High and Low Temperature Limits Debye Specific Heat
The energy expression from the Debye theory of specific heat is of the form
where TD = Debye temperature.
Even though this integral cannot be evaluated in closed form, the low and high temperature limits can be assessed.
For the high temperature case where T>>TD, the value of x is very small throughtout the range of the integral.
This justifies using the approximation to the exponential from the exponential series,
ex = 1 + x. This reduces the energy expression to
which is the Dulong-Petit result from classical thermodynamics.
For low temperatures where T
Then energy is then
and the specific heat is
This T3 dependence of the specific heat at very low temperatures agrees with experiment for nonmetals. For metals the electron specific heat becomes significant at low temperatures and is combined with the above lattice specific heat in the Einstein-Debye specific heat.
ليست هناك تعليقات:
إرسال تعليق
ملحوظة: يمكن لأعضاء المدونة فقط إرسال تعليق.