EINSTEIN'S THEORY OF HEAT CAPACITY OF SOLIDS

Assumption :

1) An ideal crystal can be considered as a system of N non- interacting particles, (atoms, molecules or ions) One mole of crystal will contain L atoms or molecules or ions exhibiting simple harmonic motion. It's assumed that there is no interaction between the oscillator and all the oscillators have isotopic frequency. In such an ideal crystals each atom has three independent vibrational degree of freedom. One mole of crystal have 3L independent distinguishable harmonic oscillators.

2) Using Planck's quantum theory, the average energy of a vibrational mode is given as

                E =     E       

                      eE/kT-1

3) For 3L vibrational modes,

E total = 3L (        E         )   (∴ E = hυ) 

                       eE/kT - 1

     = 3L (          hv.      )

                 ehυ/kT - 1

4) For one mole of solid,

Cv = (∂E)v = 3Lhʋ (−          1            )²

                                  ehʋ/kT−1

ehʋ/kT   hʋ     (−   1  )

                k            T²

=   3Lh²ʋ²   [    ehʋ/kT          ]

      kT²           (ehʋ/kT−1)²

Cv = 3R (hʋ)²   [    ehʋ/kT         ]

               kT        (ehʋ/kT−1)²

= 3R (  θE)²      [   eθE/T              ]

             T           (eθE/T − 1)²

Where θξ =  hʋ  , the characteristic 

                      k

Einstein temperature for vibration. This theory explains the variation of Cv with temperature and at high temperature 

Cv →3R, classical theory value.

Where θE =  hʋ , the characteristic

                       V

Einstein temperature of vibration. This theory explains the variation of Cv with temperature and at higher temperature Cv →3R, classical theory value.

Limitations :

                          Einstein's theory is not successful in predicting Cv, values in the lower and intermediate temperature range. The values predicted are lower than these actually observed.