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46 CHAPTER 3 The Importance of State Functions: Internal Energy and Enthalpy

relationship such as Equation (3.1) does not exist for path-dependent functions. Consider
10
y 1 mole of an ideal gas for which
20
30
Able Hill RT
40 P = f(V,T) = (3.1)
z
z
z () dx () dy V
x y y x
z
z z () dx Note that P can be written as a function of the two variables V and T. The change in P
x y
x resulting from a change in V or T is proportional to the following partial derivatives:
0P P(V +¢V,T) - P(V,T) RT
a b = lim =-
50 m 0V T ¢V:0 ¢V V 2
40 m
Able Hill
30 m a 0P b P(V,T +¢T) - P(V,T) = R
20 m 0T V = lim ¢T:0 ¢T V (3.2)
10 m
Sea level The subscript T in (0P>0V) T indicates that T is being held constant in the differen-
= 0 tiation with respect to V. The partial derivatives in Equation (3.2) allow one to
determine how a function changes when the variables change. For example, what is
FIGURE 3.1
the change in P if the values of T and V both change? In this case, P changes to P +
Starting at the point labeled z on the hill,
dP where
a person first moves in the positive x
direction and then along the y direction. 0P 0P
If dx and dy are sufficiently small, the dP = a b dT + a b dV (3.3)
0T V 0V T
change in height dz is given by
dz = a 0z b dx + a 0z b dy . Consider the following practical illustration of Equation (3.3). A person is on a hill and
0x y 0y x has determined his or her altitude above sea level. How much will the altitude (denoted
by z) change if the person moves a small distance east (denoted by x) and north
(denoted by y)? The change in z as the person moves east is the slope of the hill in that
direction, (0z>0x) y , multiplied by the distance dx that he or she moves. A similar
expression can be written for the change in altitude as the person moves north.
Therefore, the total change in altitude is the sum of these two changes or
0z 0z
dz = a b dx + a b dy
0x y 0y x
These changes in the height z as the person moves first along the x direction and then
along the y direction are illustrated in Figure 3.1. Because the slope of the hill is a
nonlinear function of x and y, this expression for dz is only valid for small changes dx
and dy. Otherwise, higher order derivatives need to be considered.
Second or higher derivatives with respect to either variable can also be taken. The
mixed second partial derivatives are of particular interest. Consider the mixed partial
derivatives of P:
RT
0c d
0 0P 0 P £ £ V ≥ Æ ≥ RT R
2
a a b b = = 0 0T = a0 c - d n0Tb =-
0T 0V T V 0T0V 0V T V V 2 V V 2
RT
0 c d
2
0 0P 0 P £ £ V ≥ Æ ≥ R R
a a b b = = 0 0V = a0 c d n0Vb =- (3.4)
0V 0T V T 0V0T 0T V T V T V 2
For all state functions f and for our specific case of P, the order in which the function is
differentiated does not affect the outcome. For this reason,
0 0f(V,T) 0 0f(V,T)
a a b b = a a b b (3.5)
0T 0V T V 0V 0T V T
Because Equation (3.5) is only satisfied by state functions f, it can be used to deter-
mine if a function f is a state function. If f is a state function, one can write
¢f = 1 i f df = f final - f initial . This equation states that f can be expressed as an
infinitesimal quantity df that when integrated depends only on the initial and final
states; df is called an exact differential. An example of a state function and its exact
differential is U and dU = dq - P external dV .