CHAPTER 3 THE FIRST LAW
3.6 WORK IN AGRAVITATIONALFIELD 79
gas vacuum gas gasFigure 3.8 Free expansion into a vacuum.negative for a spontaneous adiabatic expansion than for a reversible adiabatic expansion
with the same initial state and the same volume change.
During a rapid spontaneouscompression,pbis greater than the pressure at the stationary
wall. The work is positive and greater for a spontaneous adiabatic compression than for a
reversible adiabatic compression with the same initial state and the same volume change.
These observations are summarized by the statement that, for an adiabatic expansion
or compression with a given change of the work coordinate, starting at a given initial equi-
librium state, the work is algebraically smallest (least positive or most negative) in the
reversible limit. That is, in the reversible limit the surroundings do the least possible work
on the system and the system does the maximum possible work on the surroundings. This
behavior will turn out to be true of any adiabatic process of a closed system.
3.5.6 Free expansion of a gas into a vacuum
When we open the stopcock of the apparatus shown in Fig.3.8, the gas expands from the
vessel at the left into the evacuated vessel at the right. This process is calledfree expansion.
Thesystemis the gas. The surroundings exert a contact force on the system only at the vessel
walls, where there is no displacement. Thus, there isnowork in free expansion:∂wD 0.
If the free expansion is carried out adiabatically in a thermally-insulated apparatus, there
is neither heat nor work and therefore no change in the internal energy:ÅUD 0. If the gas
is ideal, its internal energy depends only on temperature; thus the adiabatic free expansion
of an ideal gas causes no temperature change.
3.6 Work in a Gravitational Field
Figure3.9on the next page depicts a spherical body, such as a glass marble, immersed in
a liquid or gas in the presence of an external gravitational field. The vessel is stationary on
a lab bench, and the local reference frame for work is a stationary lab frame. The variable
zis the body’s elevation above the bottom of the vessel. All displacements are parallel to
the verticalzaxis. From Eq.3.1.1, the work is given by∂wDFzsurdzwhereFzsuris the
upward component of the net contact force exerted by the surroundings on the system at the
moving portion of the boundary. There is also a downward gravitational force on the body,
but as explained in Sec.3.1.1, this force does not contribute toFzsur.
Consider first the simple process in Fig.3.9(a) in which the body falls freely through
the fluid. This process is clearly spontaneous. Here are two choices for the definition of the
system:
The system is the combination of the spherical body and the fluid. The system bound-
ary is where the fluid contacts the air and the vessel walls. Because there is no dis-