Derive the specific heat of solid using Einstein’s method and explain the result withnecessary graph

Einstein's model of specific heat of solids is based on the quantum theory of energy. According to this model, each atom in a solid is considered as an independent 3D quantum harmonic oscillator.

Step 1: Assumptions Einstein assumed that all the atoms in a solid vibrate with the same frequency (unlike Debye's model which considers a range of frequencies). He also assumed that these vibrations are quantized, meaning the energy of each oscillator (E) is given by E = nhf, where n is an integer, h is Planck's constant, and f is the frequency of the oscillator.

Step 2: Derivation The average energy of an oscillator is given by the formula E_avg = Σ(E*e^(-E/kT))/Z, where Z is the partition function given by Z = Σ(e^(-E/kT)). Substituting E = nhf into these equations and simplifying, we get E_avg = hf/(e^(hf/kT) - 1).

Step 3: Specific Heat The specific heat at constant volume (Cv) is given by the derivative of the average energy with respect to temperature. Differentiating E_avg with respect to T, we get Cv = (hf/kT^2)*(e^(hf/kT))/((e^(hf/kT) - 1)^2).

Step 4: High and Low Temperature Limits At high temperatures (hf << kT), the specific heat approaches the classical limit of 3k (Dulong-Petit law). At low temperatures (hf >> kT), the specific heat goes to zero exponentially fast.

Step 5: Einstein Temperature The temperature at which the specific heat is maximum is called the Einstein temperature (TE). It is given by TE = hf/k.

Step 6: Graph The graph of Cv vs T shows a peak at the Einstein temperature. At low temperatures, Cv increases rapidly with T, reaches a maximum at TE, and then decreases slowly to approach the classical limit of 3k.

This model, while not completely accurate (since it assumes all atoms vibrate with the same frequency), was a significant improvement over the classical theory and paved the way for the more accurate Debye's model.