Thermodynamics and Chemistry

CHAPTER 3 THE FIRST LAW

3.4 DEFORMATIONWORK 71

states approached in the limit as we carry out the compression more and more slowly are
equilibrium states, occurring in the reverse sequence of the states for expansion at infinite
slowness. The sequence of equilibrium states, taken in either direction, is areversiblepro-
cess.

The magnitude of the effect of piston velocity onpbcan be estimated with the help of

the kinetic-molecular theory of gases. This theory, of course, is not part of classical

macroscopic thermodynamics.

Consider an elastic collision of a gas molecule of massmwith the left side of the

piston shown in Fig.3.4. Assume the piston moves at a constant velocityu(positive for

expansion and negative for compression), and let thexcomponent of the molecule’s

velocity as it approaches the piston bevx(a positive value). After the collision, thex

component of the molecule’s velocity isvxC2u, and the momentum that has been

transferred to the piston is2m.vxu/. With the assumption that thexcomponent

of the molecule’s velocity remains unchanged between consecutive collisions with the

piston, the time interval between two collisions is2l=.vxu/wherelis the internal

cylinder length at the time of the earlier collision.

The average force exerted by one molecule on the piston is equal to the momentum

transferred to the piston by a collision, divided by the time interval between collisions.

The total pressure exerted by the gas on the piston is found by summing the average

force over all molecules and dividing by the piston areaAsDV=l:

pbD

nM

V

v^2 x

2uhvxiCu^2

(3.4.4)

(The angle brackets denote averages.) The pressure at the stationary wall of the cylin-

der ispD.nM=V /

v^2 x

. Accordingly, the pressure at the moving piston is given
by^9
pbDp
1 2u
h vxi
v^2 x

(^) C u
2
v^2 x
!
(3.4.5)
From kinetic-molecular theory, the averages are given by hvxiD.2RT=M/1=2and
v^2 x
DRT=M.
Suppose the piston moves at the considerable speed of 10 m s^1 and the gas in the
cylinder is nitrogen (N 2 ) at 300 K; then Eq.3.4.5predicts the pressurepbexerted on
the piston by the gas is only about 5 % lower than the pressurepat the stationary wall
during expansion, and about 5 % higher during compression. At low piston speeds the
percentage difference is proportional to the piston speed, so this example shows that
for reasonably-slow speeds the difference is quite small and for practical calculations
can usually be ignored.

3.4.2 Expansion work of a gas

We now consider the work involved in expansion and compression of the gas in the cylinder-
and-piston device of Fig.3.4. This kind of deformation work, for both expansion and com-
pression, is calledexpansion workor pressure-volume work.
Keep in mind that thesystemis just the gas. The only moving portion of the boundary of
this system is at the inner surface of the piston, and this motion is in theCxorxdirection.

(^9) An equivalent formula is derived in Ref. [ 9 ], Eq. 7. A formula that yields similar values ofpbappears in Ref.
[ 14 ].