Thermodynamics and Chemistry

CHAPTER 3 THE FIRST LAW

3.8 ELECTRICALWORK 87

one wire, and an equal number of electrons leave through a second wire. To simplify the
description, the wires are called therightconductor and theleftconductor.
The electric potentials experienced by a electron in the right and left conductors are
RandL, respectively. The electron charge ise, whereeis the elementary charge (the
charge of a proton). Thus the electrical potential energy of an electron isRein the right
conductor andLein the left conductor. The difference in the energies of an electron in
the two conductors is the difference in the electrical potential energies.
The sum of charges of a small number of electrons can be treated as an infinitesimal
negative charge. During a period of time in which an infinitesimal charge∂Qsysenters
the system at the right conductor and an equal charge leaves at the left conductor, the
contribution of the electric current to the internal energy change is the energy difference
.R∂QsysL∂Qsys/D.RL/ ∂Qsys. (The notation is∂Qsysinstead of dQsys, be-
causeQsysis a path function.) This internal energy change is calledelectrical work. Thus
the general formula for an infinitesimal quantity of electrical work when the system is part
of an electrical circuit is

∂welDÅ ∂Qsys (3.8.1)

(electrical work in a circuit)

whereÅis theelectric potential differencedefined by

Å defD RL (3.8.2)

Note that in the expression.R∂QsysL∂Qsys/for the energy difference, the term

R∂Qsysdoes not represent the energy transferred across the boundary at the right

conductor, andL∂Qsysis not the energy transferred at the left conductor. These

energies cannot be measured individually, because they include not just the electrical

potential energy but also the energy of the rest mass of the electrons. The reason we can

write Eq.3.8.1for the electrical work in a circuit is that equal numbers of electrons

enter and leave the system, so that the net energy transferred across the boundary

depends only on the difference of the electric potential energies. Because the number

of electrons in the system remains constant, we can treat the system as if it were closed.

Why should we regard the transfer of energy across the boundary by an electric

current as a kind of work? One justification for doing so is that the energy transfer is

consistent with the interpretation of work discussed on page 58 : the only effect on the

surroundings could be a change in the elevation of an external weight. For example, the

weight when it sinks could drive a generator in the surroundings that does electrical

work on the system, and electrical work done by the system could run an external

motor that raises the weight.

What is the meaning ofQsysin the differential∂Qsys? We defineQsysas the total
cumulative charge, positive or negative, that has entered the system at the right conductor

since the beginning of the process:Qsys defD

R

∂Qsys. Qsysis a path function for charge,
and∂Qsysis its inexact differential, analogous toqand∂qfor heat. Because the charge of
an electron is negative,∂Qsysis negative when electrons enter at the right conductor and
positive when they leave there.
The electric currentI is the rate at which charges pass a point in the circuit: I D
∂Qsys=dt, wheretis time. We takeIas negative if electrons enter at the right conductor