Tabak J. Mathematics and the Laws of Nature: Developing the Language of Science

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130 MATHEMATICS AND THE LAWS OF NATURE

In this experiment just as in the one described previously, gravity

does the work by pulling the weights down. This work is transmit-

ted to the water by the spinning paddles. The result is measured

by a thermometer. The increase in temperature of the water

enabled Joule to compute how much heat had gone into the water.

What Joule discovered was that as the paddles spun in the water,

the temperature of the water increased slightly. Joule was essen-

tially creating “caloric” by stirring water. The heat was the result

of friction between the paddles and the water, between the water

and the walls of the container, and inside the water itself as one

region of water flowed past another. The resulting friction raised

the temperature of the entire system. Joule had created heat, and

he had done so in a way that enabled him to state how much work

had been performed in the creation of the heat.

Joule’s experiments disproved the caloric theory. By perform-

ing variants of the same experiment he was able to show in a

rough way that the amount of heat produced for a given amount

of work is independent of the way the work is performed. It was

a fairly convincing set of experiments, although from a practical

point of view, there was still substantial doubt about the exact

value of the mechanical equivalent of heat. The reason is that

Joule’s experiments are difficult to run. Small amounts of heat

escape through the walls of the container, and there are frictional

losses in the experimental apparatus itself. Consequently the

results Joule obtained from his various experiments were only

roughly similar. His experiments led many scientists to question

the caloric theory of heat, but there was still much uncertainty

about what the truth was.

The First Law of Thermodynamics

The work of Joule and Carnot was the foundation on which the

science of thermodynamics was built. The first person to recog-

nize how Joule’s and Carnot’s ideas about work and heat could be

incorporated into a coherent theory was the German physicist and

mathematician Rudolf Clausius (1822–88). Clausius was the first

to state what is now known as the first law of thermodynamics.

Mathematics and the Laws of Thermodynamics 131

As a child Rudolf Clausius attended a small school at which his

father was principal. As he became older he was drawn equally

to history and mathematics. He eventually chose to study

mathematics and physics. His approach to physics was always

a very mathematical one. He was a student at the University

of Berlin and Halle University, where he received a Ph.D. His

dissertation was about the color of the sky: He sought a sci-

entific explanation for why the sky is blue during the day and

red and orange around sunrise and sunset. Clausius described

the phenomenon in terms of the reflection and refraction of

sunlight. His explanation was not quite correct: The sky’s col-

ors are actually caused by the scattering of light. The correct

explanation for the sky’s color was later proposed by the British

mathematician and physicist William Thomson (1824–1907),

also known as Lord Kelvin, who along with Clausius developed

the fundamental ideas that lie at the heart of the science of heat,

work, and energy.

Clausius was not one for ease. He worked for the advancement

of science throughout his life, occasionally moving from one uni-

versity to another in search of the best place to do his research.

But scientific research was not his only passion. As well as a math-

ematician and scientist, Clausius was a staunch German national-

ist. When the Franco-Prussian War began in 1870, Clausius was

already well into middle age. He was too old for the rigors of

fighting, but he volunteered to serve in the ambulance corps along

with some of his students. While helping to carry the wounded off

the field during battle, he was severely wounded in the leg. The

wound bothered him for the remainder of his life. At the conclu-

sion of his military service in 1871, Clausius returned to academia,

where he was as determined to complete his academic service as he

had been to complete his war service. It is said that even toward

the end of his life—even from his deathbed—he was still engaged

in his work as a teacher.

When Clausius was a student, the field of thermodynamics was

still dominated by the ideas of Joseph Black. In the mid-1700s

Black had classified heat by its effects on matter rather than on

a deeper analysis of its nature. This type of analysis probably

132 MATHEMATICS AND THE LAWS OF NATURE

advanced understanding when Black proposed it, but by the 1840s

it had become a hindrance.

Part of the difficulty that Clausius faced in identifying what we

now call the first law of thermodynamics was understanding the

role of latent heat. When a heat engine completes a cycle, three

quantities can be measured: (1) the work performed by the engine,

(2) the amount of heat transferred from the higher-temperature

reservoir to the engine, and (3) the amount of heat transferred from

the engine to the lower-temperature reservoir. When the engine is

running, some heat is always transferred to the lower-temperature

reservoir. The problem was that the amount of heat transferred to

the lower-temperature reservoir seemed smaller than the amount

of heat that had been drawn from the high- temperature reservoir.

