Muller A History of Thermodynamics The Doctrine of Energy and Entropy phần 10 pot

In this way he provided an application for Jüttner’s formulae.Thermal equation of state inside a white dwarf In relativistic thermodynamics the conservation of mass is replaced by the c

Table 10.1

U A

The only remaining source of energy for a white dwarf is gravitational contraction, Helmholtz fashion That keeps the star hot in the centre, perhaps hot enough – a thousand times as hot as the sun – that it must be considered a relativistic gas Note that the small electronic mass helps in this respect, because the relativistic coldness kTcc2 is more than 103timessmaller for electrons than for nuclei, or atoms at the same temperature Now, large speeds make for small de Broglie wave lengths, so that quantum effects should be small However, the large gravitational pressure compresses the star to such a degree that even the small de Broglie wave lengths interfere and thus produce quantum degeneration Therefore in a white dwarf the electron gas can perhaps be both: a relativistic gas and a quantum gas Chandrasekhar adopted this assumption as the basis for his theory of white dwarfs In this way he provided an application for Jüttner’s formulae.

Thermal equation of state inside a white dwarf

In relativistic thermodynamics the conservation of mass is replaced by the conservation of the number of particles, and momentum and energy conservation are combined in a vector equation We have

where 0

following table

A A

c U N

B A

1

Y F

296 10 Relativistic Thermodynamics

22

21with

p c

µ o p o

– determines the cell of the phase space.

For a strongly degenerate Fermi gas we thus have, cf Table 10.1

z x

o

x z

z Y c µ ʌ c 3

1 p z

z Y c µ ʌ

n

o

d21

44

)(4and

d3)(

c

explicit form of the relation – the thermal equation of state – can be obtained, if the

If relativistic effects were ignored, the square root in the integrand for p would

strongly degenerate relativistic Fermi gas.11 In that case it was fairly easy to

consider the limit of the ultimate white dwarf characterized by an infinite

mass density at the centre and zero radius Surely no other star could be denser and, presumably, have more mass That ultimate white dwarf came

11 S Chandrasekhar: “The maximum mass of ideal white dwarfs.” Astrophysical Journal 74 (1931) p 81.

S Chandrasekhar: “The highly collapsed configurations of a stellar mass, I and II.” Monthly Notices of the Royal Astronomical Society 91 (1931) p 456 and 95 (1935) p 207.

See also: S Chandrasekhar: “An Introduction to the Study of Stellar Structure” University

of Chicago Press (1939) This book is available in a Dover edition, first published in 1957

In part of his work Chandrasekhar assumed that the electron gas is a

out to have a mass of approximately 1.4 solar masses, cf Insert 10.2 This

limiting mass for white dwarfs became known as the Chandrasekhar limit.

It was confirmed by observation in the sense that no white dwarf was ever seen that has more than Chandrasekhar’s limit mass

The Chandrasekhar limit

Since the mean value of the relative molecular mass is 2, by Insert 10.1 the mass density and the pressure are given by

Y.

c µ c

ʌ B z

x

0 1 z

z B

p

Y c µ

ʌ

o

µ A Ax

ȡ

4 ) ( 3

4 with

d

and 3

) ( 3

4 2 with

o ȡ r ʌ r M r

r M ȡG r

p

c c

2 d

2

2

) ( 1

2/3 ) ( 1 2/3

) ( 1

2/3 ) ( 1 d

d d

d

2

1

2/3 2

/A c ȡ ȡ/A

Ș ĭ

/A c ȡ

ȡ/A Ș

Ș

Ș

Ș





,

characterizes the ultimate white dwarf in the sense that no other one could be

denser and have more mass In that case it is easy to solve – numerically – the

the graph shown in Fig 10.1 On the surface of the star, at r = R, we must have

is zero, but the mass is not It can be calculated as follows:

Ș

c

298 10 Relativistic Thermodynamics

Insert 10.2

Fig 10.1 A kind of density distribution in

the ultimate white dwarf

The last step makes use of the differential equation in the form

d

2/3 ) ( 1 d 2 d

d 2

1 2

r r AL ȡ

Obviously, degeneration of the electron gas has played a decisive role in the forgoing analysis It is less clear that the relativistic square root in the equation for

p is essential for the result However, it is! Without that relativistic contribution

there is no mass limit.

