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Entropy and Boltzmann Chance

Thermodynamic Entropy

If a system is not exchanging work or heat with other parts of the universe for a sufficiently long time, it ceases to change in any of the causal ways of change in classical mechanics. The mass of such a system will have become unchanging, if it were previously changing, for it will no longer exchange mass with its environment. The angular momentum of such a system will have become unchanging, if it were previously changing, for it will no longer receive torque from its environment. Such a system will no longer become more liquid and less solid nor continue any other phase transitions it might have been undergoing in the past. The pressure of such a system will have become a steady constant value in time, and it will be that value throughout the system. Chemical reactions in the system will have stopped.

An overtly unchanging state such as that is called an equilibrium state. Notice that an isolated system that has reached its equilibrium state is yet a system meeting Ayn Rand’s measurement-identity criterion for existence. “If anything were actually ‘immeasurable’, it would bear no relationship of any kind to the rest of the universe, it would not affect nor be affected by anything else in any manner whatever, it would enact no causes and bear no consequences—in short it would not exist” (ITOE 39). I say that the isolated system in its equilibrium state satisfies Rand’s measurement-identity criterion of existence: The isolated system can become not isolated and resume measurement-enabling causal interactions. Meanwhile, the isolated system continues to possess its measurable characteristics and to stand in measurable relations to its environment, continues to stand in relations such as spatial location. The concept of a thermodynamically isolated system is something of an idealization, an important one. Some have argued that to properly explain time-asymmetric thermodynamic changes, one must explicitly factor in certain ways in which the idealized reference system is not isolated from its environment (Sklar 2000, 47–48).

If two systems, each in isolated equilibrium states, are brought into contact or otherwise allowed to exchange energy with each other, they might begin to change. If they do not, they are said to be in the same equilibrium state; if they do begin to change, their two previous equilibrium states are said to have been different from each other. Two systems in the same equilibrium state can be enormously different from each other. Nevertheless, if they do not begin to change when they are allowed to exchange energy, then they had been (and still are) in the same equilibrium state.

Consider two isolated systems that have each reached an equilibrium state, where those two states are not the same state. The two systems are not in the same thermodynamic state. Let them be brought into contact and allowed enough time to reach a new state of equilibrium. Now let the two systems again be isolated. Each simply remains in this new equilibrium state. This state for each is different from its equilibrium state prior to the contact. It needs to be understood that when we speak of an isolated system’s equilibrium state we are not talking about some single self-same state across occasions of a system’s isolations. Rather, it is as if when visiting a neighbor’s home (or when he visits your home) it gets renovated, duplicated, and separated such that you and your neighbor each has a new home. 

The thermodynamic state of a system is characterized by a definite internal energy of the system and by the temperature of the system. This is so for all states accessible to a system, both its equilibrium states and its non-equilibrium states. Systems that have been brought into contact and allowed to reach a common equilibrium state have the same temperature.

Heat is energy transferred due to a temperature difference. The first law of thermodynamics is a law subsuming the conservation of energy. A system performing work without exchange of heat would be losing internal energy equal to the amount of work performed; an approximation of such a system would be steam passing through a turbine. A system to which work is applied without exchange of heat would be gaining internal energy equal to the amount of work being applied; an approximation of such a system would be water that is being pumped. The first law says that the amount of work exchanged by a system not exchanging heat depends only on the initial and final states of the system.

A cyclic thermodynamic system is one that returns to a certain thermodynamic state repeatedly. Applied to a cyclic system, the first law means that the work put into or performed by the system during one cycle will be proportional to the amount of heat added to or released from the system during the cycle. Applied to any system whatever undergoing some change in thermodynamic state—any system whose boundary is being crossed by heat or work—it means the change in the energy of the system will equal the net energy crossing the boundary.

The second law of thermodynamics says that if a system in its present state were isolated from other systems, there is exactly one state of equilibrium associated with its present state (or with any other of its allowed states having the same internal energy and number of particles and subject to the same external constraints [such as volume]) to which the system will transition if not already in that equilibrium state. This transition in isolation leaves no effects on the state of the system’s environment. The second law cannot be derived from the laws of classical mechanics or from the laws of quantum mechanics.

A reversible process is one in which the system and its environment can be restored to their initial states (except for differences that are smaller than any changes that occurred during the restoration). The transition of an isolated system in its equilibrium state back to the state from which it approached equilibrium would change the state of the system’s environment in contravention of the second law and the thermodynamic concept of equilibrium. 

The second law implies that it is impossible to construct a cyclic device whose cycle produces no effect other than yielding work and exchanging heat with a single heat reservoir. A heat engine is a cyclic system that yields work by transfer of heat from a higher-temperature reservoir to a lower-temperature reservoir. An example of a heat engine would be a steam-turbine power plant. Water receives heat from a boiler, is converted to steam, hits the blades of a turbine, cools, condenses, and is pumped back to the boiler. The blades of the turbine, when hit by the steam, cause the shaft to which they are connected to rotate. Heat is put into the system at the boiler and work is gotten out at the rotating shaft.

