## Third Law of Thermodynamics

### Introduction to the Third Law of Thermodynamics

The third law of thermodynamics states that the entropy of a pure perfect crystal is 0°K. At 0°K the atoms in a perfectly pure crystal are aligned perfectly in patterns and do not move. Moreover, there is no entropy and there is no mixing since the crystal is pure. For a mixed crystal containing the atomic or molecular species A and B, there are many possible arrangements of A and B and entropy is associated with the arrangement of the atoms/molecules.

As with any good scientific law, there are exceptions. The following are exceptions to the third law of thermodynamics:
C=O:
Carbon monoxide molecules can align in various ways in a crystal, so there is entropy associated with C=O crystals at 0 K. In this case, A and B are equally probable and the following equation can be used to calculate the entropy (called residual energy):
S = nR [ 1/2 ln(1/2) + 1/2 ln (1/2) ] = -nRln2

Glasses:
Glass is actually unordered microscopically, so it's not a perfectly pure crystal.

### Temperature Dependence of S

It's useful to know the temperature dependence of S (along with other thermodynamic parameters) so you can do experiments at a few temperatures and calculate the S for any temperature.
S(T2) = S(T1) + Cp/T dT where (Cp) is the change in heat capacity with temperature.

### S for Phase Changes/Transitions

At constant pressure,
H = q(rev)
S = q(rev)/T = H/T
H = T S
Example: the entropy of a liquid to solid phase change at 25 °C and 1 atm:
S (25 °C) = S (0 K) + Cp/T dT + H /T(rxn)

### Gibbs Free Energy (G)

The value of G determines the spontaneity of a change in the state of a system. At equilibrium, the following holds:
S = q/T = ( E - w)/T
S = ( E + P V)/T
S = (E2 - E1 + P2V2 - P1V1)/T
S = [(E2 + P2V2) - (E1 + P1V1)]/T
S = (H2 - H1)/T = H/T
A new variable of state is defined as Gibbs free energy, G = H - TS. At constant temperature G = H - TS. G can be used to determine the spontaneity of a change in state as follows:
G < 0 for a spontaneous reaction
G = 0 at equilibrium
G > 0 for no reaction

### Temperature Dependence of Gibbs Free Energy (G)

G(T1) = H(T1) - T1 S
G(T2) = H(T2) - T2 S
The temperature dependence of H and S can be approximated to zero:
H(T1) H(T2)
S(T1) S(T2)

G = H - TS = -RT ln K
ln K = (-H/R) 1/T + S/R
Therefore if ln K is plotted vs. 1/T (a can't Hoff plot), the slope equals -H/R and the intercept equals S/R.