Entropy

Entropy is a thermodynamic property which comes about as a result of the second law of thermodynamics. To demonstrate its existence, following Zemansky (1943) and Mooney (1953), consider a reversible process from an initial state i to a final state f and use the first law to give Q — Wif = (Uf — U{)] , Fig. 2.3. From i and f draw two reversible adiabatic lines. Then construct a reversible isotherm (ab) so that the area above and below the isotherm and between the original process ( i-f) and the adiabatic lines is equal. Thus we obtain that (Wif = Wiabf). Therefore, now the heat terms give (Qif = Qiabf) since (Uf - U) does not change because of the general character of a thermodynamic property. Also Qia and Qbf are equal to zero since they are adiabatic processes resulting in

(Wff) = (Wiabf) = (Wia + Wb + Wbf). Therefore, the result becomes

[Qab - Wf = (Uf - U;)] giving the final result that (Qab = Qif). In general, therefore, an arbitrary reversible process can always be replaced by a zigzag path between the same state points consisting of a reversible adiabatic line, a reversible isotherm, and another reversible adiabatic line, such that {Q , = Q },

now the heat terms give (Qif = Qiabf) since (Uf - U;) does not change because of the general character of a thermodynamic property. Now, to reach the definition of thermodynamic quantity entropy, consider a smooth reversible cycle as shown in Fig. 2.4. On it inscribe reversible adiabatic lines of thickness A. For each slice or arc, which is a reversible process, inscribe an isotherm so that the condition given above is satisfied. The cycles thus formed are all Carnot cycles with the characteristic relationship obtained between heat transfer and absolute temperature ratios. Thus for the first cycle drawn, <j Qhi = —Sl I or J Qhi + Qli = o i In a

I -QLi —LI J 1 —Hi TlI f similar fashion, for the second cycle we have J Qh2 + Ql2 = o i Adding these 