The first law tells us two things about this system: the rotary work yielded at the turbine shaft will not be greater than the heat taken in at the boiler, and the rotary work yielded at the turbine shaft equals the difference of the heat taken in at the boiler and the heat given up at the condenser (and all the other heat losses). It is the second law, not the first, which says our cyclic system must have a second, cooler heat reservoir in order to extract work from our hotter heat reservoir. It is the second law, not the first, that says the proportionality of the work delivered by the turbine and the heat taken in at the boiler will be less than unity, that the heat given up to the cooler reservoir will not be none at all.

The second law also implies that heat will not flow directly from a medium at lower temperature to a medium at higher temperature. A household refrigerator is a cyclic system that causes a flow of heat from a cooler medium to a hotter medium by work of a compressor. The second law means there can be no refrigerator without a component that is doing work, such as a compressor, or without some other energy-consuming components, such as those in a thermoelectric refrigerator. 

Let the temperatures of the two heat reservoirs for the steam-turbine power plant each be constant. The temperature of the water we would make into steam cannot be higher or equal the temperature of the boiler heat reservoir; otherwise, there will be no extraction of heat from the reservoir. Let the system water enter the boiler at some temperature below the temperature of the boiler heat reservoir, let the system water receive heat, and let it be converted to steam. At the boiling point, water changes from liquid to gas (steam), and during this conversion, heat continues to flow into the water-steam without increasing its temperature. Let the steam temperature at the condenser be some value above its condensation temperature, which condensation temperature must be higher than the cooling reservoir. At the condensation point, water changes from steam to liquid, and during this conversion, heat is given up by the steam-water without lowering its temperature.

For given temperatures of the two heat reservoirs, the closer the temperatures our system water into the boiler and our steam into the condenser are to the reservoir temperatures, the more work can be gotten out of the system for a given amount of heat put into the system and a given amount of heat losses in the route from the boiler to the condenser. A heat transfer between media whose temperatures are infinitesimally close to each other is a reversible heat transfer. An imaginary limiting type of heat engine which releases heat only to the low-temperature reservoir and in which the heat transfers at the reservoirs are accomplished across infinitesimal temperature differences, that limiting heat engine would get the most work possible from the heat drawn in from the high-temperature heat reservoir.

This limiting type of heat engine is conceptually useful. Nonetheless, we would get no coin for the work it would produce. Between an infinitesimal difference of temperature, there can flow only an infinitesimal amount of heat in a finite time. The first law says the work we could get from our limiting heat engine in a cycle is the difference between the infinitesimal amount of heat taken in at one reservoir and the infinitesimal amount of heat released at the other reservoir.

The cycle lately described as a limit of the heat-engine cycle is called the Carnot cycle. It is also the limit of another device in which the direction of heat flow and the direction of system-water flow are the reverse of their directions in the heat engine. That device would be a refrigerator.

The thermal efficiency of a heat engine is the ratio of work output to heat input. The Carnot cycle is independent of what particular working substance is circulated in a particular cyclic device. The efficiency of the Carnot cycle associated with a particular real heat engine depends only on the temperatures of the two heat reservoirs being employed by the particular heat engine. For a steam-turbine heat engine, its associated Carnot thermal efficiency is that proportion of the heat taken in at the boiler that would be delivered as rotary work from the turbine, assuming the only heat leaving the system to be at the condenser. Real heat engines give up heat not only at the condenser, but elsewhere along the way. The conceptual artifice of the Carnot cycle establishes the maximum possible efficiency for a cyclic device such as a heat engine or refrigerator.

Steam hitting turbine blades will have its own state changed. Hitting the turbine blades, delivering work in a reversible way in an imagined Carnot cycle, steam will have its own state changed. For such a change of state, the work that could be delivered (all too slowly) in a reversible process would be greater than the work deliverable irreversibly by that very same change of steam state. This is a consequence of the second law.

We call the amount of work that could be gotten from a given change of state under a reversible process the available work. The first and second laws taken together imply that when a system undergoes a change of state, in a reversible process, the change in the system’s internal energy will equal the available work associated with that reversible change of state. When a system undergoes a change between those same two end states, but by an irreversible process, the change in the system’s internal energy will be greater than the available work associated with that change of state. We say the difference between the change of internal energy and available work is “wasted” work. Entropy is the thermodynamic property whose change in value is a measure of that difference. The magnitude of the change in the entropy of a system changing states is directly proportional to the change in the system’s internal energy minus the available work associated with change between those two end states. 

According to the laws of thermodynamics, the entropy of an isolated system is never decreasing, always increasing or constant. Any process in any system can be conceptually changed into a subsidiary process in an isolated system that includes all the systems with which the subsidiary system is interacting. For this larger isolated system, the total entropy never decreases. For every process of nature, the total entropy of all the systems involved never decreases.

* Sklar, L. 2000. Theory and Truth – Philosophical Critique within Foundational Science. Oxford.

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