FreeEnergy

e PV = RT for an ideal gas.
Therefore,
?S= S2 – S1 = ?_(T_1)^(T_2)??C_v dT/T?+ ?_(V_1)^(V_2)?RdV/V.
If a phase transformation occurs in the temperature range being considered, there will be an entropy change accompanying that transformation as for zinc this will be equal to the heat of transformation (Lt) divided by the transformation temperature Tt








??S?_t= L_t/T_t

Calculation of entropy:-
To calculate the change in entropy accompanying a finite change in the system, we must integrate between the limits of temperature involved in the change, because ?S is the sum of all the entropy increments. For ?S to be calculated, each increment of heat absorbed ?q must be a reversible heat change, and
?S= S2 – S1 = ?_(T_1)^(T_2)??q/T, (2.7)
Where S2 and S1 are entropies of the system at temperatures T2 and T1 respectively.
at constant pressure C_P= ??q?_p/dT.
Therefore dS = ??q?_p/T= C_pdT/T

Thus, at constant pressure,
?S= S2 – S1 = ?_(T_1)^(T_t)??C_p/T dT?+ L_t/T_t + ?_(T_1)^(T_2)??C_p/T dT?
Where a phase transformation is involved.
Over short temperature ranges, Cp can be assumed to remain constant, but over larger temperature ranges, the relationship between Cp and T must be considered:
Cp = a + bT + cT-2.
*Why there is need for new function?
All spontaneous processes have a tendency to achieve a state of minimum energy and maximum randomness (entropy). But in actual practice it is not possible to achieve both minimum energy and maximum entropy for a system. For example
(i) there are many endothermic reactions which are spontaneous. Such as evaporation of a liquid.
(ii) there are some a thermal process where ?H= 0, which are spontaneous. Such as
AgClO2(s) ?Ag(s)+ 1/2 Cl2(g) + O2(g)
In such processes entropy factor overweings the enthalpy or energy factor.
(iii)there are many spontaneous processes where ?S of the system is negative.
H2(g) + 1/2 O2(g) ? H2O(l)
In such type of processes the enthalpy change overweighs the entropy factor (?S= negative and ?H = negative).
From the above examples, it is clear that none of ?E,?H or ?S system alone can decide the direction of a spontaneous process.