1. The Carnot cycle T a b T2 T1 c d s

2. Entropy changes in reversible processes
Various cases:
Adiabatic process: đqr = 0, ds = 0, s = constant. A reversible adiabatic process is isentropic. THIS IS NOT TRUE FOR AN IRREVERSIBLE PROCESS!
Isothermal process:

3. Entropy changes in reversible processes
Various cases:
Isothermal: ideal gas case (du = 0; đq = -đw; Pv = RT )

4. Entropy changes in reversible processes
Various cases:
Isochoric process: We assume u = u(v,T) in general, so that u = u(T) in an isochoric process. Therefore, as in the case for an ideal gas, du = cvdT. Thus,
provided cv is independent of T over the integration (really only true for ideal gas, but often good approx.).

5. What about reversible paths?
DS > 0
(1)
Isobaric (P = const)
Isothermal (PV = const)
Adiabatic (PVg = const)
Isochoric (V = const)
(2)
(3)
(4)
DS = 0
DS < 0 V
For a given reversible path, there is some associated physics.

6. The combined 1st and 2nd Laws
The 2nd law need not be restricted to reversible processes:
đQ is identifiable with TdS, as is đW with PdV, but only for reversible processes.
However, the last equation is valid quite generally, even for irreversible processes, albeit that the correspondence between đQ & TdS, and đW & PdV, is lost in this case.

7. Entropy associated with irreversible processes
Universe at
temperature T2 Q T1
T2 > T1

8. Entropy associated with irreversible processes
Isobaric
Cp1, T1i
Cp2, T2i dQ

9. Entropy associated with irreversible processes
Isobaric
Cp1, T1i
Cp2, T2i
dQ2
dQ1 Tf
Infinite heat reservoir

10. Entropy associated with irreversible processes
Isobaric
Cp1, T1i
Cp2, T2i
dQ2
dQ1 Tf
Finite heat reservoir
Qin = Qout, otherwise Tf would not remain constant

11. Entropy of mixing n1 T, P n2 T, P n = n1 + n2 T, P = P1 + P2
initial state i
final state f n1
T, P n2
T, P
n = n1 + n2
T, P = P1 + P2
Definition of partial pressure Pj of a constituent in a mixture:
Here, P is the pressure of the mixture, and xj is the kilomole fraction of the jth constituent gas:

12. Entropy of mixing n1 T, P n2 T, P n = n1 + n2 T, P = P1 + P2
initial state i
final state f n1
T, P n2
T, P
n = n1 + n2
T, P = P1 + P2
P is the pressure of the mixture; Pj is the partial pressure and xj the kilomole fraction of the jth constituent gas, where
Then,

13. Entropy of mixing

14. Maxwell’s demon