Physics: Thermodynamics

Introduction

Beginning in the nineteenth century as the study of heat energy and transfer, thermodynamics established a new model for understanding the natural world. Instead of viewing the universe as a giant clockwork mechanism, it became more common to view it as a heat engine in which the conservation and transformation of energy determined the function of everything from atoms, molecules, and fields to chemical and geological phenomena, living processes, and possibly even the birth and death of the universe itself. Thermodynamics also plays an important role in the study of such current problems as global warming, pollution, and increased demands for efficient energy. Through its connection to the theory of information, thermodynamics is also a central part of computer science and globalized information systems such as the Internet.

Historical Background and Scientific Foundations

Early Theories of Heat

The ancient Greeks thought of heat as an actual substance. The philosopher Empedocles (c.490–430 BC) considered fire one of the four basic elements, along with water, earth, and air. By the fourth century BC Aristotle (384–322 BC) began to interpret heat as a quality that could be added to matter in different quantities. Warm bodies, he believed, had more of this quality than cold ones. Aristotle's ideas dominated natural philosophy throughout the Middle Ages, but by the end of this period scientists began to question the true nature of heat. Was it a separate material substance, as Empedocles claimed, or simply a state (condition) of ordinary matter?

By the seventeenth century several natural philosophers suggested that heat might be associated with some type of motion, but there was no consensus as to how this happened. Some believed that heat was the motion of the particles that made up ordinary matter, while others suggested that it was the motion of a subtle fluidlike material distinct from ordinary matter. Searching for the answer, natural philosophers made significant advances in the study of heat.

British natural philosopher and theologian Robert Boyle's (1627–1691) experimental studies of the relationship between heat and the volume of a gas led to what is now known as Boyle's law, which states that the pressure and volume of a gas varies directly with temperature. Galileo Galilei's (1564–1642) invention of the thermometer around 1592 allowed researchers to measure heat more precisely, but it also resulted in a certain confusion between temperature (the intensity of heat) and the total quantity of heat in a body. Edmond Halley (1656–1742), of Halley's comet fame, noted that the temperature of boiling water was the same as steam. Though it took additional heat to convert boiling water into steam, he didn't seem to understand the importance of this observation to a comprehensive theory of heat, however.

By the eighteenth century scientific ambiguity about the nature of heat had led to two distinct hypotheses: a material theory, in which heat was considered the motion of the ordinary particles of matter, and a material theory, eventually labeled caloric, in which heat was perceived as a subtle fluid that was distinct from ordinary matter but was attracted to and could combine with it. Many eighteenth-century advances in heat theory were discovered by chemists and physicians who supported the caloric theory. This led to the idea that heat was conserved, since it was believed that matter, even a subtle matter, could not be created or destroyed.

The material theory, however, was less accepted until the theory of the conservation of energy emerged in the middle of the nineteenth century.

The idea that heat was always conserved led some researchers to distinguish between its intensity and its quantity. In the middle of the eighteenth century, Scottish chemist and physicist Joseph Black (1728–1799) discovered that heat had different effects on various substances. For example, a 1-pound (0.45-kg) iron bar at 212°F (100°C) had more ability to burn a person's hand than a 1-pound block of wood at the same temperature. From this Black discovered that different substances had different “heat capacities,” a concept that he later refined into the concept of specific heat: the quantity of heat, now measured in calories, required to raise one gram of water 1° Celsius (33.8°F).

Black's idea of heat capacity also led him to explain Halley's observation that steam and boiling water both existed at the same temperature. Black argued that the heat required to change a substance from one state (in this case a liquid) to another (a gas), is called latent heat. Steam carries this heat energy as it evaporates, then releases it as it condenses. This concept was successfully applied in the development of steam engines.

By the end of the eighteenth century, natural philosophers began to criticize the caloric theory of heat. One of the first to do so was Benjamin Thompson (1753–1814), an expatriate American who had

supported the Royalist cause during the Revolutionary War. After spending time in England, where he was knighted by George III, he settled in Bavaria where he became minister of war and police and received the title Count von Rumford. While supervising cannon manufacture at the Munich arsenal, Rumford noticed that a seemingly inexhaustible amount of heat was generated as the boring tool rubbed against the inside of the cannon.

Since there seemed to be no limit to the amount of heat generated in this process, he argued that it was inconsistent with the caloric theory of heat, which postulated that only a fixed amount of heat existed in any material body. In a 1798 paper he argued that heat was a form of motion, since the motion of the boring tool was the only thing touching the cannon. Although it may seem obvious that Rumford's discovery played a crucial role in establishing a mechanical theory of heat, at the time most natural philosophers rejected his work and continued to support the caloric theory.

