MAXIMUM ENTROPY AND MINIMUM ENERGY

J. MAXIMUM ENTROPY AND MINIMUM ENERGY

The concept of mechanical states is illustrated in Figure 38 using the example of balls placed on a surface of arbitrary topography. Position A represents a nonequilibrium condition, whereas B, C, D, and F are different equilibrium states. A small disturbance conveyed to the ball placed at position C will cause it to move and come to rest at either position B or D. Therefore, position C is an unstable equilibrium state for the ball. Small disturbances that move the ball to positions B and D will dissipate, and the ball will return to rest. If the ball is placed at B, a large disturbance may cause it to move to position D (which is associated with the lowest energy of all the states marked in the figure). If the ball is placed at position

D and then disturbed, unless the dis- turbance is inordinately large, it will return to its original (stable) position. Position D is an example of a stable equilibrium state, whereas position B represents a metastable equilibrium state. Also it is noted that the change Figure 38: Mechanical equilibrium states.

of states say from (C) to (B) or (D) are irreversible. A criteria for “stability”can also be de- scribed based on the potential energy associated with the various states depicted in Figure 38 . If the potential energy decreases (i.e., δ(PE) < 0) as a system is disturbed from its initial state,

that state is unstable (state C). On the other hand, if the potential energy increases once the system is disturbed ( δ(PE) >0), that initial state is stable (states B and D).

Similarly, the stability of matter in a system can be described in terms of its thermo- dynamic properties. A composite system containing two subsystems is illustrated in Figure 39 . The first subsystem consists of an isolated cup of warm coffee or hot water (W), while the room air surrounding it is the other subsystem (A). The internal constraints within the com- posite system are the insulation (which is an adiabatic constraint) around the coffee mug and the lid (which serves as a mechanical constraint) on the cup. The two subsystems will eventu- ally reach thermal equilibrium state, once the constraints are removed. The magnitude of the equilibrium temperature will depend upon the problem constraints. For example, if the walls of the room are rigid and insulated, the temperature of the room air will increase as the coffee cools. Consequently, the air pressure will increase, but the internal energy of the combined system will not change.

If the mechanical constraint is still in place, it is only possible to reach thermal equilib- rium. If the mechanical constraint is removed, say, by using an impermeable but movable pis- ton placed on top of the water, then thermo-mechanical (TM) equilibrium is achieved. If the piston is permeable, in that case the water may evaporate and also reach phase (or chemical) equilibrium. Therefore, the conditions of a system depend upon the constraints that are im- posed. If the walls of the composite system are uninsulated, heat losses may occur from the system, thereby reducing the internal energy. The equilibrium temperature will be lower for this case. Therefore, equilibrium may be reached in a variety of ways so that various scenarios may be constructed, depending on the constraints, as follows.

The room may be insulated, impermeable, and rigid so that the composite system is isolated. In this case entropy generation will occur due to irreversible processes taking place inside the system.

The room may be diathermal, rigid and impermeable. Interactions with the environ- ment (which serves as a thermal energy reservoir at a temperature T 0 ) are possible. In this case the combined entropy of the coffee and room air may not change as the two subsystems undergo the equilibration process. The coffee will transfer heat to the room air, which in turn will transfer it to the environment. Consequently, the internal energy of the composite system will decrease.

The room might be diathermal with a flexible ceiling allowing the pressure to be con- stant during the process. In this case, we can show that the enthalpy decreases while the en- tropy, pressure, and mass are fixed.