absolute temperatures as a consequence

8/8/2019 Absolute Temperatures as a Consequence

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Abso lu te Temperatures as a Consequence

of Carnot's General Axiom

C . T R U E S D E L L

Dedicated to JAMES SERRIN

C ont en t s

1. P rogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

2. T he General C arnot-C lapeyron T heorem . . . . . . . . . . . . . . . . . 360

3. Apparatus from Concepts and Logic . . . . . . . . . . . . . . . . . . . 360

4. H otness and Em pirical Tem perature, I. Continuity . . . . . . . . . . . . . 361

5. Hotness and Empirical Temp erature, I I . Differentiabil ity . . . . . . . . . . 3676. Kelvin 's A bsolute Tem peratures . . . . . . . . . . . . . . . . . . . . . 369

7. Consequences o f Part I I o f C arnot 's G eneral A x iom . . . . . . . . . . . . 372

8. T herm om etric A xiom . . . . . . . . . . . . . . . . . . . . . . . . . 373

9. T he Physical D im ension o f A bso lute Tem pera ture . . . . . . . . . . . . . 375

10. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

1 . Appendix. A xiom s o f Classical T herm odynam ics . . . . . . . . . . . . . . 377

1 . P r o g r a m

I n o u r b o o k , Conc epts and Lo gic of Class ical Thermodynamics , 1 Mr. B H A R -

A T HA a n d I d e v e l o p e d c l a s s ic a l t h e r m o d y n a m i c s o n t h e b a s is o f P a r t I o f

CARNOT'S G e n e r a l A x i o m , n a m e l y , the motive power of a Carnot cycle is pos i t ive

and is determined by i ts operat ing temperatures and by the amount o f heat i t

absorbs. For a g i v . en body , t hen , t he r e i s a f unc t i on G such t ha t f o r any C arno t

cycle cg

Li ~) = G(0 +, 0- , c +(~')) > o. (1)

T h e d o m a i n o f G is t h e se t o f o p e r a t i n g t e m p e r a t u r e s a n d h e a t s a b s o r b e d t h a t

m a y a p p e r t a i n t o C a r n o t c y c l es f o r t h e b o d y i n q u e st i on . I t is p a r t o f t h e

d e f i n it i o n o f a C a r n o t c y c le t h a t 0 ÷ > 0 - a n d t h a t C ÷ ( ~ ) > 0 . T h i s d e f i ni t io n a n d

t C. TRUESDELL& S. BHARATHA,Concepts and Logic of Classical Thermodynam ics as a

Theory o f He at Engines. Rigorously Constructed upon the Foundation Laid by S. Carnot and

F. Reech, N.Y. etc., Springer-Verlag, 1977.



358 C. TRUESDELL

t h is a x i o m m a k e s e n se in t e r m s o f u n d e r l y i n g a x i o m s a c c e p t e d t a c i t l y o r

e x p r e s s l y b y t h e p i o n e e r s r e g a r d i n g p i e c e w i s e s m o o t h p r o c e s s e s u n d e r g o n e b y a

b o d y w h i c h e x e r t s p r e s s u r e p a n d r e c e i v e s p o s i t i v e o r n e g a t i v e o r n u l l h e a t i n g Q

i n re s p o n s e t o i ts t e m p e r a t u r e 0 , i ts v o l u m e v . a n d t h e i r t i m e - r a t e s o f c h a n g e Pa n d 0 :

grvp = ~ ( K 0 ), y g - < 0 , (2 )

Q = A v ( V ,O ) P + K v ( V ,O ) O , K v > 0 . ( 3 )

T h e t i m e r a t e s a r e a s s u m e d t o e x i st a t a ll b u t a f i n i t e n u m b e r o f t im e s i n t h e

i n t e r v a l o f t i m e o v e r w h i c h t h e p r o c e s s i s d e f i n e d . T h e t h r e e cons t i t u t i ve

func t ions ra , Av , a n d K v a r e d e f i n e d a n d . c o n t i n u o u s l y d i f f e r e n t i a b l e * o v e r t h e

b o d y ' s cons t i t u t i ve domain ~ , w h i c h i s a n o n - e m p t y , c o n n e c t e d , o p e n s u b s e t o f ar e a l h a l f - p la n e . T h e n a m e s o f rv , A v , a n d K v a re pres sure f unc t ion , l a t en t hea t

wi th respect to volume, a n d spec i f i c hea t a t cons tan t vo lume . B y d e f i n i t i o n , t h e

h e a t a b s o r b e d C + i n a n y p r o c e s s is th e v a l u e o f t h e i n t e g r a l o f Q w i t h r e s p e c t t o

t i m e o v e r t h e s e t o f t im e s o n w h i c h Q > 0 .

F o r f u t u r e r e f e r e n c e I r e m a r k h e r e t h a t b y s u b s t i t u t i n g (2 ) i n t o ( 3) w e o b t a i n

Q = Av{V, 0)/~ + Kp(V,. 0) 0,

/ & v ? r ~ / & v (4)

A , = A v / ~ --~ , K , - K v = - A v - - ~ / ~ V .

Kp i s the spec i f i c hea t a t cons tan t p res sure .