The usual interpretation of the Chandrasekhar limit is that the electron gas cannot

under great pressure the electrons are pushed into the protons of the iron nuclei to

form neutrons The star thus becomes a neutron star, with a truly enormous mass

times the already large density of a white dwarf Neutron stars have

Oppenheimer (1904–1967) in 1939 If a star is bigger than that, – and does not get rid of the excess mass by nova- or supernova-explosions – it collapses into a black hole, at least according to current wisdom There seems to be no conceivable mechanism to stop the collapse It is tempting to pursue the matter further in this book However, there is a touch of science fiction in the subject and I desist, – with regret.

Chandrasekhar has left his mark in several fields of physics In his

autobiography he says that he was … motivated, principally, by a quest

after perspectives…compatible with my taste, abilities and temperament.

Stellar dynamics was the subject of only the first such quest Others followed:xBrownian motion, xradiative transfer, xhydrodynamic stability,

xrelativistic astrophysics, xmathematical theory of black holes Whenever

Ɓ

The maximal mass of a white dwarf is not alone in having been named after

Chandrasekhar There is also the NASA X-ray observatory which is called Chandrasekhar

observatory, and a minor planet,– one of about

15000 – which was named Chandra in 1958

Fig 10.2 Subrahmanyan Chandrasekhar

Maximum Characteristic Speed

After Jüttner there was a period of stagnation in the development of

rela-tivistic thermodynamics To be sure, there was some interest, and in 1957

John Lighton Synge (1897–1995) streamlined Jüttner’s results in a neat small book12 which, however, did not significantly add to previous results Also Eckart provided a relativistic version of thermodynamics of irrever-sible processes,13 in which he improved Fourier’s law of heat conduction by accounting for the inertia of energy, cf Chap 8 However, his differential

equation for temperature was still parabolic so that the paradox of heat

conduction persisted Understandably that paradox has irritated relativists

more than it did non-relativistic physicists After all, if no atom, or molecule can move faster than the speed of light, heat conduction should

12 J.L Synge: “The Relativistic Gas.” North Holland, Amsterdam (1957)

13 C Eckart: “The thermodynamic of irreversible processes III: Relativistic theory of the simple fluid.” loc cit

he found that he understood the subject, he published one of his highly

readable books, – in his words: a coherent account with order, form, and

structure Thus he has left behind an admirable library of monographs for

students and teachers alike His work on white dwarfs, but also his lifelong physics in 1983, fifty years after he discovered the Chandrasekhar limit.exemplary dedication to science, was rewarded with the Nobel prize in

300 10 Relativistic Thermodynamics

not be infinitely fast This problem was the original motive for Müller to develop extended thermodynamics, cf Chap 8, and its relativistic version.14

Shortly afterwards, Israel15 published a very similar theory and, eventually,

it was shown by Boillat and Ruggeri16 that extended thermodynamics of infinitely many moments predicts the speed of light for heat conduction Thus the paradox was resolved; the field is conclusively explained by Müller in a recent review article.17

14 I Müller: “Zur Ausbreitungsgeschwindigkeit ” Dissertation (1966) loc cit.

A streamlined version of relativistic extended thermodynamics may be found in:

I-Shih Liu, I Müller, T Ruggeri: “Relativistic thermodynamics of gases.” Annals of Physics 169 (1986).

15 W Israel: “Nonstationary irreversible thermodynamics: A causal relativistic theory.” Annals of Physics 100 (1976).

16 G Boillat, T Ruggeri: “Moment equations in the kinetic theory of gases and wave velocities.” (1997) loc.cit

17 I Müller: “Speeds of propagation in classical and relativistic extended thermodynamics.” http:/www.livingreviews.org/Articles/Volume2/1999-1mueller.

18 N.A Chernikov: “The relativistic gas in the gravitational field.” Acta Physica Polonica 23 (1964).

N.A Chernikov: “Equilibrium distribution of the relativistic gas.” Acto Physica Polonica

A decisive step forward in the general theory was done by N.A Chernikov in 1964 18 when he formulated a relativistic Boltzmann equation Let us consider this now

Boltzmann-Chernikov Equation

I have already mentioned the elegant four-dimensional formulation which is now standard in relativity It was introduced by Hermann Minkowski (1864–1909) Minkowski had taught Einstein in Zürich and later he became the most eager student of Einstein’s paper on special relativity He sugge-sted that the theory of relativity makes it possible to take time into account

as a kind of fourth dimension and he introduced the distance ds between two events at different places and different times19

