Thermodynamics and Chemistry

CHAPTER 3 THE FIRST LAW

3.4 DEFORMATIONWORK 70

x

Fgas Ffric

Fext

Figure 3.4 Forces acting on the piston (cross hatched) in a cylinder-and-piston device

containing a gas (shaded). The direction ofFfricshown here is for expansion.

Letpbbe the average pressure of the gasat the piston—that is, at the moving portion
of the system boundary (the subscript “b” stands for boundary). Then the force exerted by
the gas on the piston is given by
FgasDpbAs (3.4.2)

whereAsis the cross-section area of the cylinder.
The component in theCxdirection of the net forceFnetacting on the piston is given by

FnetDFgasFextCFfric (3.4.3)

Here,FgasandFextare taken as positive. Ffricis negative when the piston moves to the
right, positive when the piston moves to the left, and zero when the piston is stationary.
Suppose the system (the gas) initially is in an equilibrium state of uniform temperature
T 1 and uniform pressurep 1 , and the piston is stationary, so thatFfricis zero. According to
Newton’s second law of motion, the net forceFnetis also zero, because otherwise the piston
would be accelerating. Then, from Eqs.3.4.2and3.4.3, the external force needed to keep
the piston from moving isFextDFgasDp 1 As.
To avoid complications of heat transfer, we confine our attention to a system with an
adiabatic boundary. By reducingFextfrom its initial value ofp 1 As, we cause spontaneous
expansion to begin. As the piston moves to the right, the pressurepbexerted on the left face
of the piston becomes slightlylessthan the pressure on the stationary cylinder wall. The
molecular explanation of this pressure gradient is that gas molecules moving to the right
approach the moving piston at lower velocities relative to the piston than if the piston were
stationary, so that they collide with the piston less frequently and with a smaller transfer of
momentum in each collision. The temperature and pressure within the gas become nonuni-
form, and we cannot describe intermediate states of this spontaneous process with single
values ofTandp. These intermediate states are not equilibrium states.
The more slowly we allow the adiabatic expansion to take place, the more nearly uni-
form are the temperature and pressure. In the limit of infinite slowness, the gas passes
through a continuous sequence of equilibrium states of uniform temperature and pressure.
Letp 2 be the pressure in the final state of the infinitely-slow expansion. In this state,
Fextis equal top 2 As. ByincreasingFextfrom this value, we cause spontaneous compres-
sion to begin. The gas pressurepbat the piston now becomes slightlygreaterthan at the
stationary cylinder wall, because the piston is moving to the left toward the molecules that
are moving to the right. A different pressure gradient develops than during expansion. The