Thermodynamics and Chemistry

CHAPTER 3 THE FIRST LAW

3.8 ELECTRICALWORK 90

I

wel

bc

bc

Figure 3.17 Electrical work of a galvanic cell for a fixed magnitude ofQsysas a

function of the electric currentID∂Qsys=dt. Open circles: reversible limits.

circuit with an electric current passing through the cell, as in Fig.3.16(b),Ecellis different
fromEcell, eqon account of the internal resistanceRcellof the cell:

EcellDEcell, eqCIRcell (3.8.7)

The sign of the currentIis negative when electrons enter the cell at the right terminal, and
positive when electrons leave there.
In the circuit shown in Fig.3.16(b), the cell does electrical work on the resistor in the
surroundings. The energy for this work comes from the cell reaction. The formula for the
electrical work is given by Eq.3.8.1withÅreplaced byEcell:

∂welDEcell∂Qsys (3.8.8)

The figure showsEcellas positive and∂Qsysas negative, so for this arrangement∂welis
negative.
When current passes through the cell, the work done is irreversible because the internal
resistance causes energy dissipation. We can make this work approach a finite reversible
limit by replacing the external resistor shown in Fig.3.16(b) with an adjustable voltage
source that we can use to control the cell potentialEcelland the currentI. According to
Eq.3.8.7,Ecellis greater thanEcell, eqwhenIis positive, and is less thanEcell, eqwhen
Iis negative. This behavior is shown graphically in Fig.3.17. In the limit as the electric
current approaches zero from either direction and the external adjustable voltage approaches
Ecell, eq, the electrical work approaches a reversible limit given by

∂wel, revDEcell, eq∂Qsys (3.8.9)

Note that the electrical work is the least positive or most negative in the reversible limit.
Thus, unlike the dissipative work of stirring and electrical heating, electrical work with
a galvanic cell has a nonzero reversible limit, as reflected by the difference in the appear-
ance of Fig.3.17compared to Figs.3.11and3.15. During irreversible electrical work of
a galvanic cell, there is onlypartialdissipation of energy within the cell: the energy trans-
ferred across the boundary by the work can be partially recovered by returning the work
coordinateQsysto its initial value.

On page 84 the observation was made that the work coordinate of work with a re-

versible limit is always a state function. Electrical work with a galvanic cell does