2 3 2

2 2

1 2

2d (d ) (d ) (d )

dd

CD B D A

C

x

x x

x

w

cww

cw

In particular, for a rotating frame – on a carousel (say) – with coordinates

0 2 0

0 0 1 0

0 0

2

2 2 1

r c

Ȧr

c

Ȧr c

r Ȧ

d

d d

d 2

w

 w

w



D x AC g A x DC g C x DA g BD g B AC ī ,

IJ

B x IJ

A x B

ī = 0, the solution of this equation is

a motion in a straight line with constant velocity, which is the defining feature of an inertial frame The parameter IJ is usually chosen as the proper

time of the moving particle, i.e the time read off from a clock in the

momentarily co-moving Lorentz frame With that, the equation of motion may be written in the form

IJ

x p ,

p p ī µ IJ

A B

A B AC B

d

dwhere

1d

d

c



is the four-momentum of the particle as before

In this manner the tensor g ƍAB, whose invariance defines the Lorentz

frames, may be interpreted as a metric tensor of space-time Its components

in a arbitrary frame x A = x A (x ƍ B

) can be calculated from

The equation of motion represents the equation of a geodesic in space-time This is

a nice feature, much beloved by theoretical physicists, because it supports their predilection for a specious geometrical interpretation of the theory of relativity The notion was useful for Einstein, when he developed the theory of general relativity;

but most often it is used to confuse laymen with talk about curved space, etc.

p (F(

p

F p p ī

x

F

d B A d AB

Maxwell-mentum vector p Ain the collision

Chernikov uses the equation for the formulation of equations of transfer for moments of the distribution function and he concentrates on 13 moments, which is rather artificial for a relativistic theory; it is more appropriate to include the dynamic pressure and thus come up with a theory

of 14 moments.21 But we shall not pursue this question here, because so far – apart from the finite characteristic speeds – the multi-moment theory

Seeing that the collision term vanishes for the Maxwell-Jüttner distribution, we must ask whether the Boltzmann-Chernikov equation is

satisfied by that distribution, or what conditions on the fields a(x B ), T(x B),

w

;B A A

;

B

U kT

U 0

x

a

,where the semi-colon denotes covariant derivatives

20 The possibility of such a term was ignored in Chap 4, because I wished to be brief The term is only present in a non-inertial frame

21 See: I Müller, T Ruggeri: “Rational Extended Thermodynamics.” loc cit.

Of course, nobody will try to solve the equation of the geodesic in its general form

in order to calculate the orbit of a free particle It is so much easier to solve it in a Lorentz frame and transform the straight line obtained there to an arbitrary frame.

has not provided any suggestive results that go beyond Eckart’s ulation of the Fourier law, see Chap 8 Let us concentrate on equilibrium instead:

reform-Since a is a function of n and T, it follows that a temperature gradient

must exist in equilibrium, if there is a density gradient That conclusion may

be made more concrete by exploiting the second condition for the special case of a gas at rest on a carousel We obtain

This result is eminently plausible, because it reflects the inertia of the thermal energy in the field of the centrifugal potential Ȧ2

r2 Indeed, if energy has mass – and weight – it should be subject to sedimentation, as it were, by centrifugation

Einstein has postulated – in his general theory of relativity – that inertial forces and gravitational forces are equivalent Accordingly non-homo-geneous temperature fields are also created by gravitational fields – not only by centrifugal fields – because they lead to stratification of mass density I have already commented on that aspect in the context of Eckart’s relativistic paper

In view of the following argument, I should like to stress that the last

relation does not imply a transformation formula for the temperature It

represents a property of the scalar temperature field as a solution of the energy balance equation in a centrifugal force field

Ott-Planck Imbroglio

In 1907 the theory of relativity was new A fundamental change had occurred in mechanics, and physics in the immediate aftermath was in a state of flux The extension of the new concepts to thermodynamics was clearly desirable Everything seemed possible and so Planck22 came up with the idea to modify the Gibbs equation Einstein23 elaborated on that idea and

introduced a working term –qdG into the heating of a body moving with the

In the reprinting the modified Gibbs equation is misprinted: It says dQ instead of dG.

