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Fig. 1.1 Kelvin and the gas thermometer

### 1.1.2 Heat and Work

Work in a mechanical sense was a well-known concept very early in history. Perhaps the most fundamental form of work is that done against or by gravity in raising or lowering weights. Other devices that produce (or accept) work include springs, piston/cylinders, moving an object against a resisting force, turbines, electrons moving in a electric potential gradient and muscle contraction. Heat, however, is none of these and cannot be classified as work.

Heat, energy and temperature are closely related but strictly distinct concepts. Briefly but sufficiently, heat is energy in transit from a body to another at a lower temperature. The key to the definition is "in transit". One cannot refer to the "heat content" of an object. A body contains energy but not heat. The latter is manifest only as energy moving in response to a temperature difference. Nor is heat equivalent to temperature; "hot", although sounding similar to heat, refers to temperature.

An earlier, and erroneous, conception of heat was prevalent at the end of the 18th century. Heat was viewed as a "fluid" (called caloric) that is contained within a body and moves from a body at high temperature to one at low temperature. The similarity to the flow of a real fluid due to a difference in pressure is obvious. The problem with the theory is that caloric was thought to be something contained in a finite quantity by a body. If caloric flows out of a body, there should be a limit to the extent of heat transfer. This is obviously not so. Rubbing an eraser on paper heats the paper slightly, but in no way removes anything from the inside of the eraser. What is actually happening is conversion of the work expended by the scribbler in moving the eraser over the paper into heat. About 75 years passed before the caloric idea of heat was finally rejected in favor the temperature-difference-driven energy-in-transit notion described in the preceding paragraph.

The person most responsible for the refutation of the caloric theory was an American, William Rumford, in the employ of a German nobleman. His job was fabrication cannons for the Prince, a task which involved boring out large ingots of iron. As shown in Fig. 1.2, this involved horsepower delivered to the boring operation by a series of gears and pulleys. Rumford noticed that the water in which the cannon was immersed continued to heat up and boil as long as the boring operation continued. This, he reasoned, could not be explained by the caloric theory of heat, which posited a fixed quantity of the ineffable fluid in a body. He recognized the work of the horses, transmitted to the boring tool, was the source of the heat entering the water. Thus was the equivalence of heat and work first dimly recognized and heat given its current interpretation as energy leaving or entering a body.

If the flow of energy in Rumford's apparatus is followed back from the contraction of the muscles of the horses, the next transfer mechanism is utilization of the chemical energy carried in the horse's blood in the molecular form of glucose. This chemical is in turn produced from carbon dioxide and water by absorption of visible light from the sun in the grass eaten by the animals. The electromagnetic radiation is a byproduct of the fusion reactions in the sun. In a sense, then, Rumford's cannon boring operation is nuclear-powered.

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Plant food (glucosa)

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Fig. 1.2 Rumford's cannon-boring job

Although heat and work were recognized as manifestations of energy transfer between bodies, they could not be quantitatively coupled unless they could be expressed in common units. Typically, work is expressed as the product of force and distance, or Newton-meters, which equals a Joule. The conventional unit for heat, on the other hand, is calories, which is the quantity of heat that needs to be delivered to one gram of water to raise its temperature by one degree Celsius. The person most associated with solving this problem is James Joule.

Along with many serious experiments, Joule is reported to have conducted one that is both correct technically but seemingly extraordinarily difficult to render quantitative. He is supposed to have measured the temperature of the water flowing over the top of a waterfall and of the pool at the bottom (Fig. 1.3). The latter is slightly higher than the former because of the conversion of potential energy to kinetic energy and finally to heat. Whatever the validity of this tale, the correspondence of the units of heat and work, also known as the mechanical equivalent of heat, was established as 4.184 Joules per calorie3.

3 For a 100-meter-high waterfall, the temperature rise is ^ oC.

Fig. 1.3 Joule and the waterfall experiment

1.1.3 Carnot, Clausius and the Second Law

At the beginning of the 19th century, improvement of steam engines was a frustrating trial-and-error operation; eventually the practitioners began to realize that no matter how clever the improvement, there appeared to be a limit to the efficiency of these devices. In 1824, a young (28 year old) military engineer named Sadi Carnot produced a fundamental analysis of any engine that converted heat to work that was to become the basis of the Second Law of Thermodynamics4. The working fluid did not have to be steam, the theory did not depend on the source of heat or the details of the engine. The efficiency depended solely on the temperature from which the engine received heat and the temperature of the reservoir to which it rejected unusable heat. This deduction was remarkable because it was proposed well before two major features of thermodynamics with which it deals were understood. The first is the nature of heat and the second is the notion of absolute temperature.

As shown in Fig. 1.4, what Carnot conceived was an idealized cycle which represented any continuous engine. With the steam engine as the motivation, the engine receives heat from a high-temperature reservoir (e.g., hot steam), produces mechanical work (e.g., driving a piston) and very important, rejects heat to a cold reservoir (e.g., condenser water or the atmosphere). Carnot drew his far-reaching conclusions from two critical insights. First, the process in the engine had to be cyclic, in which the working fluid endlessly circulated through the device but periodically returned to its initial state. Second, and more profound, the steps through which the engine and working fluid passed had to be reversible. Reversibility is a concept that is difficult to define precisely, but is readily recognized when observed or described. Some of the more obvious requirements are the absence of dissipative processes such as friction between moving parts or

4 "Reflections on the Motive Power of Fire and on Machines Fitted to Develop that Power"

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Fig.1.4 Carnot's heat engine turbulence in the working fluid. Not so obvious is the requirement that the temperature differences between the hot reservoir and the fluid in the engine at the point of receipt of the input heat flow and between the engine at its coolest point and the low-temperature reservoir accepting reject heat had to be infinitesimally small. Seen another way, only infinitesimally-small changes in the temperatures of the two thermal reservoirs are required to reverse the entire operation (that is, for the engine to consume work and act as a refrigerator).

