# Molar specific heat of Carbon

Gold Member
why is the molar specific heat of carbon(=6.1JMol-1K-1) so different from the predicted value of 3R≈25??

DrClaude
Mentor
Is the specific heat independent of temperature?

Gold Member
Is the specific heat independent of temperature?
strictly speaking it does depend on temperature, but is often ignored due to the insignificance of the deviation.

DrClaude
Mentor
strictly speaking it does depend on temperature, but is often ignored due to the insignificance of the deviation.
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.

Gold Member
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' ?

DrClaude
Mentor
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' ?
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.

Gold Member
Oh okay, now i get it. So then, is there any way to find the molar specific heat capacity of carbon, theoretically ??

DrClaude
Mentor
Oh okay, now i get it. So then, is there any way to find the molar specific heat capacity of carbon, theoretically ??
Not that I know. You can do it empirically, by finding a function that fits the observed heat capacity, or computationally.

Gold Member
ohh! okay, thank you for your help.

The specific heat goes to the Dulong-Petit limit at "high temperature".
You can think in terms of room temperature not being a "high temperature" for diamond. This is suggested for example by the value of Debye temperature, which is over 2000 K. For metals the same value is just a few hundred K.