I n o u r b o o k w e t o o k 0 a s b e i n g w h a t i s s o m e t i m e s c a l l e d " t h e i d e a l - g a s

t e m p e r a t u r e " . O u r a t t i t u d e t o w a r d i t I l a t e r f o u n d d e s c r i b e d w e l l a n d w i t h

e v i d e n t d i s a p p r o v a l b y M A C H 2 :

I t is r e m a r k a b l e h o w m u c h t i m e p a s s e d b e f o r e i t w a s u n d e r s t o o d t h a t t o

r e p r e s e n t h o t n e s s b y a n u m b e r r e s t e d u p o n a conven t ion . I n n a t u r e t h e r e a r e

h o t n e ss e s , b u t t h e c o n c e p t o f t e m p e r a t u r e e x is ts o n l y t h r o u g h o u r a r b i t r a r y

def in i t ion , w h i c h m i g h t h a v e t u r n e d o u t o t h e rw i s e . U n t i l v e r y r e c e n t l y th o s e

w h o w o r k e d i n th i s fi el d s e e m t o h a v e l o o k e d m o r e o r l es s u n c o n s c i o u s l y f o ra n a t u r a l m e a s u r e o f t e m p e r a t u r e , f o r a r e a l t e m p e r a t u r e , f o r a s o r t o f

P l a t o n i c id e a o f t e m p e r a t u re , o f w h i c h t h e t e m p e r a t u r e r e a d o n a t h e r m o -

m e t e r w o u l d b e o n l y a n i n c o m p l e t e , i n e x a c t e x p r e s s i o n .

A s a m o n g t h e p r o p o n e n t s o f t h is t o h i m r e p r e h e n s i b l e id e a M A C H w a s a b l e t o

c i t e C LA U S IU S , I d o n o t t h i n k w e w e r e a t f a u l t i n h o l d i n g t o i t, e s p e c i a l l y s in c e

b y i m p u t i n g t o o n e i d ea l g a s a si m p l e p r o p e r t y l o n g k n o w n t o h o l d v e r y n e a r l y

f o r m a n y n a t u r a l g a s e s w e w e r e a b l e t o p r o v e a s t h e o r e m s n o t o n l y a t r a d i t i o n a l

" S e c o n d L a w " b u t a l s o a t r a d i t i o n a l " F i r s t L a w " .

2 E. MACH. Die Prinzipien der W 'drmelehre, Historisch-kritisch entwicke lt, Leipzig,Barth, 1896. M ost of the quo tation s below are f rom the chap ter called "K riti k desTem peraturbeg rif fes", hereinaf ter referred to a Temperaturbegrif f The quo ta t ion a bove i sfrom § 14.

* See the note added in proof, p. 380.



Abso lu t e Tem pera tu re s f rom Carno t ' s Ax iom 359

N e v e r t h e l e s s , it w a s n o t n e c e s s a r y f o r u s to i n t r o d u c e t h e c o n c e p t o f i d e a l g a s

a t a ll . A f t e r h a v i n g s e e n M r . S ER R IN 'S b e a u t i f u l w o r k o n h o t n e s s a n d a b s o l u t e

t e m p e r a t u r e 3 a n d a f t e r h a v i n g b e e n s t i m u l a t e d b y h i s l e tt e rs a n d l e c tu r e s a n d b y

o u r c o n v e r s a t i o n s o n t h e s u b j e c t f o r t h r e e y e a r s , I h a v e p e r c e i v e d t h a t t o

d e v e l o p t h e c o n c e p t o f a b s o l u t e t e m p e r a t u r e u p o n t h e b a s is o f C A R N O T ' s i d e a s

w o u l d b e n o t o n l y e a s y b u t e v e n w o r t h w h i l e . K E L VI N'S w o r k 4, a l s o , I h a v e

s t u d i e d i n d e t a il , a n d I g i v e a f u ll a c c o u n t o f i t i n m y Tragicomical History of

Thermodynamics, 1822-1854, n o w b e i n g p o l i s h e d f o r t h e p r e s s . H e r e I w i s h t o

r e m a r k o n l y t h a t K E L V IN 'S w o r k r e s ts e s s e n t ia l ly u p o n C A R N O T ' s i d e as a l o n e

bu t i s ne i the r c l ea r no r exp l i c i t no r en t i r e no r r i gh t i n a l l de t a i l s . CLAUSIUS

a t t e m p t e d t o p e r s u a d e h i s r e a d e r s t h a t h e h a d e s t a b l i s h e d a n a b s o l u t e t e m p e r a -

t u r e i n c i r c u m s t a n c e s f a r m o r e g e n e r a l t h a n w h a t c a n b e d e s c r i b e d i n C A R N O T ' s

t e r m s ; h i s w o r k , t o o , i s d e s c r i b e d i n m y Tragicomical History, b u t I c a n n o t

f o l l o w t h e a r g u m e n t s t h a t h e p r e s e n t s ; i n d e e d , I c a n n o t e v e n d i s t i n g u i s h w h a t h e

a s s u m e s f r o m w h a t h e c l a i m s t o p r o v e . It is m y i m p r e s s i o n t h a t M r . S E R R I N ' s

t h e o r y a c h i e v e s w h a t CLAUSIUS a t t e m p t e d , o r , r a t h e r , t h e a c h i e v a b l e p a r t o f

CLAUSIUS' i n c h o a t e p r o g r a m . T h e t r e a t m e n t I g i v e b e l o w s e e m s t o m e t o c l a r i f y ,

c o n s o l i d a t e , c o n f i r m , c o r r e c t , a n d c o m p l e t e K E L V I N 's . B e c a u s e I o ff e r i t o n l y a s

a m o d e s t p r o h i s t o r i c a l s t u d y , I p r e s e n t i t i n f o r m a l l y , u s i n g o n l y s u c h m a t h e m a t i -

c a l c o n c e p t s a s w e r e a c c e s s ib l e to a n y c o m p e t e n t m a t h e m a t i c a l s c i e n t is t o f t h e

