Standard molar entropy

The standard molar entropy at pressure = ${\displaystyle P^0}$ is usually given the symbol S°, and has units of joules per mole per kelvin (J⋅mol⁻¹⋅K⁻¹). Unlike standard enthalpies of formation, the value of S° is absolute. That is, an element in its standard state has a definite, nonzero value of S at room temperature. The entropy of a pure crystalline structure can be 0 J⋅mol⁻¹⋅K⁻¹ only at 0 K, according to the third law of thermodynamics. However, this assumes that the material forms a 'perfect crystal' without any residual entropy. This can be due to crystallographic defects, dislocations, and/or incomplete rotational quenching within the solid, as originally pointed out by Linus Pauling. These contributions to the entropy are always present, because crystals always grow at a finite rate and at temperature. However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics.

Thermodynamics

If a mole of a solid substance is a perfectly ordered solid at 0 K, then if the solid is warmed by its surroundings to 298.15 K without melting, its absolute molar entropy would be the sum of a series of N stepwise and reversible entropy changes. The limit of this sum as ${\displaystyle N\to \mathrm{\infty }}$ becomes an integral:

${\displaystyle S^{\circ }=\sum _{k=1}^N\mathrm{\Delta }{S}_k=\sum _{k=1}^N\frac{d{Q}_k}{T}\to {\int }_0^{T_2}\frac{dS}{dT}dT={\int }_0^{T_2}\frac{C_{p_k}}{T}dT}$

In this example, ${\displaystyle T_2=298.15K}$ and ${\displaystyle C_{p_k}}$ is the molar heat capacity at a constant pressure of the substance in the reversible process k. The molar heat capacity is not constant during the experiment because it changes depending on the (increasing) temperature of the substance. Therefore, a table of values for ${\displaystyle \frac{C_{p_k}}{T}}$ is required to find the total molar entropy. The quantity ${\displaystyle \frac{d{Q}_k}{T}}$ represents the ratio of a very small exchange of heat energy to the temperature T. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process.