Physical Chemistry , 1st ed.

and want to get to the top of the mountain, there are many ways to go about

it. You can go straight up the mountain, or you can spiral about the mountain,

as two possibilities. The advantage to going straight up is that the path is

shorter, but it is also steeper. A path spiraling around the mountain is less

steep, but much longer. The amount of walking you end up doing is depen-

dent on the path you take. Such quantities are considered path-dependent.

However, whichever path you take, you ultimately end up at the top of the

mountain. Your altitude above the starting point is the same at the end of the

climb regardless of which path you take. Your final altitude is said to be path-

independent.

The change in altitude for your mountain climb can be considered a state

function: it is path-independent. The amount of walking up the mountain is

not a state function, because it is path-dependent.

Consider a physical or chemical process that a system undergoes. The process

has initial conditions and final conditions, but there are any number of ways

it can go from initial to final. A state functionis any thermodynamic property

whose value for the process is independent of the path. It depends only on the

state of the system (in terms of state variables like p,V,T,n), not on its his-

tory or how the system got to that state. A thermodynamic property whose

value for the process does depend on the path is not a state function. State

functions are symbolized by capital letters; non-state functions are symbolized

by lowercase letters. Internal energy is a state function. Work and heat are not.

State functions are different in another way. For an infinitesimal change in

a system, the infinitesimal changes in the work, heat, and internal energy are

represented as dw,dq, and dU. For a complete process, these infinitesimals are

integrated from initial to final conditions. However, there is a slight difference

in notation. When dwand dqare integrated, the result is the absolute amount

of work wand heat qfor the process. But when dUis integrated, the result is

not the absolute Ubut the changein U,U, for the process. Mathematically,

this is written as

34 CHAPTER 2 The First Law of Thermodynamics

Finish

Altitude

Start

(a) (b)

Figure 2.7 Analogy for the definition of a state function. For both path (a) straight up a

mountain and path (b) spiraling up the mountain, the overall change in altitude is the same and

so is path-independent: the change in altitude is a state function. However, the overall length of

the path is path-dependent, and so would not be a state function.