Modern Engineering Thermodynamics

4.5 Point and Path Functions 107
4.5 POINT AND PATH FUNCTIONS
A quantity, say y, that has a value at every point within its range is called a point function. Its derivative is written
as dy, and its integral from state 1 to state 2 is
Z 2
1
dy = y 2 − y 1
Thus, the value of the integral depends only on the values of y at the end points of the integration path and is
independent of the actual path taken between these end points. This is a fundamental characteristic of point
functions. All intensive and extensive thermodynamic properties are point functions. Therefore, we can write
Z 2
1
dE = E 2 − E 1 ;
Z 2
1
du = u 2 − u 1 ;
Z 2
1
dm = m 2 − m 1
and so forth.
Aquantity,sayx, whose value depends on the path taken between two points within its range is called a path
function. Since path functions do not differentiate or integrate in the same manner as point functions, we cannot
use the same differential and integral notation for both path and point functions. Instead, we let dx denote the
differential of the path function x, and we define its integral over the path from state 1 to state 2 as
Z 2
Z 2
dx = 1 x 2 Note: dx ≠ ðx 2 − x 1 Þ
(4.15)
1
A path function does not have a value at a point. It has a value only for a path of points, and this value is
directly determined by all the points on the path, not just its end points. For example, the area A under the
curve of the point function w = f(y) is a path function because
and
Z 2
1
dA = 1 A 2 =
Z Y2
Y1
dA = wdy = f ðyÞ dy
1
f ðyÞ dy = area under fðÞbetween y the points y 1 and y 2
Clearly, if the path f(y) is changed, then the area 1 A 2 is also changed. Consequently, we say that 1 A 2 is a path function.
We see in the next sections that both the work and heat transports of energy are path functions. Therefore, we write
the differentials of these quantities as dW and dQ, and their integrals as
and
Z 2
1
Z 2
1
dW = 1 W 2 (4.16)
dQ = 1 Q 2 (4.17)
Since the associated rate equations contain the time differential, we define power as the work rate, or
_W = dW/dt (4.18)
and, similarly, the heat transfer rate is
_Q = dQ/dt (4.19)
Each of the different types of work or heat transport of energy is called a mode. A system that has no operating work
modes is said to be aergonic. Similarly, a system that changes its state without any work transport of energy having
NOTE!
Since work and heat are not thermodynamic properties and therefore not point functions,
Z 2
dW ≠ ΔW: Similarly,
Z 2
1
1
1
path function integrals.
dQ ≠ Q 2 − Q 1 , and
Z 2
Z 2
1
dW ≠ W 2 − W 1 and
dQ ≠ ΔQ: Equations (4.16) and (4.17) are the only correct ways to write these