Classical Theory of Specific Heat (Dulong–Petit Law)

Introduction

The classical theory of specific heat of solids was proposed by Dulong and Petit in 1819.

It explains the molar specific heat of crystalline solids at high temperatures using classical statistical mechanics.

Basic Assumptions

A solid consists of N atoms arranged in a lattice.
Each atom vibrates about its equilibrium position.
Each atom behaves like a three-dimensional harmonic oscillator.
Equipartition theorem is valid.

Energy of One Harmonic Oscillator

For one atom vibrating in one direction:

\[ E = \frac{p^2}{2m} + \frac{1}{2} k x^2 \]

There are two quadratic terms:

Kinetic energy term
Potential energy term

By equipartition theorem, each quadratic term contributes:

\[ \frac{1}{2} k_B T \]

Thus, energy per direction:

\[ E = k_B T \]

Energy per Atom

Since each atom vibrates in three dimensions:

\[ E_{atom} = 3 k_B T \]

Total Energy of Solid

For N atoms:

\[ U = 3 N k_B T \]

Specific Heat at Constant Volume

Specific heat is defined as:

\[ C_V = \left( \frac{\partial U}{\partial T} \right)_V \]

Therefore,

\[ C_V = 3 N k_B \]

For one mole (N = N_A):

\[ C_V = 3 N_A k_B \]

Since $N_A k_B = R$,

\[ C_V = 3 R \]

Dulong–Petit Law

The molar specific heat of a solid at high temperature is approximately:

\[ C_V \approx 3 R \approx 25 \, \text{J mol}^{-1} \text{K}^{-1} \]

Success of the Theory

Correctly predicts specific heat of many solids at high temperature.
Simple and based on equipartition theorem.

Failure of Classical Theory

Experimental observations show:

At low temperatures, $C_V$ decreases sharply.
As $T \to 0$, $C_V \to 0$.

Classical theory predicts constant $C_V = 3R$ at all temperatures, which contradicts experiments at low temperature.

This failure led to quantum theories of specific heat (Einstein and Debye models).

Important Result

Dulong–Petit law is valid only in the high-temperature (classical) limit. Quantum effects dominate at low temperatures.

WHAT are PHONONS

Phonon is the quantum of lattice vibrational energy (elastic wave).
It has integral spin.
It obeys Bose-Einstein statistics and is made of indistinguishable particles.
Phonons travel with the speed of sound in a solid, unlike photons which travel at the speed of light.
The number of phonons is not conserved; phonons can be created and destroyed.
They are neutral.
Phonons exhibit both wave and particle characteristics.
Vibrational spectrum of a phonon varies in the range $ 10^4 $ to $ 10^{12} $ Hz.
Energy of a phonon is given by: $ E = \hbar \omega $
Phonons play a major role in many physical properties of condensed matter, such as thermal and electrical conductivity. The study of phonons is a crucial part of condensed matter physics.

(The concept of the phonon was introduced by Igor Tamm (1895–1971, Vladivostok, Russia) in 1932, for which he was awarded the Nobel Prize in 1958.)

Classical Theory (Doulong and Petit's Law ) 1819

Specific heat is same for all the elementary solids which is 6 calories or 3 R at room tmperature .Specific heat capacity is defined as the amount of heat requred to raise the temperature of a uniit mass of body through a temperature difference of one \href{http://unit.In}{unit.In} solids while ernergy is given it is spent by the atoms which vibrate about lattice simple harmonically and the remaining part by the free electrons to move to the excited states. As the energy abosrbed by the electrons is very small . Thus the Heat capacity of the solid is due to atomic vibrations at the lattice sites only.

Atoms in a solid vibrate freely about their mean positions like harmonic oscillators.
All the oscillators vibrate with the same frequency.
Their energies are different as they vibrate with different amplitudes.
The energy of an oscillator can take values from 0 to infinity.
The internal energy of a solid is due to the sum of vibrational motion of atoms and thermal excitation of electrons to higher energy states.
\[ U_{\text{solid}} = U_{\text{lattice}} + U_{\text{electron}} \] Thus, \[ C_V = \left( \frac{dU}{dt} \right)_V = \frac{dU_{\text{lattice}}}{dt} + \frac{dU_{\text{electron}}}{dt} \] \[ = C_{\text{lattice}} + C_{\text{electron}} \] The contribution from electrons is usually very small and can often be neglected.

$$ C_{V}=C_{\text {lattice }} $$

DERIVATION OF SPECIFIC HEAT :( DOULONG-PETIT-CLASSICAL METHOD

Let us derive the MEAN ENERGY of the solid first.

