# What does zero heat capacity mean?

What is the physical meaning of 0 heat capacity? Does it mean that no matter how much heat I put in, I cannot raise the temperature? If so, does this means that if a system is at absolute zero, it cannot be raised in temperature?

Zero heat capacity means an infinitesimally small amount of energy will increase the temperature by an infinitely large amount.

An object has a non-zero heat capacity because it has internal degrees of freedom that it can channel absorbed energy into. The more internal degrees of freedom the higher the heat capacity.

Zero heat capacity emerges as the zero temperature limit of an Einstein solid. We get this limit because the energy spacing of quantum states has to be less than $kT$ for those states to be excited and participate in determining the specific heat. As $T \rightarrow 0$ the spacing is greater than $kT$, and the end result is the solid has no accessible degrees of freedom and therefore a zero heat capacity.

However the moment you add the smallest amount of heat the temperature increases, some states become accessible and the specific heat becomes non-zero. The zero specific heat is more of an idealised limit than a real phenomenon.

• So does this mean that no physical system can have zero heat capacity at $T>0$? – JonTrav1 Dec 3 '16 at 8:32
• Another way to say the same thing is that if $C=0$, when you raise the temperature the energy remains constant, because $kT$ is still less than $\epsilon$, where $\epsilon$ is the energy difference between the first excited and ground state. Therefore, as John correctly said, no transition can occur and the system stays in the ground state. This is a purely quantum phenomenon: classical systems have $C>0$ at $T=0$ (es. the Dulong-Petit model). – valerio Dec 3 '16 at 8:51
• Any system undergoing an isothermal change with associated heat transfer? – Farcher Dec 3 '16 at 12:45
• @John Rennie: Minor correction: Your first sentence should read: "Zero heat capacity means an infinitesimally small amount of energy will increase the temperature by a finite amount." – Pirx Dec 3 '16 at 13:07
• @Pirx Depends what you mean by "infinitesimal" and which infinitesimals you use, really :) – Patrick Stevens Dec 3 '16 at 14:13