Determine the Absolute Entropies of Solids

Absolute entropies of solids:
Entropy change for an infinitesimally small change of a state of a substance is given as:
dS = dq / T

If the changes take place at constant pressure, then,
(∂S) _P = (∂q) _P / T

Multiplying both the sides by ∂T, we will get:
(∂S / ∂T) _P = [(∂q / ∂T) _P] X 1 / T

We know that, (∂q / ∂T) _P = C_P
Therefore, (∂S / ∂T) _P = C_P X 1 / T

Hence, at constant pressure, dS = C_P dT / T
The substance which is perfectly crystalline, the absolute entropy S = 0 at temperature T = 0.
Therefore,
^S∫₀ dS = ^T∫₀ (C_P / T) dT
S_T = ^T∫₀ C_P dT / T
S_T = ^T∫₀ C_P d (ln T)
Where S_T is the absolute entropies of crystalline solid at constant temperature T