Clausius’s contemporaries debated about what had happened to the

“missing” heat. Had the missing heat been converted into work, as

Joule had asserted, or had it become latent and so not detectable as

a change in temperature? Some scientists of the time, reluctant to

abandon the idea of conservation of caloric, used the idea of latent

heat to prop up the old caloric model.

Clausius, however, rejected the theory of conservation of heat.

Missing heat had been detected in the operation of every heat

engine, but there was no reason to suppose that it had “gone

latent.” Clausius found ample proof in Joule’s experiments that

the missing heat had been converted into work. Incorporating the

ideas of his predecessors into a single unified concept, Clausius

asserted that it is not heat that is being conserved in a given sys-

tem but rather energy. According to Clausius heat does not just

flow through engines in the way water flows through turbines;

heat is converted by engines during the process of doing work. In a

closed system (a system cut off from its surroundings) the energy

of the system cannot change. If the system does interact with its

surroundings, then the energy of the system fluctuates as heat is

transmitted across the boundary and work is done.

Work and heat are related through the energy of the system.

In the absence of work the energy of a system—for the moment,

imagine the system as a cylinder of gas sealed with a piston—can

be changed only by the transmission of heat across the system

Mathematics and the Laws of Thermodynamics 133

boundary. This was the type of phenomenon that had originally

caught the attention of early scientists. This is also why recogniz-

ing their mistake was difficult for them: In the absence of work no

heat is converted, so the caloric theory appears valid. Alternatively

the energy of the system can be changed by work. If the system

performs work on its environment—for example, the system may

raise a weight—then the energy of the system decreases by an

amount equal to the amount of work performed. This conversion

into work had been incorrectly interpreted as heat’s becoming

latent. Furthermore both processes—the transmission of heat and

the production of work—can occur simultaneously. In this case

the change in energy of the system is equal to the sum of the work

done and the heat transmitted. Clausius asserted that the change

in energy of a system can be completely accounted for by the sum

of the heat flow into or out of the system and the work performed.

This result, called the first law of thermodynamics, is often stated

like this:

(Change in energy of a system) =

– (work done by the system) + (heat flow across the boundary)

(The negative sign preceding the work term indicates that when

the system performs work, the energy decreases. If heat flows out

of the system, we give it a minus sign; if heat flows into the system,

the sign is reversed.) Here is how Clausius expressed the first law:

The energy of the universe is constant.

Clausius’s statement of the first law of thermodynamics strikes

many people as more philosophical than mathematical. Our

statement of the first law in terms of “the change in energy of

the system” is meant to convey more of the mathematical flavor

involved. Often the first law of thermodynamics is described

in the language of cylinders and pistons. Clausius’s description,

however, does the most justice to the importance of the first law,

because it conveys some sense of the scope of the discovery. The

discovery of energy and its relationship to heat and work is one of

134 MATHEMATICS AND THE LAWS OF NATURE

the great milestones in the history of science. It is a cornerstone

of modern scientific thought.

The first law is more than an important principle. The state-

ment that the rate of change of energy equals the rate of heating

and working of the system allows the mathematically inclined

scientist to express his or her ideas in terms of very specific math-

ematical equations. Like the conservation of momentum and con-

servation of mass, the first law of thermodynamics can be written

in terms of a differential equation. Scientists can tailor these types

of equations to the particular situations in which they are inter-

ested. A solution to this type of differential equation is a function

that represents the energy of the system.

Clausius established a firm link between mathematics and sci-

ence, but was he right? How do we know that the first law of

thermodynamics is valid everywhere and in all situations? This is,

after all, what Clausius was hoping to convey with his extremely

broad statement about energy and the universe. The answer to the

question about the universal validity of the first law is surprisingly

simple (and to some people not altogether satisfying): We “know”

that the first law is true because no one has ever observed a situ-

ation in which the first law is false. No one, for example, has ever

seen energy consumed or created. Of course only a few hundred

years ago many scientists spent their working life studying phys-

ics without ever observing the consumption or the creation of

heat (caloric), whereas today these kinds of observations pose no

problem at all to the curious high school student. Will we one day

be able to point to exceptions to the first law of thermodynamics?

The best that can be said about the validity of the first law is that

since it was first formulated by Clausius, no scientist or engineer

has ever derived a contradiction by assuming its validity.

Despite the importance of his discovery and despite his recogni-

tion of its importance to science, Clausius also recognized that the

first law is, in a sense, deficient. The first law is true, he knew, but

it is not true enough. In his study of heat Clausius had realized

that there is another law of nature, now known as the second law

of thermodynamics, that further constrains the types of processes

that can occur.