Printed version: Annalen der Physik 26 (1908) p 1

2 2 2



304 10 Relativistic Thermodynamics

speed q G is the momentum; it includes a relativistically small term,

because the mass is c2

U

m c  The modified Gibbs relation thus reads

G q V p U Q S

The transformation of dU, p,dV, and dG between the moving body and

the body at rest were known and thus Einstein produced the relation

2 0 2

transfor-Now Planck had already argued that the entropy of the body should be

unaffected by motion, and therefore the second law written as dq = TdS

seemed to require

2 0 2

c

That relation was later rephrased by epigones of the argument in the

words: A moving body is cold

On the surface the argument appears plausible It does ignore, however, the fact

that the Gibbs relation is a relation for a body at rest: The heating consists of the

non-convective part of the energy flux and the internal energy is the non-convective

part of the energy The power, or working of the force dG has no place in the Gibbs

equation therefore, or it should not have.

Also, the heating of a body in the Gibbs equation is the integral of the heat flux over the surface And relativistically the heat flux forms three components of the energy-momentum tensor It is that fact which determines the transformation of the heating, not its position in the Gibbs equation

None of the serious physicists in the following years and decades followed Planck and Einstein in this precarious thermodynamic argument, neither Eckart, nor Synge, nor Chernikov Consequently one might have thought that the argument was discarded as a valiant, perhaps, though erroneous early attempt on relativistic thermodynamics

Not so, however! In 1962, H Ott24 revisited the argument on a slightly different basis involving Joule heating, and he came to the conclusion that

such that: A moving body is hot.

Serious people in the field ignore the subject, which was appropriately

termed the Ott-Planck imbroglio by Israel and Stewart.25 However, the farce continues and Peter Thomas Landsberg26 – himself an enthusiastic contributor to the imbroglio – cites papers on the subject of temperature transformation in special relativity as recent as the late 1990’s.27

11 Metabolism

If the truth were known, thermodynamics would be seen as explaining little about the details of life functions in animals and plants, at least compared to what there is to be explained This is no different than with engines: Thermodynamics cannot provide a recipe for their construction, or give information about where and how to arrange seals and boreholes for lubrication, and how to operate the valves and where to install them What

thermodynamics can do about engines is to give an account of the balance

of in- and effluxes of mass, momentum, energy and entropy, and that is essentially what it can also do about life For the engine that task has been done satisfactorily; for animals and plants maybe there remains something

to be done

Having said this, I hasten to stress that, what thermodynamics is able to

provide, is good enough to refute esoteric theories, and to convince people with an open mind that nothing unnatural occurs in the living body: No

vitalistic force of old, nor Niels Bohr’s complimentarity of life and physics,

akin to the wave-particle dualism of quantum mechanics.1

I have previously – cf Chap 4 – warned against an over-interpretation of entropy as a measure of disorder and I stress that caution again To be sure,

an animal definitely seems more ordered than the sum of its atoms, loosely

distributed, and it does probably have a lower entropy But then, what is the

entropy of an animal? Or let us ask the easier question: What is the entropy

of a molecule like hemoglobin, one of the simpler proteins with only about

500 amino acids? Maybe molecular biologists can come up with an answer;

if so, I do not know about it But I do know that surely it must be a case of simplism when Schrödinger says2 that animals maintain their highly ordered state, because they eat highly ordered food Indeed, before the animal body makes use of the food in any way, – and sets about to create order – it breaks the food down to much less ordered fragments than those which it ingests

1 In his later years Bohr expressed doubts that life functions can be reduced to physics and chemistry See: N Bohr: ‘‘Atomphysik und menschliche Erkenntnis.” [Atomic physics and human knowledge] Vieweg Verlag, Braunschweig (1985).

2 E Schrödinger: ‘‘What is life? The physical aspect of the living cell” Cambridge: At the University Press, New York: The Macmillan Company (1945) p 75

In writing this chapter on metabolism I disregard Schrödinger’s warning

that a scientist is usually expected not to write on any topic of which he is

the subject is interesting, and it seems to be replete with unsolved problems

of a quantitative nature Therefore it is easy to yield to the temptation to write about it, albeit in a layman’s manner

Carbon Cycle

One of the truly mind-expanding discoveries of all times, concerning life and life functions, was the observation that carbon, hydrogen and oxygen cycle through living organisms, driven by solar radiation: Plants use water from the soil and carbon dioxide from the air to produce their tissue and they release oxygen Animals on the other hand breathe oxygen and use it to break down plant tissue In the process they release carbon dioxide and water The plants perform their task only in the light