During the early years of the nineteenth century, while chemists focused on the distribution of heat, physicists focused on its transmission or transfer. The leading figure in this effort was the French scientist Joseph Fourier (1768–1830). Ignoring the material vs. mechanical theories debate, Fourier used mathematics to develop differential equations that described heat flow through bodies of different shapes. His theories found almost immediate use among scientists studying the effects of solar and geothermal heat on Earth's surface.

The Steam Engine

Along with early theories of heat, the other important influence on thermodynamics was the invention of the atmospheric steam engine in 1712 by British engineer Thomas Newcomen (1663–1729). Working from earlier discoveries that the atmosphere exerted a force and that condensing steam created a vacuum, Newcomen invented an engine in which steam condensed inside a cylinder fitted with a piston to create a vacuum; atmospheric pressure then pushed the piston to the bottom of the cylinder. At this point in time, however, scientists thought that the steam engine's power source was atmospheric pressure, not heat.

One of the first to recognize the heat's primary role in the steam engine was James Watt (1736–1819), a Scottish inventor and instrument maker at Glasgow University. While attempting to get a small Newcomen engine to work, he realized that the steam had to reheat the cylinder each time after it condensed—an inefficient process. Using Black's concept of specific heat, Watt knew that the iron cylinder absorbed much of the heat on each alternate cycle. Watt added a second cylinder in which the steam could condense and cool, allowing the main cylinder to stay hot. The engine's power was now clearly associated with the flow of heat from the hot boiler to the cool condenser.

Watt's 1769 patent included another important improvement that influenced the development of thermodynamics. He realized that the pressure of the steam entering the cylinder was usually higher than atmospheric pressure; this allowed him to extract extra work from the steam by using its expansive power to push the piston for part of its cycle. Understanding the role of expansion in the cycle of a heat engine would prove a crucial element of the foundation of thermodynamics.

Until this point, the work done by a steam engine was calculated simply by multiplying the volume of the cylinder by the atmospheric pressure. Because Watt's engine allowed the steam to expand as it pushed against the piston, pressure dropped throughout the stroke; this made calculating the work performed difficult. In 1796 one of Watt's assistants attached a small pressure gauge to the main cylinder. As the piston moved and the pressure dropped, a pen attached to the gauge marked a pressure-volume (PV) curve on a piece of paper. The area under the curve delineated the work done, and PV indicator diagrams became a fundamental element of the new science of thermodynamics. Although Watt's invention of the separate condenser made the role of heat in the steam engine seem obvious, conventional wisdom still considered the steam engine primarily a pressure engine rather than a heat engine.

The Beginnings of Thermodynamics

In 1804 Arthur Woolf (1776–1837) patented a new high-pressure steam engine that was much more efficient than Watt's. After the Napoleonic Wars ended in 1815 one of Woolf's partners in France began building compound steam engines, which allowed steam to expand in two stages. French engineers, many with significant backgrounds in science, were intrigued by the new design and sought to understand the reasons behind its dramatically increased efficiency. The leading figure in this development was French physicist and engineer Nicholas Léonard Sadi Carnot (1796–1832), who had been trained at the École Polytechnique, France's leading engineering school. Carnot, a supporter of the mechanical theory of heat, also believed that heat was the universal motive force responsible for such phenomena as wind and ocean currents. This led him to speculate that just as water could generate power by falling from a height, heat flow from a higher temperature to a lower one could also be used to generate power.

IN CONTEXT: THE MOTIVE POWER OF HEAT

Sadi Carnot's (1796–1832) Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance (Reflections on the motive power of fire and on machines fitted to develop that power, 1824) is one of the fundamental works of thermodynamics. In it, he argues that heat is not only the source of the steam engine's power, but is the motive power behind all natural phenomena. As such, heat must be studied as a general phenomena and not one simply limited to steam engines. Although others had noted the role of heat in various geological and meteorological phenomena, Carnot was the first to see it as a universal motive force:

“Every one knows that heat can produce motion. That it possesses vast motive-power no one can doubt, in these days when the steam-engine is everywhere so well known.

To heat also are due the vast movements which take place on the earth. It causes the agitations of the atmosphere, the ascension of the clouds, the fall of rain and of meteors, the currents of water which channel the surface of the globe, and of which man has thus far employed but a small portion. Even earthquakes and volcanic eruptions are the result of heat.”

