Einstein's theory of heat capacity

Einstein's quantum theory of heat capacity was created by Einstein in 1907 in an attempt to explain the experimentally observed temperature dependence of heat capacity .

In developing the theory, Einstein relied on the following assumptions:

Atoms in the crystal lattice behave like harmonic oscillators that do not interact with each other.
The oscillation frequency of all oscillators is the same and equal $\nu =\omega /2\pi$ ${\ displaystyle \ nu = \ omega / 2 \ pi}$ ${\ displaystyle \ nu = \ omega / 2 \ pi}$ .
The number of oscillators in 1 mole of matter is $3N_{a}$ ${\ displaystyle 3N_ {a}}$ ${\ displaystyle 3N_ {a}}$ where $N_{a}$ ${\ displaystyle N_ {a}}$ ${\ displaystyle N_ {a}}$ - Avogadro number .
The energy of their quantization: $\varepsilon =n\hbar \omega$ ${\ displaystyle \ varepsilon = n \ hbar \ omega}$ ${\ displaystyle \ varepsilon = n \ hbar \ omega}$ where $n\in N$ ${\ displaystyle n \ in N}$ ${\ displaystyle n \ in N}$ , $\hbar$ ${\ displaystyle \ hbar}$ $\ hbar$ - the reduced Planck constant (Dirac constant) .
The number of oscillators with different energies is determined by the Boltzmann distribution

N_{n}=N_{0}\exp \left(-{\hbar \omega  \over kT}n\right),

{\ displaystyle N_ {n} = N_ {0} \ exp \ left (- {\ hbar \ omega \ over kT} n \ right),}

{\ displaystyle N_ {n} = N_ {0} \ exp \ left (- {\ hbar \ omega \ over kT} n \ right),}

Where $k$ ${\ displaystyle k}$ $k$ - Boltzmann constant $T$ ${\ displaystyle T}$ $T$ - thermodynamic temperature .

Internal energy of 1 mole of substance:

{\bar {U}}_{\mu }=3{\bar {\varepsilon }}N_{a}.

{\ displaystyle {\ bar {U}} _ {\ mu} = 3 {\ bar {\ varepsilon}} N_ {a}.}

{\ displaystyle {\ bar {U}} _ {\ mu} = 3 {\ bar {\ varepsilon}} N_ {a}.}

The average energy of one oscillator ${\bar {\varepsilon }}$ ${\ displaystyle {\ bar {\ varepsilon}}}$ ${\ displaystyle {\ bar {\ varepsilon}}}$ is found from the ratio for the average value:

{\bar {\varepsilon }}={\sum _{n=0}^{\infty }{\varepsilon _{n}N_{n}} \over \sum _{n=0}^{\infty }{\ N_{n}}}

{\ displaystyle {\ bar {\ varepsilon}} = {\ sum _ {n = 0} ^ {\ infty} {\ varepsilon _ {n} N_ {n}} \ over \ sum _ {n = 0} ^ { \ infty} {\ N_ {n}}}}

{\ displaystyle {\ bar {\ varepsilon}} = {\ sum _ {n = 0} ^ {\ infty} {\ varepsilon _ {n} N_ {n}} \ over \ sum _ {n = 0} ^ { \ infty} {\ N_ {n}}}}

and is:

{\bar {\varepsilon }}={\hbar \omega  \over \exp \left({\hbar \omega  \over kT}\right)-1},

{\ displaystyle {\ bar {\ varepsilon}} = {\ hbar \ omega \ over \ exp \ left ({\ hbar \ omega \ over kT} \ right) -1},}

{\ displaystyle {\ bar {\ varepsilon}} = {\ hbar \ omega \ over \ exp \ left ({\ hbar \ omega \ over kT} \ right) -1},}

from here:

{\bar {U}}_{\mu }=3N_{a}\hbar \omega {1 \over \exp \left({\hbar \omega  \over kT}\right)-1}.

{\ displaystyle {\ bar {U}} _ {\ mu} = 3N_ {a} \ hbar \ omega {1 \ over \ exp \ left ({\ hbar \ omega \ over kT} \ right) -1}.}

{\ displaystyle {\ bar {U}} _ {\ mu} = 3N_ {a} \ hbar \ omega {1 \ over \ exp \ left ({\ hbar \ omega \ over kT} \ right) -1}.}

Defining heat capacity as a derivative of internal energy with respect to temperature, we obtain the final formula for heat capacity:

C={dU \over dT}=3R\left({\hbar \omega  \over kT}\right)^{2}{\exp \left({\hbar \omega  \over kT}\right) \over \left(\exp \left({\hbar \omega  \over kT}\right)-1\right)^{2}}.

{\ displaystyle C = {dU \ over dT} = 3R \ left ({\ hbar \ omega \ over kT} \ right) ^ {2} {\ exp \ left ({\ hbar \ omega \ over kT} \ right) \ over \ left (\ exp \ left ({\ hbar \ omega \ over kT} \ right) -1 \ right) ^ {2}}.}

{\ displaystyle C = {dU \ over dT} = 3R \ left ({\ hbar \ omega \ over kT} \ right) ^ {2} {\ exp \ left ({\ hbar \ omega \ over kT} \ right) \ over \ left (\ exp \ left ({\ hbar \ omega \ over kT} \ right) -1 \ right) ^ {2}}.}

According to the model proposed by Einstein, at absolute zero temperature the heat capacity tends to zero, at high temperatures, on the contrary, the Dulong – Petit law is satisfied. Value $\theta _{E}={\hbar \omega \over k}$ ${\ displaystyle \ theta _ {E} = {\ hbar \ omega \ over k}}$ ${\ displaystyle \ theta _ {E} = {\ hbar \ omega \ over k}}$ sometimes called Einstein's temperature .

Theory Deficiencies

The divergence of the theories of Einstein and Debye

Einstein's theory, however, does not agree well with experimental results due to the inaccuracy of some of Einstein's assumptions, in particular, the assumption that the oscillation frequencies of all oscillators are equal. A more accurate theory was created by Debye in 1912 .

Sources

Sivukhin D.V. General course of physics. - T. II. Thermodynamics and molecular physics.

Einstein's theory of heat capacity

Theory Deficiencies

See also

Sources

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