1 8 7 0 s - c o n c e p t s , h o w e v e r , s o m e o f w h i c h w e r e n o t a t t h e d is p o s a l o f K E L V IN

and CLAUSIUS i n 1 8 4 8 - 1 8 5 4 a n d h e n c e a r e n o t t o b e f o u n d i n t e x t b o o k s o f

t h e r m o d y n a m i c s t o d a y . U s e o f t h e m o d e r n t h e o r y o f m a n i f o l d s w o u l d e f fe c t

b r i e f er d e d u c t i o n o f r e s u l ts m o r e g e n e r a l t h a n t h o s e b e lo w , b u t a l l I a t t e m p t

h e r e i s t o m o t i v a t e t h e a s s u m p t i o n s a n d t o s h o w t h a t e l e m e n t a r y a r g u m e n t s ,

c a l l i n g u p o n n o g e n e r a l t h e o r e m s a b o u t m a n i f o l d s , w o u l d h a v e s u f f i c e d t o p r o v e

e v e r y t h i n g t h a t t h e p i o n e e r s s o u g h t i n t h i s d o m a i n . F o r a f u l l y m a t h e m a t i c a l

t r e a t m e n t m e e t i n g m o d e r n s t a n d a r d s o f r i g o r a n d g e n e ra l it y th e r e a d e r s h o u l d

c o n s u l t M r . S E R R I N 's w o r k a n d s t u d i es b y o t h e r s w h i c h I e x p e c t t o b e c o m p l e t e

s o o n .

3 j .B . SERRIN,Foundations o f Classical Thermodynamics, Lec tu re No tes , Depa r tmen tof Mathema t ics , Univers i ty of Chicago, 1975, and " Th e concepts of therm odyn am ics" , pp .

411-451 of Contemporary Developments in Continuum Mechanics & Partial D ifferentialEquations (P roceed ings o f the In t e rna tiona l Sym pos ium on Con t inuum M echan ics andPart ia l Differential Equ ations, Rio de Janeiro, A ugu st 1977), edited by G .M. DE LA PENHA& L.A . M EDEIROS, Am sterdam , N or th-H ol lan d Publi sh ing Co., 1978. Mr . SERRINpresented fur ther result s in his Batem an lec tures a t The Johns Ho pkins Univers i ty in M ay,1978, and in subsequen t lectures elsewhere. His w ork applies to a class of m aterials m oregenera l than tha t cons idered here .

4 W . THOMSON la ter Lo rd KELVIN), O n the absolu te therm om etr ic sca le founded onCarno t ' s t heo ry o f the m ot ive power o f hea t ; and ca lcu l at ed f rom R egnau l t 's obse rva t ions" .Proceedings of the Cam bridge Philosophical Society 1 (1843/1866), No. 5, 66-71 (1849)=Philosophical Magazine (3) 33 (1849), 31 3-3 17 =(w ith added notes) pp. 100-112 of W.THOMSON'S Ma thematical and P hysical Papers, Volume 1, 1882.

J. P. JOULE & W. THOMSON, "O n the therm al effects of f luids in m otion , Par t II" ,Philosophical Transactions of the Ro ya l Society ( lo nd on ) 144 (1854), 321-364 = (with pp.362-~-364 excised), pp. 357-400 of W . THO MSON 'S Mathematical and Physical Papers,Vo lum e 1, 1882 =p p. 247-299 of JOULE'SScientific Papers,Volum e 2, 1887. See Section I V of" Theore t i ca l D educ t ions" .



360 C. TRUESDELL

E s s e n t i a l t o m y a r g u m e n t i s t h e a p p a r a t u s p r o v i d e d b y Concepts and Logic.

O f c o u r s e o n l y t h o s e p a r t s o f it t h a t s u r v i v e w h e n a l l m e n t i o n o f i d e al g a s e s is

e x c is e d c a n b e u s e d fo r t h e p u r p o s e . T h u s I h a v e p h r a s e d ( 1 ) - ( 4 ) i n t e r m s o f

s o m e o n e e m p i r i c a l sc a le o f t e m p e r a t u r e , v a l u e s o f w h i c h a r e d e n o t e d b y 0, 0 -~.a n d 0 - . T h e r a n g e o f th i s s c a le is s o m e o p e n r e a l i n te r v a l. U n t i l § 7 all reasoning

will be local, w i l l r e f e r o n l y t o t e m p e r a t u r e s i n s o m e s u b i n t e r v a l , p e r h a p s v e r y

s m a l l , a n d o n l y t o some one body.

2 . T h e Gen era l C a rn o t - C la p ey ro n T h eo rem

CLAPEYRON,a p p l y i n g to (1) a n a r g u m e n t t h a t C A R N O T h a d i n v e n t e d b u t

h a d u s e d o n l y f o r i d e a l g a s e s , d e t e r m i n e d a s f o l l o w s t h e w o r k d o n e b y a C a r n o t

cyc le ~ wi th in f in i t e s im a l d i f fe rence A O o f o p e r a t i n g t e m p e r a t u r e s :

L((ff)~/~(0) C + (~)A 0. ( 5 )

H e r e ~t is " C a r n o t ' s f u n c t i o n " , d e r i v e d f r o m G i n ( 1) t h r o u g h a l i m i t i n g p r o c e ss .

F r o m ( 1 ) a n d ( 5 ) w e m i g h t t h i n k t o c o n c l u d e t h a t / 1 > 0 , a n d t h e e a r l y a u t h o r s .

K E L VI N a m o n g t h e m , se e m t o h a v e d o n e s o . H o w e v e r . e x a m i n a t i o n o f th e l i m i t

p r o c e s s s h o w s t h a t o n l y a w e a k e r s t a t e m e n t f o l l o w s :

~ > 0 . ( 6 )

W e s h a l l s e e t h a t t h e d i s t i n c t i o n , a l t o g e t h e r u n n o t i c e d b y t h e p i o n e e r s , b e a r su p o n t h e c o n c e p t o f a b s o l u t e t e m p e r a t u r e s.

