Physical Chemistry , 1st ed.

The negative sign indicates that the work done contributes to a decreasein the

amount of energy of the system.* If the piston moves inward, Figure 2.2c, then

the surroundings are doing work on the system,and the amount of energy in

the system is increased. The infinitesimal amount of work done on the system

is defined by equation 2.2, but because the volume change dVis in the oppo-

site direction, the work now has a positive value. Notice that our focus is the

system. The work is positive or negative with respect to the system, which is

the part of the universe of interest to us.

If we add up all of the (infinite) infinitesimal changes that contribute to an

overall change, we get the total amount of work done on or by the system.

Calculus uses the integral to add up infinitesimal changes. The total amount

of work,w, for a change as represented in Figure 2.2 is therefore

wpextdV (2.3)

Whether this integral can be simplified or not depends on the conditions of

the process. If the external pressure remains constant throughout the process,

then it can be removed to outside the integral and the expression becomes

wpextdVpextdVpextVVVif

In this case, the limits on the integral are the initial volume,Vi, and the final

volume,Vf, of the process. This is reflected in the last expression in the equa-

tion above. Evaluating the integral at its limits, we get

wpext(VfVi)

wpextV (2.4)

If the external pressure is not constant throughout the process, then we will

need some other way of evaluating the work in equation 2.3.

By using pressures in units of atm and volumes in units of L, we get a unit

of work in Latm. This is not a common work unit. The SI unit for work is the

joule, J. However, using the various values ofRfrom the previous chapter, it

can be shown that 1 Latm 101.32 J. This conversion factor is very useful to

get work into its proper SI units. If volume were expressed in units of m^3 and

pressure in pascals, units of joules would be obtained directly since

Pa m^3 

m

N

 2 m

(^3) N m J
Example 2.1
Consider an ideal gas in a piston chamber, as in Figure 2.2, where the initial
volume is 2.00 L and the initial pressure is 8.00 atm. Assume that the piston
is moving up (that is, the system is expanding) to a final volume of 5.50 L
against a constant external pressure of 1.75 atm. Also assume constant tem-
perature for the process.
a.Calculate the work for the process.
b.Calculate the final pressure of the gas.
26 CHAPTER 2 The First Law of Thermodynamics
*It is easy to show that the two definitions of work are equivalent. Since pressure is force
per unit area, equation 2.2 can be rewritten as work faorrecaevolume force distance,
which is equation 2.1.