## (1) Specific Heat :

Specific Heat: According to the first law of thermodynamics, when a certain quantity of heat is subjected to a system, it is used in two parts. The first part is used in increasing the internal the internal energy of the system. Which is manifested by a rise in its temperature. The second part is used in furnishing the work to the system when it does in the process of its expansion during heating. If dQ is the absorbed energy of the system, if an internal energy is increased by dU and external work done in dW then the first law can be written as

dQ = dU + dW

But external work dW = PdV

∴ dQ = dU + PdV

If the temperature is increased by dT by the energy absorped in the system then due to unit mass of the system, the increase in temperature per kelvin for the amount of heat absorbed is dQ/dT and is known as specific heat of system.

∴ Specific Heat c = dQ/dT

Generally there are two types of specific heat.

- Specific heat at constant volume Cv = (dQ/dT)v = (dU/dT)v
- Specific heat at constant pressure Cp = (dQ/dT)p

### (2) Molar specific heat :

The necessary amount of heat dQ for one kilo mole of the substance the increase temperature dT is given by

Molar specific heat C = (dQ/dT) = Mc

Where M is the molecular of substance.

If volume is constant during the increase of temperature then molar specific heat at constant volume

Cv = (dQ/dT)v = (dU/dT)v

If the pressure is constant during the increase of temperature then molar specific heat of constant pressure,

Cp= (dQ/dT)p

Generally for most of the solids show negligible change in volume with respect to increase of temperature that why difference between the two specific heat is a these solid is quite low.

#### (3) Lattice specific heat :

When certain thermal energy is added to solid, then ther increase in its internal energy and temperature. The increase in internal energy of a solid may occur in two ways :

- The atoms assumed to be free to vibrate about their equilibrium positions are drawn into vigorous vibration.
- The free electron may be excited thermally to higher energy states.

Thus there are two contributions of the internal energy of solids with respect to atomic vibration in the crystal and free electron thermal excited in the crystal is Ulattice and Uelectric respectively, then

Internal energy of solids U = Ulattice + Uelectronic

∴ Specific heat by solids at constant volume

Cv = dU/dT = dUlattice /dT + dUelectronic/ dT

And, Cv = Cvlattice + Cvelectronic

Cv is known as lattice specific heat of solid lattice or lattice heat capacity. Generally from experimental from observed that no free electrons are present and the specific heat of a crystal is only due to the excitation of thermal vibrations in the lattices i.e., only the lattice specific heat is t considered. Therefore mostly Specific heat of solids.

Cv = Cvlattice

##### (4) Dulong and Petit law :

According to classical theory, for solids atoms are freely viberate about the mean position. For one dimensional vibrational motion is a quadratic equation of the two realtive variables (velocity and displacement respectively) of kinetic and potential energy. According to the theory of energy of equitirtition for the average kinetic energy is equal to (1/2)KbT and that the average potential energy is kbT/2 per degree of freedom. Where Kb is Boltzmann constant and T is the temperature of solid. Hence corresponding average energy of one dimensional vibrational motion is (KbT/2)+ (KbT/2) = KbT. But vibrational of atoms is three dimensional. Hence average energy per atoms of solid is equal to 3KbT.

Average energy for the one mole of substance

U= 3 N₀KbT = 3RT

Where N is the Avagadro’s number and R is the gas constant. Thus lattice specific heat of solids

Cvlattice = dU/dT =3R

This formula has been firstly found by Dulong and Petit in 1819. And so it is known as Dulong and Petit law. Dulong and Petit law is generally correct at room temperature. And above for elements of atomic weight greater than Fourty. But experimental it is observed that as the temperature approaches absolute zero, the specific heat of simple solids approach zero as shown in fig. According to Dulong and Petit law, at high temperature must of the solids the molar specific heat is equal to 3R. At normal temperature specific heat of light element is less than 3R. Temperature between zero from 20K.