Es sen t i a l ly equ iva len t to (5 ) i s the General Carnot-Clapeyron Theorem:

8 ~ 7 3

laA v = -~-ff. (7)

H e n c e w e a t o n c e p e r c e i v e t h e c o n s t i t u t i v e i n e q u a l i t y

8~> o

Av 80 = "

I t fo l lows f rom (4 )a tha t

( 8 )

Kp > K v. (9)

W e n o t e t h a t K v = K v i f a n d o n l y i f A v 8vo/~O=O.

3 . A p p a r a t u s f r o m Concepts and Logic

Concepts and Logic p r o v i d es r i g o ro u s p r o o f s o f d e l i m i t e d s t a te m e n t s m o r e

p r e c is e a n d m o r e g e n e r a l t h a n C A R N O T'S a n d m o r e s p e c if ic th a n t h o s e o b t a i n e d

b y R E E C H i n a m u c h e a r l i e r a n d i n c o m p l e t e a t t e m p t t o d e v e l o p C A R N O T's id e a s .

T h e normal set ~n o f t h e b o d y r e p r e s e n t e d b y t h e c o n s t i t u t i v e q u a n t i t i e s c j , ~v,

A v a n d K v i s the s e t o f po in t s P o f ~ such tha t



Absolute Tem peratures from Ca rnot's Axiom 361

1. P i s a rb i t r a r i ly ne a r to a po in t o f 2 a t w h ic h A t4 :0 .

2 . T h e i s o t h e r m t h r o u g h P c o n t a i n s a p o i n t o f 2 a t w h i c h A v 4 :0 .

T h e n o r m a l s e t n e e d b e n e i t h e r c o n n e c t e d n o r o p e n ; i t m a y b e e m p t y . I t i s t h el a rge s t subs e t o f ~ fo r w h ic h CA RN O T'S ide a s se e m to be f ru i t fu l . The se t o f

t e m p e r a t u r e s t h a t c o r r e s p o n d t o p o i n ts o f ~ , is o p e n , b u t i t n ee d n o t b e

c o n n e c t e d . I t is t h e u n i o n o f a n e n u m e r a b l e c o l l e c t io n o f d i s j o i n t o p e n i n t er v a l s.

I n w h a t f o l l o w s n o w we shall presume that 0 lies in one of these intervals.

T h e o r e m 7ex i n C h a p t e r 9 a n d T h e o r e m s 9h is a n d 1 0 b i s i n C h a p t e r 1 0 o f

Concep t s and Log ic t e ll u s t h a t o n t h a t i n t e rv a l t h e r e a r e c o n t i n u o u s l y di f fe r e n -

t i a b le f u n c t i o n s g a n d h , t h e f o r m e r b e i n g a n increasing f u n c t i o n a n d t h e l a t t e r a

positive f u n c t i o n , s u c h t h a t i n a s i m p l e C a r n o t c y c le ~ i n ~ .

C

. . . . h ( O - )t ~ ) = h ( - - ~ C + ( ~ ) ' (1 0)

g(O+)-g(O - )L(rg) =

h(O +)

a n d t h a t a t e a c h p o i n t o f ~ n

g ' &~~ A v = - ~ ,

C÷(~) , (11 )

( 1 2 )

0 8

M o r e o v e r ,

u n i q u e t o w i t h i n a n a d d i t i v e c o n s t a n t ; a n d

g ' > 0 e x c e p t o n a s e t w i t h e m p t y i n t e r io r .

I n p a r t i c u l a r , C a r n o t ' s f u n c t i o n # i n ( 7 ) h a s t h e f o r m

g '# = ~ - ;

i t i s u n i q u e a n d c o n t i n u o u s ; a n d

# > 0 e x c e p t o n a s e t w i t h e m p t y i n t e ri o r .

h i s un ique to w i th in a mu l t ip l i c a t ive f a c to r ; w he n h i s a ss igne d , g i s

(14)

( 1 5 )

( 1 6 )

4 . H o t n e s s a n d E m p r i c a i T e m p e r a t u r e , I . C o n t i n u i t y

I d o n o t r e g a r d t h e c o n t e n t s o f t h i s s e c t i o n a n d t h e s u c c e e d i n g o n e a sa n y t h i n g b u t a n e x p o s i t i o n o f i d e as s o s i m p l e th a t t h e y m u s t h a v e b e e n s e n s e d

i n f o r m a l l y b y th e p i o n e e r s o f t h e r m o d y n a m i c s . T h e r e f o re , a l t h o u g h I d o n o t f i n d

t h e m e x p l i c i t l y i n a n y e a r l y s o u r c e , I d o n o t s p e c i f i c a l l y a t t r i b u t e t h e m t o a n y

l a t e r a u t h o r .



362 C. TRUESDELL

T h e c o n c e p t o f hotness is n a t u r a l a n d p r i m i t i v e , l i k e p re s e n c e , w e i g h t , a n d

p l a c e . W e p e r c e i v e h e a t a n d c o l d a s w e d o l i g h t a n d d a r k , w e t a n d d r y . fa r a n d

n e a r, t h r o u g h o u r s en se s. A s M A C H w r o t e 5

A m o n g t h e s e n s a t i o n s b y w h i c h , t t ' i r o u g h t h e c o n d i t i o n s t h a t e x c i t e t h e m , w e

p e r c e i v e t h e b o d i e s a r o u n d u s , t h e s ens a t i ons o f hea t f o r m a s p e c i a l s e q u e n c e

( c o ld , c o o l , t e p id , w a r m . h o t ) o r a s p e c i al c l a s s o f m u t u a l l y r e l a t e d e l e m e n t s .

