Constraints

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An important concept in the study of thermodynamic systems is that of constraints. This notion is best illustrated by example. Consider the gas in a cylinder whose volume is determined by the position of a piston as in Figure 1.3. Several constraints are operative in this case. Most obvious is the position of the piston. It constrains the volume to have a certain value. If the piston is removed by a small amount the constraint is said to be relaxed. Note that a force must be applied

PISTON

Figure 1.3 Schematic diagram of a gas filled cylinder with adiabatic walls and a movable piston.

ADIABATIC WALLS

Figure 1.3 Schematic diagram of a gas filled cylinder with adiabatic walls and a movable piston.

(actually relaxed, then gradually reapplied) externally to implement this change in the constraint. If the piston is removed by a small amount, some agent must perform work to restore it to its original position. Similarly, the walls that are impervious to the transfer of thermal energy form a constraint. If a leak of thermal energy were to occur, such as on bringing the system into contact with a temperature reservoir at a slightly different temperature, this constraint would be said to have been relaxed and the thermodynamic coordinates of the system will have to be changed to restore the original temperature. Thermodynamic systems are always subject to certain constraints and their nature and number are essential ingredients in the description of the system and its state.

Consider two thermally isolated chambers adjacent to one another separated by a partition. On one side is gas A and on the other is gas B. Let the chambers have the same temperature and pressure. The partition forms a constraint restricting the two gases from mixing. If the partition is removed, the constraint is relaxed and the two systems will pass through nonequilibrium states to their final well-mixed equilibrium state. The irreversible process following removal of the constraint represents one which for ideal gases involves no changes in pressure or temperature, but external work must be performed to restore the original conditions.

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