Jan Baptista van Helmont (1577–1644) was an alchemist on the verge of becoming a chemist or, perhaps, a biochemist On the one hand he claimed

to have seen and used the philosopher’s stone – the hypothetical ultimate

tool of alchemy – but on the other hand he was keen enough as an experimenter to see that water was essential for plant growth, while soil was not, or not to the same degree Helmont did not recognize the importance of carbon dioxide for plants, even though he actually discovered that gas,

which he called gas sylvestre, i.e wood gas, because he had found that it

was released by burning wood It took another hundred years before the significance of that observation was recognized by Stephen Hales (1677–1761) Carbon dioxide has originally entered the wood from the air surrounding the leaves of a plant, thus furnishing the second component – after water – that is essential for plant growth

Another generation later Joseph Priestley (1733–1804), one of the coverers of oxygen, noticed that oxygen was used up in the air by breathing giving off oxygen These observation were all couched in the language

dis-of the phlogiston theory, – even then obsolete4 –, but Jan Ingenhousz (1730–1799) was able to penetrate the verbiage and to see a broad scheme

of balance in nature: Plants consume the carbon dioxide of the air and

3 Ibidem p vi

4 The phlogiston theory is the forerunner of Lavoisier’s caloric theory, see Chap 2 In the 18th century a weightless fluid called phlogiston was supposed to flow from a body when that body burns, or rusts, or is just cooling As far as burning and rusting was concerned, Lavoisier refuted the concept, because he showed that both phenomena are due to the combination of a body with oxygen Heating or cooling was another matter Lavoisier maintained that heat was indeed a weightless fluid which he called caloric.

and that, plants can restore the freshness of used-up air, obviously by

Respiratory Quotient 309release oxygen, while animals breathe oxygen and give off carbon dioxide

In this manner there is a stable balance Ingenhousz showed that the plants need light in order to build up their tissue That is why we now call the

process photosynthesis.

Ingenhousz, who was first to discover this grand scheme, is not very much known nowadays, but he was a celebrity in his time Being a physician, he became an early expert on inoculation, particularly smallpox inoculation, and he travelled all over Europe serving the members of royal families with smallpox, as it were, – in small doses!

Respiratory Quotient

It was the eminent chemist Berzelius, cf Chap 4, who introduced the distinction of organic and inorganic substances in 1807 The former were the substances of life, and – in Berzelius’s view – they called for a separate type of chemistry from the chemistry of elements and of their simple stoichiometric compounds that were the stock-in-trade of his own work and

everybody else’s at the time There were vague notions that a vis viva, a vitalistic force, was involved in living bodies, a spark of life Berzelius

himself and his followers even conceived of a strict barrier between the chemistries of life and non-life

Seeing and appreciating the difference between rock and lizard, as it were, one must admit that there is a certain plausibility to the idea and it took at least half a century to refute it This required an improved knowledge of the life functions, and exact measurements The first organic process to be thoroughly investigated was respiration Even Lavoisier and Henry Cavendish (1731–1810) had understood that respiration supported a kind of combustion in the body of animals by which the oxygen of the air was partially consumed and converted to carbon-dioxide and water Obviously therefore, whatever substance, or substances fed the combustion had to contain carbon and hydrogen Beyond that, the substances were unknown chemically, so that no quantitative conclusions could be drawn

However, it stood to reason that, whatever it was that burned had to be

supplied to the animal – or man – with the food

Early in the 19th century it became clear upon analysis of the food of animals that there were three main types

x carbohydrates x lipids x proteins

The carbohydrates form the chief components of cereals, and of fruit and vegetables They are of different types but closely related and, for the moment, we take sugar – more precisely glucose – as their representative

