RMP Lecture Notes

Entropy

1st Law of Thermodynamics -- analyzed using U
2nd Law of Thermodynamics -- define new property S to use in analysis, to allow expansion of the 2nd law beyond cycles

Entropy (S) is a state property that serves as a measure of reversibility and of the disorder of the system. It is developed to account for losses in the potential to do useful work, and so accounts for waste in a process.

if a change is to spontaneously occur.

Note the inequality -- since a reversible system is the ideal, it serves as a bound; as a result, application of the 2nd Law often gives equations with inequalities.

The Clausius Inequality

The second law leads to the Clausius Inequality {8.1}:

if not satisfied, the 2nd law is violated
if a reversible cycle, =0
work done by an irreversible cycle is always less than that of a reversible cycle (why? Q=W by 1st law)

Now consider some cyclic PV process: If we apply the Clausius inequality to this process

we see that the path followed by the process does not change the result. Therefore, the quantity entropy

is a state property. This quantity thus behaves like we want thermodynamic properties to behave, so we use it to define Entropy{8.2}: entropy

Entropy is an extensive, state property. It can be made intensive by dividing by the mass.

The 2nd Law of Thermodynamics can now be restated in terms of the property entropy:

irreversible processes will cause an increase in entropy of the universe and in the loss of potential to do useful work
reversible processes do not change the entropy of the universe, so isentropic paths can be used to determine the minimum work requirements and maximum work output
suggestions for processes that would cause a decrease in the entropy of the universe are impossible.

As with most of the other properties we've been wrestling with, entropy is usually evaluated in terms of differences. These can be found be integrating the definition {8.3}:

Assumptions:

the integral must be evaluated along a reversible path between states 1 and 2; this "ideal" path need not be the "real" path. It will assume all mechanical energy changes are reversible, no friction, uniform temperature at a given instant, etc.
the temperature must be absolute

Observe:

As a state property, entropy fits the state postulate: values can be found if any two intensive variables are known
different paths may have different Q_reversible, but the quantity Q_rev/T will be the same for all

Why must an ideal path be used? If water is heated with electrical work, Q=0, but the temperature rises and the state changes. Consequently, some other path must be used to determine the entropy for this circumstance.

References:

Cengel, Y.A. and M.A. Boles, Thermodynamics: An Engineering Approach (3rd ed.), WCB McGraw-Hill, 1998, p. 320-23.
Elliott, J.R. and C.T. Lira, Introductory Chemical Engineering Thermodynamics, Prentice Hall PTR, 1999, pp. 10-11, 97.
Sonntag, R.E., C. Borgnakke, and G.J. Van Wylen, Fundamentals of Thermodynamics (6th ed.), John Wiley, 2003, p. 251-57.
Sonntag, R.E., C. Borgnakke, and G.J. Van Wylen, Fundamentals of Thermodynamics (5th ed.), John Wiley, 1998, p. 223-29.
Sonntag, R.E., C. Borgnakke, and G.J. Van Wylen, Fundamentals of Thermodynamics (4th ed.), John Wiley, 1994.

R.M. Price
Original: 6/19/97
Modified: 6/12/2000, 3/19/2002, 5/20/2004; 6/29/2004