QUASIEQUILIBRIUM EXPANSION OR COMPRESSION PROCESSES

An idealized type of process called a quasiequilibrium process.
quasiequilibrium process is one in which all states through which the system passes may be
considered equilibrium states. A particularly important aspect of the quasiequilibrium process
concept is that the values of the intensive properties are uniform throughout the system, or
every phase present in the system, at each state visited.
To consider how a gas (or liquid) might be expanded or compressed in a quasiequilibrium fashion, refer to Fig. 2.6, which shows a system consisting of a gas initially at an
equilibrium state. As shown in the figure, the gas pressure is maintained uniform throughout by a number of small masses resting on the freely moving piston. Imagine that one of
the masses is removed, allowing the piston to move upward as the gas expands slightly.
During such an expansion the state of the gas would depart only slightly from equilibrium.
The system would eventually come to a new equilibrium state, where the pressure and all
other intensive properties would again be uniform in value. Moreover, were the mass replaced, the gas would be restored to its initial state, while again the departure from equilibrium would be slight. If several of the masses were removed one after another, the gas
would pass through a sequence of equilibrium states without ever being far from equilibrium. In the limit as the increments of mass are made vanishingly small, the gas would
undergo a quasiequilibrium expansion process. A quasiequilibrium compression can be
visualized with similar considerations.
Equation 2.17 can be applied to evaluate the work in quasiequilibrium expansion or compression processes. For such idealized processes the pressure p in the equation is the pressure
of the entire quantity of gas (or liquid) undergoing the process, and not just the pressure at
the moving boundary. The relationship between the pressure and volume may be graphical
or analytical. Let us first consider a graphical relationship.
A graphical relationship is shown in the pressure–volume diagram ( p–V diagram) of
Fig. 2.7. Initially, the piston face is at position x1, and the gas pressure is p1; at the conclusion of a quasiequilibrium expansion process the piston face is at position x2, and the pressure is reduced to p2. At each intervening piston position, the uniform pressure throughout
the gas is shown as a point on the diagram. The curve, or path, connecting states 1 and 2 on
the diagram represents the equilibrium states through which the system has passed during
the process. The work done by the gas on the piston during the expansion is given by p dVwhich can be interpreted as the area under the curve of pressure versus volume. Thus, the
shaded area on Fig. 2.7 is equal to the work for the process. Had the gas been compressed
from 2 to 1 along the same path on the p–V diagram, the magnitude of the work would be
 
ne"> the same, but the sign would be negative, indicating that for the compression the energy transfer was from the piston to the gas.
The area interpretation of work in a quasiequilibrium expansion or compression process
allows a simple demonstration of the idea that work depends on the process. This can be
brought out by referring to Fig. 2.8. Suppose the gas in a piston–cylinder assembly goes from
an initial equilibrium state 1 to a final equilibrium state 2 along two different paths, labeled
A and B on Fig. 2.8. Since the area beneath each path represents the work for that process,
the work depends on the details of the process as defined by the particular curve and not just
on the end states. Using the test for a property given in Sec. 1.3, we can conclude again
(Sec. 2.2.1) that work is not a property. The value of work depends on the nature of the
process between the end states.
The relationship between pressure and volume during an expansion or compression process
also can be described analytically. An example is provided by the expression pVn constant,
where the value of n is a constant for the particular process. A quasiequilibrium process described by such an expression is called a polytropic process. Additional analytical forms for
the pressure–volume relationship also may be considered.
The example to follow illustrates the application of Eq. 2.17 when the relationship between
pressure and volume during an expansion is described analytically as pVn constant.