ν and ν+dν ……………… 1 (from Rayleigh – Jeans law)

If G be the velocity of transverse wave, then the no. of modes of transverse vibrations per unit volume with frequencies between ν and ν+dν …………………. 2

2 is multiplied as there are two transverse modes (y, z).

Therefore, total number of independent mode of vibrations per unit volume with frequencies ν and ν+dν ……………… 3

If V be the volume of 1 mole of solid; then total number of independent mode of vibrations of 1 mole of solid with frequencies ν and ν+dν ………………… 4

Debye assumes that this result holds for all the frequencies from the lowest  ν = 0  to the highest possible ν = νD,

called Debye the frequency characteristic of the substance.

On this assumption, as there are N atoms in 1 mole of solid each having 3 degrees of freedom. …………… 5

Debye assumes that mean energy of each vibration has the quantum value Total energy of one mole of solid  …………………. 6

Molar specific heat at constant volume,    ………………….. 7

#### I. At high temperature :

T → large → hν/kT = x is small, thus  #### II. At low temperature :

T → small → hν/kT = x is large, also  is called Debye temperature. This is known as Debye’s T^3 law. 