SOURCE: Carnot, Sadi. Reflections on the Motive Power of Fire.

Edited by E. Mendoza, New York: Dover Publications, 1960.

During the eighteenth century, scientists and engineers developed a theory for the motive power of water that led to the discovery of optimum conditions for its most efficient use. Drawing an analogy between water and heat, Carnot developed a similar set of conditions for the most efficient use of the motive power of heat. He demonstrated that an ideal engine operated in a cycle (now known as a Carnot cycle), in which some working substance such as steam or air underwent a series of expansions and compressions; during the cycle, heat flowed from a higher to a lower temperature, doing work in the process. In his pamphlet Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance (Reflections on the motive power of fire and on machines fitted to develop that power, 1824), Carnot used this analysis to draw several important conclusions:

First he argued that an engine that was more efficient than one following a Carnot cycle could be used to run a Carnot cycle in reverse; this would accumulate heat, which, in turn, could be used to run the more-efficient Carnot engine—theoretically resulting in perpetual motion. Since perpetual motion is impossible, he argued, an engine following a Carnot cycle must be the most efficient possible.

Second, because an engine following a Carnot cycle was the most efficient possible, the power produced by heat engines depended only on the temperature difference through which the heat fell, not upon the working substance that was used in an engine.

Most importantly, while Carnot based his reasoning on the material theory of heat, his conclusions turned out to be independent of any particular theory. Although his ideas would later become the fundamental elements of the new science of thermodynamics, they had little impact on most scientists and engineers until the late 1830s and early 1840s, when the French engineer Émile Clapeyron (1799–1864), produced a mathematical analysis of Carnot's work that was later translated into English and German.

Conservation of Energy

During the 1840s Carnot's theory ran into conflict with the new discovery of the mechanical equivalent of heat by English physicist James Prescott Joule (1818–1889). Carnot's theory had explained the work in a heat engine as the result of the flow of heat from a higher temperature to a lower one, but Joule's experiments showed heat could be transformed into mechanical work and vice versa. Joule's experiments were part of a movement in science that would lead to the reformulation of mechanical theories in terms of the new concept of energy.

Joule grew up near the industrial city of Manchester, England, where his grandfather and father had established a large brewery. Joule spent two years studying with the famous meteorologist and chemist John Dalton (1766–1844) from whom he learned many of the skills of experimental chemistry. Although Manchester's manufacturing and railroads ran on steam power, by the 1830s Joule was studying the newly invented electric motor, which he believed could become a more economical power source than steam.

At the time, electric motors had only one-fifth the efficiency of the best steam engines. To understand this unfavorable comparison he conducted a series of experiments during the 1840s. Joule discovered that battery-generated electrical currents produced heat, a phenomenon that was clearly generated by chemical activity inside the battery. But electricity could also be generated by a magneto, in which a coil of wires was rotated in a magnetic field. This too generated heat. Since no chemical activity occurred, and no other part of the circuit was being cooled, Joule concluded in 1843 that heat was being generated by the mechanical effort used to rotate the coil of wires. He was able to demonstrate that 838 ft-lbs (116 kg/m) of work would raise 1 pound (0.45 kg) of water 1°F (0.55°C).

To demonstrate his belief that the mechanical equivalent of heat was universal and not peculiar to electric circuits, Joule carried out a series of further experiments that did not include electricity. In the most famous of

these, conducted in 1845, he placed a paddle wheel rotated by falling weights inside a water-filled cylinder. With this apparatus, he was able to calculate that the water's temperature increase was caused by the paddle wheel's rotation. In every experiment Joule found the mechanical equivalent of heat to be very close to what he measured in his electrical experiment.

Although his experiments demonstrated that mechanical work could be transformed into heat, Joule also believed that heat could be transformed into mechanical work. This led him to formulate a new conservation law. According to the material theory of heat, heat was always conserved, but the mechanical equivalent of heat meant that heat could appear in certain situations, as it did in Joule's various experiments, and that it could also disappear in others, as it did in a heat engine. Because its appearance and disappearance were always associated with doing or creating mechanical work, Joule proposed that while heat was not conserved, a quantity that natural philosophers had once labeled vis viva, or “living force,” was. This quantity was mv², where m is an object's mass and v its velocity. Today, using the modern concept of energy developed in the nineteenth century, we would say that this quantity is twice the kinetic energy of an object with mass m and velocity v. Joule was mistaken

in thinking that kinetic energy is conserved (neither diminished nor increased): Energy as such is conserved, but not any one of its forms (e.g., kinetic energy, potential energy, electrical energy, chemical energy, etc.).