. .. T h e e s s e n c e o f t h is p h y s i c a l b e h a v i o r c o n n e c t e d w i t h t h e c h a r a c t e r i s t i c o f

s e n s a t i o n s o f h e a t ( t h e t o t a l i t y o f t h e s e r e a c t i o n s ) w e c a ll i ts h o t n e s s

[-'~W ~ r m e z u s t a n d " ] 6 .

A l s o , w r o t e M A C H 7,

T h e s e n s a t i o n s o f h e a t , l i ke t h e r m o s c o p i c v o l u m e s , f o r m a s i m p l e s e ri es , a

s imple cont inuous mani fo ld . I t d o e s n o t f o l lo w t h e r e f r o m t h a t t h e ho t nes s e sa l s o f o r m s u c h a m a n i f o l d . T h e p r o p e r t i e s o f a s y s t e m o f s i g n s d o n o t

d e t e r m i n e t h e p r o p e r t ie s o f t h a t w h i c h t h e y d e s ig n a t e . . . . T h e a s s u m p t i o n o f

a s imple con t inuous m ani fo ld o f hotnesses i s su f fi c ien t.

M A C H j u s t if i e s t h i s l a s t a s s e r t i o n o n l y b y s t a t i n g t h a t n o e x p e r i e n c e i n d i c a t e s t h e

c o n t r a r y .

T h e c o n c e p t o f " s i m p l e c o n t i n u o u s m a n i f o l d " o r " c o n t i n u o u s m a n i f o l d o f

o n e d i m e n s i o n " d e r iv e s f r o m R I E M A N N s a n d h a s b e e n m a d e p r e c is e i n m o r e

r e c e n t w o r k s o n p u r e m a t h e m a t i c s . M A C H a s s u m e s t h a t the se t f ig o f a l l

5 MACH,op. cir., § I o f " H i s t o r i s c h e U b e r s i c h t d e r E n t w i c k l u n g d e r T h e r m o m e t r ie " .

6 M r . SERRIN p re fe rs "ho tne ss l eve l" . Ea r l i e r m od ern au tho rs , I be ing one , have u sed" t e m p e r a t u r e " l o o s e ly t o m e a n s o m e t i m e s h o t n e ss , s o m e t i m e s a n u m b e r r e p r e s e n t i n g it . Fo re x a m p l e A . H . WILSON,Thermodynam ics and Statistical Mechanics, C a m b r i d g e , C a m b r i d g eUn ive rs i ty Press, 1957 , wr ites in § 1 .21 " I f we a s sum e th a t t em pera tu re i s a p r im ary idea , them e t h o d s o f m e a s u r i n g i t m u s t b e t o a c e r t a i n e x t e n t e m p i r ic a l ."

7 M A C H , Tem peraturbeg riff § 5.

s C f § 1 4 o f th e a r t ic l e b y F . E N R I O U ES, " P r i n z i p i e n d e r G e o m e t r i e " , Enzyklopiidie derMathematischen Wissenschaften, Volume 31 , Heft 1 , 1907:

T h e f u n d a m e n t a l c o n c e p t i n th e t h e o r y o f th e c o n t i n u u m is th a t o f th e continuous

manifold of one dimension. Regarded abs t rac t ly , th i s concep t may be iden t i f i ed wi th thec o n c e p t o f t h e line. i f the po in t o f the l ine is t aken a s the e lem en t o f the m an i fo ld a nd i f norega rd i s had fo r the re la t ion o f the l ine to the re s t o f space o r fo r any (me t r ic ) concep t o fthe l eng th o f segmen ts (o r a rcs ) o f the l ine . Thu s o n ly those p rope r t i e s o f the l ine a rec o n s i d e r e d t h a t a r e c o n n e c t e d w i t h g e o m e t r i c a l d e t e r m i n a t i o n o f t h e l in e t h r o u g h t h em o t i o n o f a p o i n t : t h e natural orderings of the po in t s o f the l ine and the i r continuity, th e

segments, etc.

T h o u g h t h e " c o n t i n u o u s m a n i f o ld o f o n e d i m e n s i o n " is t h u s d e f in e d p a r t l y in t e rm s o f w h a ti t is not , the explanat ion is pre t ty c lear . ENRIQUES at t r ib utes the c . .oncept to RIEMANN'Sce leb ra ted Habilitationsrortrag of 1854 , f ir st pub l i shed in 1868 : "U be r d ie Hypo these n ,w e l c h e d e r G e o m e t r i e z u G r u n d e l i e g e n ' , r e p u b l i s h e d i n a ll e d i ti o n s o f R IE MA N N 'sGesam melte Mathem atische Werke. RIEMANN's p re sen ta t ion , cond i t ioned b y the c i rcum-stances , is no t expl ic i t, bu t i t suff iced to g ive m ath em atic ian s s om e essentia l ideas which the ycould render , in t ime, prec ise .

Fo r a h i s t o r y o f t he c o n c e p t o f m a n i f o l d s ee t h e a r t ic l e b y D . M. J O HN SO N. " T h e p r o b l e mo f t h e i n v a r i a n c e o f d i m e n s i o n i n t h e g r o w t h o f m o d e r n t o p o l o g y " , Archive for History ofExa ct Sciences, Volume ~0 (1979) , pp . 97-189.



