# Why is the molar specific heat capacity of carbon unusually low?

Most of the solids approximately obey Dulong - Petit law, which says that the molar specific heat of a solid is $3R \approx 24.94 \frac{\mathrm{J}}{\mathrm{K}\cdot \mathrm{mol}}$, where $R$ is the gas constant, near room temperature and atmospheric pressure.

While this appears to hold for most of the solids, I noticed that carbon has an anomalous value of $6.1 \frac{\mathrm{J}}{\mathrm{K}\cdot \mathrm{mol}}$:
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Question: Why is the molar specific heat capacity of carbon so low?

• The low specific heat of diamond was explained by Einstein as a quantum phenomenon. Low mass together with strong bonds gives high vibrational frequencies with quanta larger than $k_BT$ at room temperature.
– user137289
Commented Oct 8, 2017 at 7:46
• Can I get any reference in support of this? Commented Oct 11, 2017 at 4:08
• It is in any textbook of solid state physics. This is a chapter from Tipler's Modern Physics: bcs.whfreeman.com/webpub/Ektron/Tipler%20Modern%20Physics%206e/…
– user137289
Commented Oct 11, 2017 at 6:32

The value is for diamond. Carbon atoms in Diamond has tightly bond to each other witch result in a hight Einstein frequency. So that the Dulong Petit law only is vailid for higher temperatur than room temperature.

What I think is diamond has C atoms closely packed, inhibiting expected vibration about the mean position for each atom. As quantum mechanical approach requires a minimum non zero amount of energy before a degree of freedom comes into play, we would not have the expected amount of KT. Subsequently, dU would be less than 3RT, and C=dU/dT = less than 3R Let me know what you think