Mathematics and the Laws of Thermodynamics 135

The Second Law of Thermodynamics

The second law of thermodynamics is the result of the work of

Rudolf Clausius and William Thomson, also known as Lord

Kelvin. Clausius was the first to state it. He noticed that the first

law was, in a sense, incomplete, because it does not make a strong

enough distinction between possible processes and impossible ones.

Clausius is right: Every system changes in such a way that the first

law of thermodynamics remains valid. The problem is that there

are many processes that never occur that nevertheless would con-

form to the first law of thermodynamics if they did occur.

To illustrate the problem, imagine using a block of ice as a heat-

er. We usually imagine ice as being without heat, but this is never

the case. Although a great deal of heat must be removed from a

large body of water to freeze it into a solid block of ice, there is

still a great deal of heat left in the body even after it has frozen

solid. This statement is easily proved. If we place a container of

liquid nitrogen on a block of ice, the liquid nitrogen boils. (The

boiling point of nitrogen at atmospheric pressure is about –321°F

(–196°C).) The heat that boils the nitrogen is the heat that has

flowed out of the ice. As the nitrogen boils, the block of ice gets

even colder, indicating that there had been heat in the block all

along. If we can easily boil liquid nitrogen by placing it on a block

of (water) ice, why cannot we also boil liquid water in the same

way? It is easy enough to imagine how this might occur: We place

a container of water on a block of ice. The heat from the block of

ice flows into the container of water, causing the liquid water to

begin to boil. Of course this never happens, but could it happen?

There is nothing in the first law of thermodynamics to rule out

this possibility.

Clausius thought about Carnot’s engine in a slightly more

technical vein. Carnot imagined his engine forming a conduit

for heat as the heat flows from the higher-temperature thermal

reservoir to the lower-temperature one. Heat, Carnot imagined,

flows from hot to cold in just the same way that water flows from

higher elevations to lower ones. This analogy between the flow

of heat and the flow of water is at the heart of the caloric theory,

136 MATHEMATICS AND THE LAWS OF NATURE

but Clausius had rejected the caloric theory. Unfortunately there

is nothing in his new theory to substitute for the idea that heat,

like water, always seeks its own level. So theoretically the possi-

bility exists that an engine, placed between a higher-temperature

and a lower-temperature thermal reservoir, can run on heat that

flows from the lower-temperature reservoir “up” to the higher-

temperature reservoir. No one has ever seen this occur, but again

there is nothing in the first law to rule it out, either. In the case

of the heat engine’s running on “backward”-flowing heat, the first

law is satisfied provided that the decrease in energy of the lower-

temperature reservoir equals the sum of the work performed by

the engine plus the amount of heat rejected to the higher-temper-

ature reservoir. An axiomatic basis for the field of thermodynamics

was still incomplete.

To complete his theory Clausius asserted the existence of a

second law of thermodynamics. The second law was to become a

major theme of Clausius’s life. He wrote a number of papers about

it, carefully stating and restating it, weighing one interpretation of

the second law against another. Part of the reason that he spent so

This nuclear reactor, under construction in Finland, is a steam engine. It

converts thermal energy into electrical energy by harnessing the power of

expanding steam. It is designed to be substantially more efficient than the

nuclear plants currently in use in the United States.

(Areva NP)

Mathematics and the Laws of Thermodynamics 137

much time writing about the second law is that he was asserting

something that is quite new in the history of science. The second

law is fundamentally different from previous natural laws. It is

stated in the negative. It states the impossibility of certain process-

es. By contrast the conservation laws are all positive. They state

that some property is conserved. The mathematical form of the

second law is also different from the form of all previous laws of

nature. Unlike conservation laws, which are written as equalities,

the mathematical statement of the second law of thermodynamics

is an inequality. The scientific, mathematical, and philosophical

implications of the second law continue to draw the attention of

thoughtful people to this day.

Clausius’s first attempt to grapple with this newfound physical

principle rested on the observation that when two objects of differ-

ent temperatures are placed in contact with each other, heat always

flows from the warmer body to the cooler one. The warmer body

always cools, and the cooler body always warms. The reverse never

happens. It is never the case that when two bodies at different tem-

peratures are placed in contact with each other, heat flows from

the cooler body into the warmer body. If this impossible transfor-

mation could happen, then the warm body would become warmer

and the cool body would become cooler still. Those observations

seem almost too obvious to bother mentioning, but Clausius car-

ried them a step further. He claimed that it is not possible to devise

a cyclic machine or system of cyclic machines whose only effect is

to transfer heat from a low-temperature reservoir to a high-tem-

perature reservoir. (A refrigerator, of course, transfers heat from

the freezer to the warmer air in the kitchen, but in doing so it also

produces a lot of additional heat of its own. The refrigerator does

not violate the second law of thermodynamics.) The second law

of thermodynamics has been phrased and rephrased a great deal

since Clausius’s time. Here is one version of what is usually called

Clausius’s statement of the second law of thermodynamics:

It is impossible to construct a cyclic engine whose only effect is

the transfer of heat from a body at a lower temperature to one

at a higher temperature.