Absolu te Tem pera tures f rom Carnot ' s Axiom 363

hotnesses is a continuous manifold. ~ m a y w e l l b e t h e f i rs t m a n i f o l d t o h a v e b e e n

s u g g e s te d b y p h y s i c a l e x p e r i e n c e n o t e x p r e s s ed i n t e r m s o f m a s s , p o s i ti o n , a n d

t i m e . T h i s c i r c u m s t a n c e l e t s m e t h i n k i t w o r t h w h i l e t o p r e s e n t t h e s i m p l e

p h y s i c a l i d e a s t h a t l e a d u s t o i m p u t e s p e c i a l i z i n g p r o p e r t i e s t o t h e h o t n e s s

m a n i f o l d .

F i r s t o f a l l t h e r e i s o u r s e n s a t i o n t h a t a t a c e r t a i n t i m e o n e b o d y i s h o t t e r

t h a n a n o t h e r . S e n s a t i o n s , i n d e e d , a r e n o t a l w a y s c o n s i s t e n t . T h i s f a c t d o e s n o t

m a k e u s r e je c t a ll t h e e v i d e n c e t h e y b r i n g u s . R a t h e r , i f t h e y s o m e t i m e s b e a r

c o n f l ic t i n g w i t ne s s, w e r e g a r d t h e m a s b e i n g p a r t l y i n e r r o r t h e n , a n d w e d o n o t

o n t h a t a c c o u n t d e n y t h e e x i s te n c e o f w h a t t h e y u s u a ll y s u gg e st . W e l et t h e

e x c e p t i o n p r o v e t h e r u le . G u i d e d b y s e n s o r y e x p e ri e n c e , w e c o n c e i v e ~ a s b e i n g

o r d e re d . U s i n g > - t o d e n o t e " h o t t e r t h a n " a n d ~ t o d e n o t e " n o t c o o l e r t h a n " ,

w e a ss u m e t h a t ~ is a s im p l e o r d e r i n g u p o n J { : I f h x so ur a n d h 2 e . H t h e n e i t h e r

hl~=h 2 o r h2~=h1, a n d t h o s e t w o r e l a t io n s b o t h h o l d i f a n d o n l y i f h i = h 2.

I f h z ~ - h t , t h e s e t o f a l l h s u c h t h a t h 2 ~ h ~ h ~ w e m a y c a l l a bounded segment

]h~,h2[ o f ._, 'f . T h e segments o f ~ f' c o n s i s t i n t h e s e t s o f t h is k i n d a n d t h e s e t s

d e f in e d b y r e l a t i o n s o f t h e f o r m s h>-h~ a n d h 1 ~ -h . T h e s e g m e n t s p r o v i d e a b a s i s

f o r a t o p o l o g y o n ~ ' , i n t e r m s o f w h i c h w e m a y d e f i n e l im i t s a n d c o n t i n u i t y .

W h e n w e r e s t r i c t a t t e n t i o n t o a b o u n d e d s e g m e n t , o n l y i t s s u b s e g m e n t s , a l s o

n e c e s s a ri l y b o u n d e d , a r e n e e d e d f o r t h is p u r p o s e .

Temperature is a n u m e r i c a l i n d i c a t o r o f h o t n e s s, j u s t a s m i l e s t o n e s a l o n g a

r o a d i n d i c a t e n e a r n e s s a n d f a rn e s s b y n u m b e r s . A s M A C H w r o t e 9,

T h e t e m p e r a t u r e i s . . . n o t h i n g e l s e t h e n t h e characterisation, th e mark o f t h eh o t n e s s b y a n u m b e r . T h i s temperature number h a s s i m p l y t h e p r o p e r t y o f a n

inventory entry, t h r o u g h w h i c h t h i s s a m e h o t n e s s c a n b e r e c o g n i z e d a g a i n

a n d i f n e c e s sa r y s o u g h t o u t a n d r e p r o d u c e d .

An empirical-temperature function o r empirical-temperature scale i s a n a s s ign -

m e n t o f c o - o r d i n a t e s t o a s e g m e n t o vg o f o vt ' o r t o ~ i ts e lf . T h e v a l u e 0 0 o f a

t e m p e r a t u r e f u n c t i o n 0 a t a p a r t i c u l a r h o t n e s s h i s t h e empirical temperature o f

h 0 on the s c a le 0 : I f h o e o~o , the n 0 o = 0 ( h o ). T ha t i s n o t a ll . As M ACH ~° wr o te ,

T h i s [ t e m p e r a t u r e ] n u m b e r m a k e s i t p o s s ib l e to re c o g n i z e a t t h e s a m e t i m e

th e order i n w h i c h t h e i n d i c a t e d h o t n e ss e s f o l lo w o n e a n o t h e r a n d [ t or e c o g n i z e ] between w h i c h o t h e r h o t n e s s e s a g i v e n h o t n e s s l i e s .

T h a t i s , t h e m a p p i n g 0 preserves the order ofhotnesses:

hl~= h 2 ~ O(hl)>=O(h2). (17)

I t is t o t h i s a s s u m p t i o n , I b e li e v e , t h a t M A C H r e f e r r e d w h e n h e w r o t e t h a t " t h e

s e n s a t i o n s o f h e a t . . . f o r m a s i m p l e s e r ie s " . H e m e r e l y p u t i n w o r d s t h e s i l en t

p r e j u d i c e a l l t h e p i o n e e r s o f t h e r m o m e t r y , c a l o r i m e t r y , a n d t h e r m o d y n a m i c s

a c c e p t e d a s a m a t t e r o f c o u r s e : Hotnesses may be represented faithfully by points

9 M ACH, Temp eraturbegriff §22.

to Temp eraturbegriff § 22 .