The idea of the conservation of energy, which would later become the first law of thermodynamics, was based on the work of several scientists, including Joule, Julius Robert Mayer (1814–1878), and Hermann von Helmholtz (1821–1894). Mayer, a German physician serving aboard a ship in the East Indies in the early 1840s, noticed that the blood of patients in the tropics was brighter red than of those in Europe. He concluded that in a warm climate blood needed less oxidation to maintain body temperature than in a cold climate. This led Mayer to argue that other physiological processes, such as muscular activity, could also be the result of the chemical combustion of food.

If this were true it meant that heat (food combustion), was being transformed into work (muscular activity). After returning to Germany in 1842 he broadened this idea to include inorganic processes as well. He noted that a gas expanding against some external pressure did work and, in the process, absorbed heat. Using published data on the specific heat of gases, Mayer was able to calculate the mechanical equivalent of heat (independent of Joule, who published his results shortly afterward).

This work led Mayer to put forward the idea that would later be labeled the conservation of energy. He published a paper in 1842 in which he concluded that forces could be converted into each other, but were indestructible, or conserved. At the time, “force” (kraft, in German), was defined as the ability to cause motion; it would later become the definition of energy. Therefore, with some hindsight, one can give Mayer credit for the discovery of the conservation of energy.

While Mayer proposed the idea of the conservation of energy and Joule provided experimental evidence that established the principle, Hermann von Helmholtz provided its mathematical foundation. Like Mayer, von Helmholtz's interest in physiology led him to the conservation of energy. Von Helmholtz also still used the German kraft for force, since the concept of energy had not yet emerged. Von Helmholtz rejected the widely held notion that heat produced by animals was based on the existence of some special vis viva, and argued instead that it could be explained by normal physical and chemical processes, such as the oxidation of food. From this he concluded that there must be some constant relationship between mechanical work and heat. To support his belief in the constancy, or conservation, of force in physiology, von Helmholtz put forward a mathematical proof that the conservation of force held for all natural processes.

In his 1847 paper “On the Conservation of Force,” von Helmholtz showed that in any system of material objects governed by attractive and repulsive forces, the sum of the quantity known as vis viva plus something he called “tensional force” was a constant. Not until the concept of energy emerged in the 1850s did people recognize that vis viva and tensional force corresponded to kinetic and potential energy, respectively, and that his proof that the sum of the two were constant corresponded to a statement of the conservation of energy.

The First Law of Thermodynamics

Although Mayer, Joule, and von Helmholtz were major contributors to the discovery of the conservation of energy, its formulation as part of thermodynamics was primarily the result of the work of a group of British scientists and engineers, including William Thomson, later Baron Kelvin (1824–1907); William John Macquorn Rankine (1820–1872); and James Clerk Maxwell (1831–1879); along with the German scientist and engineer Rudolf Clausius (1822–1888). During the late 1840s Thomson realized that Carnot's theory of heat engines, which argued that work was produced by simply transferring heat from a higher temperature to a lower one, was contradicted by Joule's theory that heat was converted into work. Thomson also noted that irreversible phenomena, such as the conduction of heat through a solid, did not produce work; according to Joule's theory, however, the dissipated heat should be able to produce work.

At about the same time, Clausius wrote a paper in which he argued that Joule's and Carnot's theories could be reconciled if, during a Carnot cycle, only some of the heat was converted into work, the remainder was transferred from a higher temperature to a lower one, and there was some fixed relationship between the two processes.

During the first half of the 1850s Thomson and Rankine began to reformulate the laws of thermodynamics in terms of the new concept of energy. Although the term had a long history, it had been often used in vague and imprecise ways. Thomson and Rankine argued that it could be used as the basis for understanding all processes in natural philosophy, including mechanics, chemistry, electromagnetism, and thermodynamics. “Energy” came to be defined as the ability to do work.

Rankine further distinguished between the actual energy found in moving things, and potential energy, such as that stored in weights positioned at some height, electric charges, and certain types of chemical energy. (In 1862 Thomson and Scottish physicist and mathematician P.G. Tait [1831–1901] substituted the term kinetic energy for actual energy.) In 1853 Rankine reformulated von Helmholtz's conservation of force as the universal principle of the conservation of energy, which stated that the total energy in the universe was a constant. This became the first law of thermodynamics.

