calculation of entropy

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
?S= S2 – S1 = ?_(T_1)^(T_2)???q?_p/T= ?_(T_1)^(T_2)?(C_p dT)/T
Similarly, at constant volume,
?S= ?_(T_1)^(T_2)?C_vdT/T.
If the temperature and volume are not constant, for a small heat change,
?q=dU+PdV
= C_v dT+PdV.
Therefore,
?q/T= C_v dT/T+ PdV/T= C_v dT/T+ RdV/V
Because 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
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. However ?S_System certainly decides the direction of a change. At constant E and V in an isolated system, S is maximum. Generally in Chemistry, reactions are rarely studied at constant energy (E) reactions are conveniently carried out either at constant temperature and volume (T and V) or constant temperature and pressure (T and P). Hence it is necessary to introduce new functions which can decide the direction of a spontaneous process. Also the new functions are helpful in deciding the feasibility of a process.
Free Energy
The factor (?H-T?S) has dimensions of energy-because ?H is an energy term and ?S is the heat absorbed divided by the absolute temperature. It has been called the change in the "free energy" of the system, "free energy" being a thermodynamic function of great importance.
Consider a system undergoing a thermodynamically reversible change at constant temperature and constant volume.
The First Law of Thermodynamics,
?U=q-w .
q is the heat absorbed reversibly by the system at temperature T. and w is the work done by the system.
?S= q/T
Thus, ?U=T?S-W,
and -W = ?U-T?S.
-W is the maximum work (whether mechanical or electrical, etc.) that can be obtained from the system, and a thermodynamic function F, the "work function" or Helmholtz free energy (after H. von Helmholtz) can be defined such that
-W = ?U-T?S= ?F.
F = U – TS,
And it is a thermodynamic variable, depending only on the state of the system-not on its history- because U, T and S are all thermodynamic variables.
At constant temperature and constant pressure, work may be done by the system as a result of a volume change. This will be P?V according to
W=P?V
and will not be "useful"work. The useful work will then be the maximum work, -W less the energy lost due to the volume change (-P?V).
Then we can define the "useful work" as ?G, the Gibbs free energy change of the system, where