Page 110 - Mechanical Engineers' Handbook (Volume 4)

3 The Second Law of Thermodynamics for Closed Systems 99

If the closed system expands or contracts quasistatically (i.e., slowly enough, in mechanical
equilibrium internally and with the environment) so that at every point in time the pressure
P is uniform throughout the system, then the work-transfer term can be calculated as being
equal to the work done by all the boundary pressure forces as they move with their respective
points of application,
2 2
W PdV
1 1
The work-transfer integral can be evaluated provided the path of the quasistatic process,
P(V), is known; this is another reminder that the work transfer is path-dependent (i.e., not a
thermodynamic property).

3 THE SECOND LAW OF THERMODYNAMICS FOR CLOSED SYSTEMS

A temperature reservoir is a thermodynamic system that experiences only heat transfer and
whose temperature remains constant during such interactions. Consider first a closed system
executing a cycle or an integral number of cycles while in thermal communication with no
more than one temperature reservoir. To state the second law for this case is to observe that
the net work transfer during each cycle cannot be positive,

W 0

In other words, a closed system cannot deliver work during one cycle, while in communi-
cation with one temperature reservoir or with no temperature reservoir at all. Examples of
such cyclic operation are the vibration of a spring–mass system, or a ball bouncing on the
pavement: for these systems to return to their respective initial heights, that is, for them to
execute cycles, the environment (e.g., humans) must perform work on them. The limiting
case of frictionless cyclic operation is termed reversible, because in this limit the system
returns to its initial state without intervention (work transfer) from the environment. There-
fore, the distinction between reversible and irreversible cycles executed by closed systems
in communication with no more than one temperature reservoir is

W 0 (reversible)

W 0 (irreversible)

To summarize, the first and second laws for closed systems operating cyclically in contact
with no more than one temperature reservoir are (Fig. 1)

W Q 0

1
This statement of the second law can be used to show that in the case of a closed
system executing one or an integral number of cycles while in communication with two
temperature reservoirs, the following inequality holds (Fig. 1)
Q H Q L
0
T H T L
where H and L denote the high-temperature and the low-temperature reservoirs, respectively.
Symbols Q and Q stand for the value of the cyclic integral Q, where Q is in one case
L
H