364 C. TRL'ESDELL

on t lre real l ine. W e r e t h i s n o t s o . t h e i n n u m e r a b l e d i a g r a m s i n w h i c h a 0 - a x i s

a p p e a r s w o u l d m i s r e p r e s e n t .

T e m p e r a t u r e s h a d b e e n i n t r o d u c e d a n d w e r e g r o w n f a m i l i a r l o n g b e f o r e t h e

a b s t r a c t c o n c e p t o f h o t n e ss , a n d i t w a s s t u d y o f o u r e x p e r i e n c e w i th t e m p e r a -

t u r e s t h a t t a u g h t u s, f in a l ly , t o is o l a t e a n d a b s t r a c t h o t n e s s . I n d e e d , t e m p e r a t u r e

a n d h o t n e s s w e r e c o n f u s e d f o r a l o n g t i m e . a n d e v e n i n M A X W E L L 'S m a g i s t e r i a l

7 7 w o ry o f H e a t t h e t w o a r e n o t d i s t i n g u i s h e d e x c e p t b y i m p l i c a t i o n x~"

T h e t e m p e r a t u r e o f a b o d y . .. is a q u a n t i t y w h i c h i n d i c a t e s h o w h o t o r h o w

c o l d t h e b o d y i s .

W h e n w e sa y t h a t t h e t e m p e r a t u r e o f o n e b o d y i s h i g h e r o r l o w e r t h a n

t h a t o f a n o t h e r , w e m e a n t h a t t h e f ir st b o d y i s h o t t e r o r c o l d e r t h a n t h e

s e c o n d , b u t w e a ls o i m p l y t h a t w e r e f e r t h e s t a t e o f b o t h b o d i e s t o a c e r t a i n

s c a l e o f t e m p e r a t u r e s . B y t h e u se , th e r e f o r e , o f t h e w o r d t e m p e r a t u r e , w e fi xi n o u r m i n d s t h e c o n v i c t i o n t h a t i t is p o s s i b l e , n o t o n l y t o f ee l, b u t t o

m e a s u r e , h o w h o t a b o d y i s .

W h e n w e c o m e t o c o n s i d e r h o w t o m e a s u r e " h o w h o t a b o d y i s " , i t s u f f i c e s

f o r p r a c t i c a l p u r p o s e s t o l i m i t t he r a n g e o f a n e m p i r i c a l sc a le t o s o m e b o u n d e d

i n t e r v a l ] 0 1 , 0 2 [ . F o r o u r p r e s e n t , l o c a l c o n s i d e r a t i o n s , t h a t i n t e r v a l i s b e s t

r e g a r d e d a s c o n t a i n e d i n o n e o f t h e i n t e r v a ls m e n t i o n e d i n § 3 in c o n n e c t i o n w i t h

s o m e o n e b o d y . I t is n a t u r a l t o s u p p o s e t h a t t h i s r e a l i n t e r v a l i s t h e i m a g e o f a

b o u n d e d s e g m e n t ] h l, h 2 [ o f ~ . A c c o r d i n g l y w e b e g i n b y c o n s i d e r i n g o n l y t h o s e

e m p i r i c a l - t e m p e r a t u r e s c a l e s t h a t a r e o n e - t o - o n e , o r d e r - p r e s e r v i n g m a p p i n g s o fb o u n d e d s e g m e n t s ] h l , h 2 [ o f ~,~ o n t o b o u n d e d i n te r v a ls ] 0 1 , 0 , [ o f ~ . T h u s t h e

f u n c t io n 0 in (17 ) is a " h o m e o m o r p h i s m " : a c o n t i n u o u s t r a n s f o r m a t i o n h a v i n g a

c o n t i n u o u s i n v e rs e . I t is p l a u s i b l e a n d c a n b e p r o v e d e a s i l y t h a t t h e s e g m e n t

] h x , h 2 [ h a s t h e " H a u s d o r f f p r o p e r t y " : A n y t w o d i s t i n c t h o t n e ss e s i n it li e i n

s u b s e g m e n t s w h o s e i n t e r se c t i o n i s e m p t y . T h a t f a ct m a k e s ] h a, h z [ a c o n t i n u o u s

m a n i f o l d i n t h e m o d e r n s e n se o f t h e t e r m ; a n e m p i r i c a l - t e m p e r a t u r e f u n c t i o n i s a

char t on 2 It" wh i ch pr es er ves t he i n t r i n s i c o r de r o f ho t nes s es .

S u p p o s e t h a t o n s o m e o n e b o u n d e d s e g m e n t ~ t h e r e a r e t w o e m p i r i c a l -

t e m p e r a t u r e s c a le s ; 0 a n d 0 ". T h e n i f h o ~ o ,

0 o = 0 ( ho ) , 0 " = 0 " ( / , o ) = 0 " o 0 - '( 0 o ) . (18)

T h e c o m p o s i t e m a p p i n g 0 * o0 - ~ m a p s t h e r a n g e o f th e s c a le 0 o n t o t h e r a n g e o f

t h e s c a l e 0 " : I f f = O * o O - 1. t h e n

0 * = f ( 0 o ) , 0 o s R a n g e 0. (1 9)

B e c a u s e o f (17 ) we s e e tha t f i s a o n e - t o - o n e , b i c o n t i m t o u s , i n c r e a s i n g m a p p i n g o f

R a n g e 0 o n t o R a n g e 0 " . It w a s th e s e m a p p i n g s o f r e al i n te r v a l s o n t o r e al

i n t e r v a l s th a t t h e p i o n e e r s o f t h e r m o m e t r y h a d t o u s e i n c o m p a r i n g t h e r e su l ts

o f u s i n g d i f f e r e n t e m p i r i c a l - t e m p e r a t u r e s c a le s .