The Second Law of Thermodynamics

While the transformation of heat into work led to the formulation of the first law of thermodynamics, the problem of heat dissipation led to the second. As we have seen, Clausius argued that only a portion of heat in an engine was converted into work, while another portion was simply dissipated. In 1854 he began to reformulate his ideas, arguing that the dissipation of heat from a warmer body to a colder one had the “equivalence value” of the work required to move that heat back to the warmer body. In 1865 Clausius introduced the term “entropy,” from the Greek word for transformation or change, to refer to the equivalence value of the transformation of heat.

Clausius and Rankine, who had developed a similar formula for the equivalence value of heat, demonstrated that for reversible processes, such as the Carnot cycle, the change of entropy is zero. Clausius also analyzed irreversible processes, such as those encountered in actual heat engines, and discovered that in those cases the entropy always increased. Using the new concept of entropy, Clausius was able to formulate the second law of thermodynamics, which stated that the total entropy in the universe is always increasing.

While the concept of entropy became fundamental to thermodynamics, there was a great deal of debate as to how it should be interpreted physically. In the 1860s Clausius suggested that entropy might be associated with the dispersal or rearrangement of the molecules that composed matter. Maxwell went further and suggested that if the molecules in a body or gas had a range of velocities, represented by something like a bell curve, the second law of thermodynamics, or the entropy principle, might be essentially a statistical law rather than one that could be explained in terms of the motions of individual molecules.

In an 1877 paper, Austrian physicist Ludwig Boltzmann (1844–1906) developed statistical mechanics, which states that the properties of a substance's atoms determine its larger physical properties. Boltzmann's statistical mechanics expressed entropy as proportional to the probability of finding a system in a state with a given distribution of molecular motions. Since disorderly states were more probable than orderly states, the second law of thermodynamics could be interpreted as simply saying that a system would tend to go from a less probable state (lower entropy and more order) to a more probable state (higher entropy and more disorder).

Thermodynamics and Science

The first and second laws of thermodynamics had emerged from the study of heat, but scientists soon recognized that energy and entropy were universal concepts that could be applied to a wide range of scientific topics. One of the first to do so was American scientist Josiah Willard Gibbs (1839–1903). During the 1870s he applied thermodynamics to the field of physical chemistry, where he used concepts such as energy and entropy to study mixtures of substances in different chemical phases, such as solid, liquid, or gas.

Gibbs introduced temperature-entropy and volume-entropy diagrams, using them to formulate his “phase rule” relating the degrees of freedom of a chemical system to the number of components and the number of phases in it. For example, in a system where water exists in all three phases (ice, water, and steam), there are no degrees of freedom, so with any temperature or pressure change one or more of the phases will disappear. Gibbs's work became so widely known that historian Henry Brooks Adams (1838–1918) attempted to apply the phase rule to the study of history, postulating that history went through phases, which he labeled religious, mechanical, electrical, and ethereal. Brooks posited that just as changes in temperature and pressure could change the phase of water from solid to liquid or to a vapor, social and political forces could change history from one phase to another.

When thermodynamics was applied to geology and biology, it conflicted with the theory of evolution. Starting with Earth's present temperature and applying the formula for the dissipation of heat, Thomson argued that Earth could not have been hospitable to life for a long enough period in the past for Darwinian natural selection to have taken place. Other critics argued that the emergence of higher forms of life by purely physical means was highly improbable and therefore violated the second law of thermodynamics.

By the twentieth century scientists discovered that the basis for this criticism was faulty. When heat generated by radioactive decay was later taken into account, Earth's age was much older than Thomson's estimate. Others argued that the second law applies to the total entropy in the universe; this meant that there could be pockets of decreasing entropy, represented by the evolution of higher forms of life, as long as this was offset by an increase of entropy in some other part of the universe.

Thermodynamics also played a significant role in the development of a new model of cosmology. An important implication of the second law was the “heat death” of the universe. If dissipation were a fundamental characteristic of nature—not just heat engines—then at some future point entropy would reach a maximum. With everything at the same temperature, there would be no possibility of extracting any work from such a system. Also, since maximum entropy would be the most probable state, the universe could not move to a more probable state. But, as James Jeans (1877–1946) would later note, it was the movement from less probable to more probable that defined “time's arrow,” or the direction of the flow of time from past to future.

Therefore at a state of maximum entropy, or probability, the concept of the flow of time would become meaningless and the universe would effectively come to an end. This idea led many late-nineteenth century writers to begin to question the idea of human progress and raise the idea that Western civilization was in a decline. Such conclusions contributed to a general mood of pessimism at the end of the century. The universal application of the laws of thermodynamics also replaced the time-honored concept of a clock as a model for the universe with the concept of a heat engine.

