Heat capacity at low temperature

Combining Eqs.  and  we have

$$dS = \frac{C}{T}dT,$$ (3.29)

where $C$ is some appropriate heat capacity which depends on the physical conditions under which the transformation is carried out. From this we can write:

$$S(T) - S(T_0) = \int_{T_0}^T\frac{C}{T^\prime}dT^\prime.$$ (3.30)

The entropy of any finite system must also be finite, as according to Eq.  it is simply related to the total number of distinct microstates available to the system ^3.7. This imposes constraints on the heat capacity $C$, and in particular that it must go to zero for $T$ going to zero, otherwise the integral  cannot be computed for $T_0 = 0$ (see also ).