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this cylindrical system is V Al Ab Ax. The change in system volume when the Section 2.2
piston moves by dx is dV d(Ab Ax) A dx. Equation (2.25) becomes P-V Work
dw rev P dV closed system, reversible process (2.26)*
The subscript rev stands for reversible. The meaning of “reversible” will be discussed
shortly. We implicitly assumed a closed system in deriving (2.26). When matter is trans-
ported between system and surroundings, the meaning of work becomes ambiguous; we
shall not consider this case. We derived (2.26) for a particular shape of system, but it can
be shown to be valid for every system shape (see Kirkwood and Oppenheim, sec. 3-1).
We derived (2.26) by considering a contraction of the system’s volume (dV 0).
For an expansion (dV 0), the piston moves outward (in the negative x direction), and
the displacement dx of the matter at the system–piston boundary is negative (dx 0).
Since F is positive (the force exerted by the piston on the system is in the positive x
x
direction), the work dw F dx done on the system by the surroundings is negative
x
when the system expands. For an expansion, the system’s volume change is still given
by dV A dx (where dx 0 and dV 0), and (2.26) still holds.
In a contraction, the work done on the system is positive (dw 0). In an expan-
sion, the work done on the system is negative (dw 0). (In an expansion, the work
done on the surroundings is positive.)
So far we have considered only an infinitesimal volume change. Suppose we carry
out an infinite number of successive infinitesimal changes in the external pressure. At
each such change, the system’s volume changes by dV and work PdV is done on the
system, where P is the current value of the system’s pressure. The total work w done
on the system is the sum of the infinitesimal amounts of work, and this sum of infin-
itesimal quantities is the following definite integral:

2
w rev P dV closed syst., rev. proc. (2.27)
1
where 1 and 2 are the initial and final states of the system, respectively.
The finite volume change to which (2.27) applies consists of an infinite number
of infinitesimal steps and takes an infinite amount of time to carry out. In this process,
the difference between the pressures on the two sides of the piston is always infinites-
imally small, so finite unbalanced forces never act and the system remains infini-
tesimally close to equilibrium throughout the process. Moreover, the process can be
reversed at any stage by an infinitesimal change in conditions, namely, by infinitesi-
mally changing the external pressure. Reversal of the process will restore both system
and surroundings to their initial conditions.
A reversible process is one where the system is always infinitesimally close to
equilibrium, and an infinitesimal change in conditions can reverse the process to
restore both system and surroundings to their initial states. A reversible process is
obviously an idealization.
Equations (2.26) and (2.27) apply only to reversible expansions and contractions.
More precisely, they apply to mechanically reversible volume changes. There could be
a chemically irreversible process, such as a chemical reaction, occurring in the system
during the expansion, but so long as the mechanical forces are only infinitesimally
unbalanced, (2.26) and (2.27) apply.
The work (2.27) done in a volume change is called P-V work. Later on, we shall
deal with electrical work and work of changing the system’s surface area, but for now,
only systems with P-V work will be considered.
We have defined the symbol w to stand for work done on the system by the sur-
roundings. Some texts use w to mean work done by the system on its surroundings.
Their w is the negative of ours.