why is the molar specific heat of carbon(=6.1JMol-1K-1) so different from the predicted value of 3R≈25??
The deviation is far from insignificant, as at low temperature the heat capacity has to go to zero. And what can be called "low" temperature is very relative. At room temperature, carbon (be it diamond or graphite) is far from the asymptotic limit given by the Dulong-Petit law.strictly speaking it does depend on temperature, but is often ignored due to the insignificance of the deviation.
The Dulong-Petit law works if you can apply the equipartition theorem, that is if all quadratic degrees of freedom have an average energy ##\langle E \rangle = k_B T / 2##. Since vibration is quantized, this can only be the case for the vibrational modes if there is enough energy to significantly populate excited states. Some solids have such a high threshold that you need to go very high temperatures before you have sufficient excitation and can neglect the discrete (quantized) aspect of vibrational energy.Yes but why?? carbon and even Beryllium don't go by the Dulong Petit law for specific heat(molar) to be 3R. at room temp.
Everywhere they say, 'due to their high energy vibrational modes not being populated at room temperature' ?