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Work at the Moving Boundary of a Simple Compressible System

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Equation

(Eq2)      1W 2 = ∫ 2   1 P dV

Work done on the air during a quasi-equilibrium compression process

Nomenclature

1W 2	work from state 1 to state 2
P	pressure
V	volume
A	area
L	length

Details

There are a variety of ways in which work can be done on or by a system. These include work done by a rotating shaft, electrical work, and the work done by the movement of the system boundary, such as the work done in moving the piston in a cylinder. Some detail of the work done at the moving boundary of a simple compressible system during a quasi-equilibrium process will be considered.

Consider as a system the gas contained in a cylinder and piston, as shown.



Let one of the small weights be removed from the piston, which will cause the piston to move upward a distance dL. This can be considered a quasi-equilibrium process and the amount of work W done by the system during this process can be calculated. The total force on the piston is PA, where P is the pressure of the gas and A is the area of the piston. Therefore, the work δW is:

δW = PA dL

But A dL = dV, the change in volume of the gas. Therefore,

(Eq1)

δW = P dV

The work done at the moving boundary during a given quasi-equilibrium process can be found by integrating Eq1. However, this integration can be performed only if the relationship between P and V is known during this process. This relationship may be expressed in the form of an equation, or it may be shown in the form of a graph.

Consider first a graphical solution. As an example, a compression process such as what occurs during the compression of air in a cylinder will be examined as shown.



At the beginning of the process the piston is at position 1, and the pressure is relatively low. This state is represented on a pressure–volume diagram (usually referred to as a P–V diagram). At the conclusion of the process the piston is in position 2, and the corresponding state of the gas is shown at point 2 on the P–V diagram. Therefore, points 1 and 2 represent individual states and the path from state 1 to state 2 represents the process. Assume that this compression was a quasi-equilibrium process and that during the process the system passed through the states shown by the line connecting states 1 and 2 on the P–V diagram. The assumption of a quasi-equilibrium process is essential here because any point chosen on line 1–2 represents a definite state, and these states will correspond to the actual state of the system only if the deviation from equilibrium is infinitesimal. The work done on the air during this compression can be found by integrating Eq1:

1W 2 =

∫

2   1

δW =

∫

2   1

P dV

or:

(Eq2)

1W 2 = ∫ 2   1 P dV

The symbol ₁W ₂ is to be interpreted as the work done during the process from state 1 to state 2. It is clear from the P–V diagram that the work done during this process:

∫

2   1

P dV

is represented by the area under the curve 1–2, area a–1–2–b–a. In this example the volume decreased, and the area a–1–2–b–a represents the work done on the system. If the process had proceeded from state 2 to state 1 along the same path, the same area would represent work done by the system.

Further consideration of a P–V diagram, such as in the figure, leads to another important conclusion. It is possible to go from state 1 to state 2 along many different quasi-equilibrium paths, such as A, B, or C as shown.



Since the area underneath each curve represents the work for each process, the amount of work done during each process not only is a function of the end states of the process but depends on the path that is followed in going from one state to another. For this reason work is called a path function or, in mathematical parlance, δW is an inexact differential.

This concept leads to a brief consideration of point and path functions or, to use another term, exact and inexact differentials. Thermodynamic properties are point functions, a name that comes from the fact that for a given point on a diagram, such as in the figure, or surface, as shown, the state is fixed, and thus there is a definite value of each property corresponding to this point. The differentials of point functions are exact differentials, and the integration is simply:

∫

2   1

P dV = V2 − V1

Thus, the volume in state 2 and the volume in state 1 can be spoken of, and the change in volume depends only on the initial and final states.

Work, however, is a path function, for, as has been indicated, the work done in a quasi-equilibrium process between two given states depends on the path followed. The differentials of path functions are inexact differentials. Thus, for work:

∫

2   1

δW = 1W 2

It would be more precise to use the notation ₁W _2A, which would indicate the work done during the change from state 1 to state 2 along path A. However, it is implied in the notation ₁W ₂ that the process between states 1 and 2 has been specified. It should be noted that work in the system in state 1 or state 2 is never spoken of, and thus it would never be written W₂ − W₁.

In evaluating the integral of Eq2, it is of desire to determine the area under the curve of the above figure. In connection with this point, two classes of problems are identified:

1. The relationship between P and V is given in terms of experimental data or in graphical form (as, for example, the trace on an oscilloscope). Therefore, Eq2 may be evaluated, either by graphical or numerical integration.
2. The relationship between P and V makes it possible to fit an analytical relationship between them. Then direct integration would apply. One common example of this type of functional relationship is a process called a polytropic process.