The chemical formula is C6H12O6, so that Gay-Lussac – one of the discoverers of the thermal equation of state of ideal gases – could assume that glucose consisted of 6 carbon atoms strung together and a water molecule attached to each one in the manner of hydrates The structure is more complex, as we know now, see Fig 11.1, but Gay-Lussac’s concept

led to the misnomer carbohydrate, which is here to stay Actually, what we

eat is not glucose itself, but rather something like starch or other substances which are built up from several or many glucose molecules The large molecules are held together by glycoside bonds, having shedded water

molecules in a process that is called condensation – obviously because it

produces liquid water

Again lipids, or fats are of varied types Their pioneer was Michel Eugène Chevreul (1786–1889) Fats are used in manufacturing soap and

as a young man Chevreul was involved in that business He was able to isolate different insoluble organic acids – also called carbonic acids, or fatty acids – like stearic acid, palmitic acid and oleic acid Lipids themselves

result from the carbonic acids by esterification with glycerol C3H8O , 3

giving off water, i.e undergoing condensation cf Fig 11.1 A typical representative is oleine C57H104O6, an ingredient of olive oil, or also of blubber, i.e whale oil

Fig 11.1 Left: Two glucose molecules combining by a glycoside bond Right: Olein

Glycerol combining with oleic acids

Respiratory Quotient 311While carbohydrates and lipids contain only carbon, hydrogen and oxygen, the third type of food-stuff – of which egg-white is the best-known representative – also contains nitrogen, a little sulphur and, sometimes, still less phosphorus The molecules are polymers formed from amino-acids which are bound together by a peptide link, again a bond formed by

condensation The detailed structure is too complex and varied to be easily

characterized In 1838 Gerardus Johannes Mulder devised a model molecule of 88 individual atoms which he hoped might be used to build up

other albuminous substances The word albuminous is derived from albus =

white in Latin; it is sometimes used as a generic name for substances like egg-white.5 More often these substances are called proteins in English,

because Mulder called his model molecule Protein, from Greek, meaning of

first importance Otherwise the model sank into oblivion; it was too simple

Now, if indeed food was involved in a combustion inside animals, and if

CO2 and H2O were the reaction products, the reactions for carbohydrates

and lipids had to obey the stoichiometric formulae

1

6

57 52 1

The volume ratio of exhaled CO2 to inhaled O2 is called the respiratory

quotient, abbreviated as RQ Thus the stoichiometric formulae imply

RQ = 1 for the carbohydrate

RQ = 0.71 for the lipids,

since both CO2 and O2 are ideal gases The value for proteins lies between, at roughly RQ = 0.8

in-So, if chemistry is involved in respiration, the RQ should lie between 0.7 and 1 And indeed, the chemist Henri Victor Regnault6 put animals in a cage and carefully measured the oxygen input and the carbon-dioxide output and found the ratio to be right What is more, if he fed the animals a diet of carbohydrates, the RQ tended to one, while on a fat-rich diet it

tended to 0.7 This was later confirmed for a man in a cage by the chemist

Max von Pettenkofer (1818–1901) – the founder of scientific hygiene All

of this provided strong evidence that there was no vis viva involved, at least

not in respiration

5 Actually, in German proteins are called ‘‘Eiweisse” [egg whites].

6 We have met him before in connection with his 700 page-long memoir of careful measurements of vapour properties, cf Chap 3.

Metabolic Rates

So what about the energy to be gained from food? Was the first law

satisfied, or did the intervention of a vitalistic force render thermodynamic

laws invalid in the field of nutrition?

If sugar and fat and the mix of proteins normally eaten by an animal are burned in a calorimeter they provide heats of reaction as follows7

gkJ

105.39

106.23

101.17

lipidsproteinssugar

3 3 3

The person who did all this carefully was the physiologist Max Rubner (1854–1932) He presented his findings in a report8 in which he came to the conclusion that the law of conservation of energy was maintained in nutrition just as punctiliously as in ordinary combustion By now scientists were ready to believe that physical laws govern both: life and non-life Once this was understood, the distinction between organic and inorganic chemistry began to lose its original meaning Organic chemistry became the branch that deals with carbon compounds

The chemical changes that take place in animals and humans are called

metabolism; from Greek: to rearrange The metabolic rate may be

measured in Watt – just like the power of a heat engine The maximal metabolic rate that a person can achieve is approximately 700W, but that

can only be sustained for a few minutes So what is the minimum, the basal

metabolic rate?

The basal metabolic rate is abbreviated as BMR; it can be achieved by a person lying down in a comfortably warm room, having fasted for some

7 We are now back from the mol to ordinary mass units The use of the mol in organic chemistry with it huge molecules would be totally impractical Not so, however, for the glucose synthesis and the glucose decomposition, see below.

8 M Rubner: ‘‘Gesetze des Energieverbrauchs bei der Ernährung.” [Laws of energy consumption in nutrition] (1902)