By the end of the nineteenth century a number of scientists and philosophers, particularly in Germany, began to see the concept of energy as independent of any mechanical hypothesis. This group, led by the physical chemist Friedrich Wilhelm Ostwald (1853–1932), and known as “energeticists,” came to reject all material explanations of natural phenomena, including atomism, arguing instead that everything could be explained using the concept of energy. While most scientists continued to accept atomism, the energeticists' ideas influenced the philosophy of science, especially ideas put forward by the Austrian physicist Ernst Mach (1838–1916). Using Carnot's idea that the theory of heat engines was independent of any particular hypothesis about the nature of heat, Mach argued that thermodynamics should become the model for all scientific theories since it was independent of any hypothesis, such as atomism, that could not be confirmed by the senses and therefore might not be true.

Thermodynamics had a significant impact on twentieth century science. Applying its laws to electromagnetic radiation during the late nineteenth century led German physicist Max Planck (1858–1947) to suggest in 1900 that radiation was not emitted and absorbed in a continuous way but in discrete packages, called quanta. Shortly thereafter, Albert Einstein (1879–1955) applied Boltzmann's statistical mechanics to light, concluding that light exists as discrete particle-like packages of energy, later called photons. These discoveries became the basis for quantum mechanics during the early part of the twentieth century.

Modern Cultural Connections

In 1948 Claude Shannon (1916–2001), an American mathematician and electrical engineer working at Bell Labs, helped establish modern information theory by drawing on thermodynamics. He noted that the more disorder there was in an information system the less information it contained. This led him to see information as the negative of entropy and to develop a mathematical equation for it that was similar to Boltzmann's equation of entropy. This work has been one of the foundations of the modern information age.

About the same time, Ilya Prigogine (1917–2003), a Russian-born chemist living in Belgium, applied thermodynamics to systems that were very far from equilibrium, which he labeled dissipative. This work, which led him to become one of the founders of chaos theory, earned him the 1977 Nobel Prize for chemistry in 1977.

Thermodynamics' most recent role has been in the understanding of black holes. Between 1972 and 1974 British theoretical physicist Stephen Hawking (1942–) and Mexican-born theoretical physicist Jacob Bekenstein (1947–) used a combination of quantum mechanics, the general theory of relativity, and thermodynamics to predict that black holes would slowly radiate away. They discovered that in this process, the black hole's event horizon (a line that, if crossed, will cause an object to fall into a black hole) can never decrease. Since this was similar to the second law, in which entropy can never decrease, they proposed that black holes have entropy that is proportional to the area of the event horizon.

While many scientific theories were overthrown by the theory of relativity and quantum mechanics during the early twentieth century, the laws of thermodynamics remained unchanged and were recognized as universal laws essential to understanding a wide range of scientific and technological phenomena, many of which were far removed from their original study of heat and steam engines.

See Also Physics: Fundamental Forces and the Synthesis of Theory; Physics: Wave-Particle Duality.

bibliography

Books

Cardwell, D.S.L. From Watt to Clausius: The Rise of Thermodynamics in the Early Industrial Age. Ithaca, NY: Cornell University Press, 1971.

Carnot, Sadi. Reflections on the Motive Power of Fire, by Sadi Carnot; and other Papers on the Second Law of Thermodynamics, by É Clapeyron and R. Clausius. E. Mendoza, ed. New York: Dover Publications, 1960.

Harman, P.M. Energy, Force, and Matter: The Conceptual Development of Nineteenth-Century Physics. Cambridge: Cambridge University Press, 1982.

Kuhn, Thomas S. “Energy Conservation as an Example of Simultaneous Discovery.” In Critical Problems in the History of Science. Edited by Marshall Clagett. Madison: University of Wisconsin Press, 1959.

Layton, Edwin T., Jr., and John Lienhard, eds. History of Heat Transfer. New York: American Society of Mechanical Engineers, 1988.

Prigogine, Ilya, and Isabelle Stengers. Order Out of Chaos: Man's New Dialogue with Nature. Boulder, CO: New Science Library, 1984.

Smith, Crosbie. The Science of Energy: A Cultural History of Energy Physics in Victorian Britain. Chicago: University of Chicago Press, 1998.

Smith, Crosbie, and M. Norton Wise. Energy and Empire: A Biographical Study of Lord Kelvin. Cambridge: Cambridge University Press, 1989.

David F. Channell