Since each atom is a simple harmonic oscillator. It's Total energy is $$ E=\frac{p^{2}}{2 m}+\frac{1}{2} m \omega^{2} q^{2}=f(p)+f(q) $$ Since $n(E)=e^{-\alpha-\beta E}=A e^{-\beta E}$ Applying Boltzman Distribution of energy, Average Energy of the oscillator is :

$$ \begin{aligned} \Rightarrow\langle E\rangle&=\frac{\int E e^{-\frac{E}{k T}} d E}{\int e^{-\frac{E}{k T}} d E}=\frac{\int(f(p)+f(q)) e^{-\left(\frac{f(p)+f(q)}{k T}\right)} d p d q}{\int e^{-\left(\frac{f(p)+f(q)}{k T}\right)} d p d q} \\ & =\frac{\int f(p) e^{-\frac{f(p)}{k T}} d p \int e^{-\frac{f(q)}{k T}} d q \int f(q) e^{-\frac{f(q)}{k T}} d q \int e^{-\frac{f(p)}{k T}} d p}{\int e^{-\frac{f(p)}{k T}} d p \int e^{-\frac{f(q)}{k T}} d q}\\ & =\frac{\int f(p) e^{-\frac{f(p)}{k T}}}{\int e^{-\frac{f(p)}{k T}} d p}+\frac{\int f(q) e^{-\frac{f(q)}{k T}} d q}{\int e^{-\frac{f(q)}{k T}} d q} \\ & =\frac{\int_{-\infty}^{\infty} \frac{p^{2}}{2 m} e^{-\frac{p^{2}}{2 m k T}} d p}{\int_{-\infty}^{\infty} e^{-\frac{p^{2}}{2 m k T}} d p}+\frac{\int_{-\infty}^{\infty}\left(\frac{1}{2} m \omega^{2} q^{2}\right) e^{-\frac{m \omega^{2} q^{2}}{2 k T}} d q}{\int_{-\infty}^{\infty} e^{-\frac{m \omega^{2} q^{2}}{2 k T}} d q} \\ & =\frac{k T \int_{-\infty}^{\infty} x^{2} e^{-x^{2}} d x}{\int_{-\infty}^{\infty} e^{-x^{2}} d x}+\frac{k T \int_{-\infty}^{\infty}\left(y^{2}\right) e^{-y^{2}} d y}{\int_{-\infty}^{\infty} e^{-y^{2}} d y} \quad\left(\text { taking } x^{2}=\frac{p^{2}}{2 m k T}, y^{2}=\frac{m \omega^{2} q^{2}}{2 k T}\right) \\ & =k T \frac{\frac{1}{2} \sqrt{\pi}}{\sqrt{\pi}}+k T \frac{\frac{1}{2} \sqrt{\pi}}{\sqrt{\pi}} \quad\left(\because \int_{-\infty}^{\infty} e^{-a x^{2}} d x=\sqrt{\frac{\pi}{a}}, \int_{-\infty}^{\infty} x^{2} e^{-a x^{2}} d x=\frac{1}{2 a} \sqrt{\frac{\pi}{a}}\right) \\ & =\frac{k T}{2}+\frac{k T}{2}=k T \\ \Rightarrow E_{\text {total }}&=3 \mathrm{NkT} \end{aligned} $$

Specific heat of the solid is :

\[ \begin{align} C_V &= \frac{dE}{dT} \\ &= \frac{d}{dT}(3NkT) \\ &= 3Nk = 3R \end{align} \]

CONCLUSION:

Specific heat is constant at all temperatures and is equal to $ 25~\mathrm{J/mol{\cdot}K} $.
Later, it was found to be $ 3R $, as $ R $ had not been defined at that time.
This law is valid at high temperatures for most solids.
It is valid for elements of atomic weight greater than 40

Limitations:

It fails for light elements such as Boron.Beryilium, Carbon or diamond at low temperature 2. $\mathrm{C}_{\mathrm{v}}$ apprcoches to zero at low temperature found experimentaly where it is not so .