A far-reaching consequence of the seemingly-innocuous claim that mechanical work can be extracted from a reversible engine connecting two bodies at different temperatures is a qualitative statement of the Second law of thermodynamics. This connection was enunciated not by Carnot but by Rudolph Clausius some two decades after Carnot's epic work. The argument, based on Fig.1.5, is as follows. A corollary of Carnot's assertion that a temperature difference must be capable of producing work is the impossibility of the reverse process (transferring heat up a temperature difference) without supplying external work. Engine #1 is perfectly reversible but operated as a refrigerator. The objective is to prove that no other engine operating between the same two temperatures can have a higher efficiency of converting heat to work than that of engine #1.

Suppose that engine #2 is more efficient than engine #1. Then less heat is required for #2 to produce the work required to operate #1, or Qi > Q2. Taken together, the two engines are transferring Q1 - Q2 of heat from the low-temperature reservoir to the high-temperature reservoir, without assistance from external work. This violates the corollarly to Carnot's assertion, so the supposition that engine #2 can have a higher efficiency than engine #1 is incorrect. Two conclusions are drawn from this exercise: a) No engine can have a higher efficiency than a reversible engine when the operating between the same two temperature reservoirs; b) It is impossible to transfer heat from low to high temperature without doing work.

Fig. 1.5 Clausius' demonstration of the maximum efficiency of Carnot's engine

Conclusion b) is a qualitative statement of the Second law of thermodynamics. Conclusion a) is the logical basis for a quantitative statement of the Second law. In his time, Carnot could not have converted his analysis into quantitative form for several reasons.

First, heat (as caloric) was thought to be conserved. Just as the amount of water passing through a waterwheel is not diminished, so the amount of caloric (heat) rejected by an engine was believed to be the same as the amount entering the engine. The work performed by the engine was at the expense of the quality of the caloric, not its quantity.

Second, the First law of thermodynamics had not been formalized. Had it been, Carnot would have recognized that the work done by the cyclical engine subtracted from the heat it received. In fact, the notion of energy was fuzzy (as it is today).

Third, the concept of a thermodynamic temperature (with an absolute zero value) was yet to be formulated, so Carnot was constrained to asserting that the efficiency of the ideal engine increased as the hot reservoir became hotter and as the cold reservoir became colder.

Lastly, the concept of entropy and its quantitative connection to heat was decades away. Carnot would have needed this to formulate the efficiency of the ideal engine even if the thermodynamic temperature scale had been established.

### 1.1.4 The First law and energy

About the middle of the 19th century, it all came together. Rumford's cogitations on the boring of cannons suggested that heat and work were somehow interconvertible and that the work of the horses corresponded to the heat generated by the cannon-boring operation. The concept of temperature was well-established and recognized as a different entity from heat. An essential step was the quantitative understanding of the relation of heat and work, as furthered by the work of Joule.

Assembling all of these hints into a formula for the First Law is attributed to the German physicist Rudolph Clausius, who, in 1850, more or less said that in a particular process, the difference between heat and work is a property of the substance involved called energy. More specifically, the type of energy was termed "internal energy", in order to differentiate it from potential and kinetic forms of energy as treated in mechanics. A body contains a certain amount of internal energy, and when some of it leaves the body, it does so as either heat or work, or both.

There is no clear definition of internal energy. It is somehow contained in the motion of molecules of a gas, in the vibrations of atoms in a solid, the attraction between atoms in molecules and in the forces that bind the nucleus. All we really know are that it is "conserved", in the sense that it can never be created or destroyed. In a particular process, its change is the difference between heat input to and work done by the body. Richard Feynman said "We have no knowledge of what energy is.. it is an abstract thing...". Fortunately, none of these uncertainties prevent attaching firm numerical values to the internal energy of all sorts of substances, at least to within an additive constant. In transferring from one place to another, energy takes on a variety of forms, all of them collectible under the rubric of either heat or work.

Although internal energy appears to be intimately connected to heat and work via the First Law of Thermodynamics, its essential quality is closer to those of temperature and pressure; all three are thermodynamic properties, or attributes of a body held in well-defined conditions.

### 1.1.5 Entropy and the Second Law of Thermodynamics

The idea that attributes of a body constitute its thermodynamic properties has been alluded to in the preceding section. Some properties, such as volume, pressure, and even temperature, are clearly understandable and easily measurable. There is no meter for internal energy yet it is a property as surely as the preceding three. What then is a metric for a property? The most reliable seems to be the following: in moving from one condition to a new one, a quantity is a property if its change does not depend on the path taken. On this basis, neither heat nor work is a property. However, their difference is path-independent, and the property represented by this difference is the internal energy - this is just the First Law.

Unlike internal energy, entropy has no connection to ordinary human experience. It is a near-totally abstract quantity. Yet it is a property on the same level as temperature, pressure, volume, and internal energy. The path-independence criterion was utilized by Clausius in discovering (for that is the only word for it) entropy. His method was the same as that described above for the internal energy, namely, to search for a combination of heat and work in processes between the same two states that is independent of the path between the two states..Such a combination is a property . Clausius found that the heat divided by the absolute temperature (or more generally, the integral of the heat divided by the temperature) is the same for all process routes between fixed beginning and end states. He called this odd property entropy, and with it, quantitative forms of the First and Second Laws were complete.

Unlike internal energy, entropy is a concept totally unfamiliar to human experience. Two decades following its discovery by Clausius, Ludwig Boltzmann derived an equation relating entropy to the degree of order of a system. Order and disorder are comprehensible to the mind, so entropy finally had an anchor in human experience.