138 MATHEMATICS AND THE LAWS OF NATURE

It is, admittedly, a very peculiar natural law. First, it is, as we

have already pointed out, a negative statement. Negative state-

ments cannot be proved experimentally: There is no experiment

that can rule out the existence of another experiment that does

not conform to the second law. Nevertheless, no experiment that

violates the second law has ever been devised. In fact as we soon

see, any experiment that did violate the second law would have very

peculiar implications for the universe. Second, though the second

law is held up as a universal law of nature, it is often stated in terms

of the impossibility of designing a certain type of refrigerator. (It

is, after all, the function of a refrigerator to transfer heat from a

body at low temperature, such as a container of ice cream, to one at

higher temperature, the air in the kitchen.) There is nothing else in

science to compare with the second law of thermodynamics.

In addition to ruling out the possibility of certain types of physi-

cal transformations of heat and work, the second law of thermody-

namics has profound philosophical implications. A good example

of this type of implication is the concept of heat death. An exposi-

tion of the idea can be found in the work of the other founder of

thermodynamics, William Thomson (Lord Kelvin).

William Thomson was born in Belfast, Ireland. His mother

died when he was six. His father, a mathematician, taught William

and his older brother, James. Their father must have been a good

teacher, because James enrolled in the University of Glasgow at

age 11 and William enrolled at age 10. William Thomson pub-

lished his first mathematics papers when he was still a teenager.

He wrote about a mathematical method for describing the flow of

heat through solids, a method that had been recently pioneered by

the French mathematician Jean-Baptiste-Joseph Fourier (1768–

1830). Later Thomson enrolled in Cambridge University, and it

was from Cambridge that he graduated.

During his years at Glasgow and Cambridge Thomson had

proved himself an able mathematician, but he wanted to learn

more about the experimental side of science. After graduation he

moved to Paris, where he worked with the French physicist and

chemist Henri-Victor Regnault (1810–78). Regnault was a tireless

experimentalist. He tested and measured the physical properties

Mathematics and the Laws of Thermodynamics 139

of various gases and liquids. Thomson’s stay at Regnault’s labora-

tory resulted in a paper, published in 1849, “Account of Carnot’s

Theory of the Motive Power of Heat; with Numerical Results

derived from Regnault’s Experiments on Steam.” Thomson even-

tually joined the faculty at the University of Glasgow, where he

remained the rest of his working life.

Scientists today best remember Thomson—or Kelvin—for his

contributions to the theory of thermodynamics and his work in

electricity and magnetism. (The temperature scale most com-

monly used in science, the Kelvin scale, is named after him.)

During his own life, however, Kelvin was at least as famous for

his work on telegraph cables. By the middle of the 19th century,

many European cities were linked by the telegraph. The telegraph

had also become an important part of life in North America.

People began to consider the possibility of linking North America

to Europe via a 3,000-mile (4,800-km) undersea cable. Kelvin

understood the physics of sending an electrical signal through a

long cable. He was hired as a consultant on the project, but his

ideas were not incorporated into the first attempt to lay the cable.

The attempt failed and Kelvin’s ideas were used during later suc-

cessful attempts. His contributions made Britain a world leader in

electronic communications.

Kelvin was a more conservative scientist than Clausius. He had

recognized for some time that there were problems in the theory

of heat, but knowing that something is wrong is not the same

as knowing what is right. Kelvin recognized that Joule’s experi-

ments indicated that heat is not conserved, and he knew that the

development of the science of thermodynamics depended on the

identification of a conserved property, but at the time he could

think of no alternative to the conservation of caloric. Kelvin did

not entirely abandon the theory of the conservation of caloric until

he read Clausius’s paper.

Kelvin formulated his own version of the second law of ther-

modynamics, and today both versions are taught side by side.

Although they look different, one can prove that Clausius’s version

is true if and only if Kelvin’s version of the second law is true, so

they are, logically speaking, completely equivalent. What makes