Thermodynamics and Chemistry

CHAPTER 3 THE FIRST LAW

3.6 WORK IN AGRAVITATIONALFIELD 80

Fbuoy Ffric

mg

fluid

z

(a)

Fbuoy Ffric

mg

Fstr

fluid

z

(b)

Figure 3.9 Spherical body (shaded) in a gravitational field. The arrows indicate the

directions and magnitudes of contact and gravitational forces exerted on the body.

(a) The body falls freely through a fluid in a vessel.

(b) The body is lowered on a string through the fluid.

placement of this boundary,nowork is being done on or by the system:∂wD 0.

(We ignore expansion work caused by the small temperature increase.) If the process

is adiabatic, the first law tells us the system’s internal energy remains constant: as

the body loses gravitational potential energy, the system gains an equal quantity of

kinetic and thermal energy.

 The system is the body; the fluid is in the surroundings. The upward components

of the forces exerted on the body are (1) a gravitational forcemg, wheremis the

body’s mass andgis the acceleration of free fall; (2) a buoyant force^13 FbuoyDV^0 g,

whereis the fluid density andV^0 is the volume of the body; and (3) a frictional drag

forceFfricof opposite sign from the velocityvDdz=dt. As mentioned above, the

gravitational force is not included inFzsur. Therefore the gravitational work is given

by

∂wDFzsurdzD

FbuoyCFfric

dz (3.6.1)

and is negative because dzis negative: the body as it falls does work on the fluid.

The positive quantity

(^) Fbuoydz
(^) is the work of moving displaced fluid upward, and
jFfricdzjis the energy dissipated by friction to thermal energy in the surroundings.
This process has no reversible limit, because the rate of energy transfer cannot be
controlled from the surroundings and cannot be made to approach zero.
Next, consider the arrangement in Fig.3.9(b) in which the body is suspended by a thin
string. The string is in the surroundings and provides a means for the surroundings to exert
an upward contact force on the system. As before, there are two appropriate choices for the
system:
 The system includes both the body and the fluid. The moving part of the boundary
is at the point where the string is attached to the body. The force exerted here by the
string is an upward forceFstr, and the gravitational work is given by∂wDFzsurdzD
(^13) The buoyant force is a consequence of the pressure gradient that exists in the fluid in a gravitational field (see
Sec.8.1.4). We ignore this gradient when we treat the fluid as a uniform phase.