1~ j.C. MAXWELL,Theory o f H eat, 10 h (la st)e dit ion , 1891, p. 2.



Absolute Tem pera tures f rom Carnot ' s Axiom 365

I f 0 a n d 0 * a r e d e f i n e d o n b o u n d e d s e g m e n t s ~ a n d 9 f , w h o s e i n t e rs e c t io n

is t h e s e g m e n t ~ f'0 , t h e f o r e g o i n g a r g u m e n t s h o w s t h a t 0 a n d 0 * p r o v i d e

c o m p a t i b l e c h a r t s w h i c h t o g e t h e r c o v e r ~ w o ~ , . . T h u s ._,ug~w ~ _ , i s a c o n t i n u o u s

m a n i f o l d . W e c a n g o o n i n th i s w a y t o a n y c o l l e c ti o n o f o v e r l a p p i n g s e g m e n t s o f

. ~ . F o r t h e t i m e b e i n g , h o w e v e r , w e s h a l l re v e r t t o t h e s tu d y o f a si n g le b o u n d e d

s e g m e n t o f J~ .

T h e h o m e o m o r p h i s m s o f a b o u n d e d s e g m e n t o f .3 f w i th a b o u n d e d r ea l

i n t e r v a l f a l l i n t o t w o c l a s s e s : t h o s e t h a t p r e s e r v e t h e o r d e r o n o ~ , a n d t h o s e t h a t

r e v e rs e it. H o w d o w e d e c id e w h i c h o f t h e t w o c l as se s o f h o m e o m o r p h i s m s d o e s

c o n s is t o f e m p i r i c a l - te m p e r a t u r e s c a le s a n d w h i c h d o e s n o t ?

I t is c u s t o m a r y h e r e t o a p p e a l a g a i n t o t h e d i re c t e v i d e n c e o f o u r s e n se s : W e

f ee l t h a t b o i l i n g w a t e r is m u c h h o t t e r t h a n f r e e z in g w a t e r, etc. T h e m a p p i n g s

t h a t a s s ig n t o b o i l i n g w a t e r a h i g h e r t e m p e r a t u r e t h a n t o f r e e z in g w a t e r a r e

t h e r e f o r e r e g a r d e d a s f a it h fu l to t h e i n t ri n s ic o r d e r i n g o f ~ , a n d m a p p i n g s o f

t h e o t h e r c la ss a r e r e je c te d . W h i l e o f c o u r s e a n y c o m p a r i s o n o f t h e o r y w i th

e x p e r i e n c e r e s ts u l t i m a t e l y u p o n h u m a n s e n s a t io n s , w e p r e f e r t o e l im i n a t e a s

m a n y a p p e a l s t o t h e m a s w e c a n . T o d i s c e r n w h i c h o f t h e t w o c l as s es o f

h o m e o m o r p h i s m s r e s p e c t s (1 7) a n d h e n c e c o n s is t s o f e m p i r i c a l -t e m p e r a t u r e

s c a le s , w e m a y u s e h e a t e n g i n e s a s th e y a r e i d e a l i z e d b y C a r n o t c y c l e s. A C a r n o t

c y c l e a b s o r b s h e a t a t t h e higher o f i ts t w o o p e r a t i n g t e m p e r a t u r e s : 0 + > 0 - . A

p a r t i c u l a r c y c l e t h a t is a C a r n o t c y c l e a c c o r d i n g t o t h e h o m e o m o r p h i s m s o f o n e

c la s s is t h e r e v e r s e o f a C a r n o t c y c l e a c c o r d i n g t o t h o s e o f t h e o t h e r c l as s. T h e

w o r k d o n e b y a c y c le is a p u r e l y m e c h a n i c a l p r o p e r t y o f t h a t c y c le , i n d e p e n d e n t

o f t h e m e t h o d s o f m e a s u r i n g h o t n e s s . T h a t w o r k is p o s it i v e , n e g a t i v e , o r n u l l. I f

a c y c l e d o e s p o s i t i v e w o r k , i ts r e v e r s e d o e s n e g a t i v e w o r k . C A R N O T 's G e n e r a l

A x i o m ( 1 ) a s s e r t s t h a t every C arno t c yc le does pos i t i v e w ork . M o r e o v e r , this

s ta temen t i s invar ian t u n d e r t h e t r a n s f o r m a t i o n s ( 1 9 ) d e f i n e d b y i n c r e a s i n g

f u n c t i o n s f T h u s C A RN O T 's G e n e r a l A x i o m serve s to distinguish the class o f

empirical - temperature scales f r o m t h e c la ss o f h o m e o m o r p h i s m s t h a t r e v e rs e t h e

i n tr in s i c o r d e r o f h o tn e s se s . S u p p o s e t h a t a b o d y u n d e r g o a c y c le s u c h t h a t

1. it a b s o r b s a n d e m i t s h e a t o n l y o n t w o d i s t i n c t is o t h e r m s , w i t h w h i c h w e

h a v e s o m e h o w a s s o c i a te d n u m b e r s 0 + a n d 0 - , r e s p e c ti v e ly ;

2 . t h e a m o u n t o f h e a t i t a b s o r b s i s p o s i t i v e ;

3 . i t does pos i t i ve w ork .

T h e n t h e n u m b e r s 0 + a n d 0 - c a n n o t r e s u l t fr o m a n e m p i r i c a l - te m p e r a t u r e

f u n c t i o n u n l e s s 0 + > 0 - . T o g i v e a p r a c t i c a l e x a m p l e o f th i s d i s t in c t i o n , w e

p r o c e e d n o w t o t h e s t u d y o f th e r m o m e t e r s .

A the rmome te r i s a b o d y u s e d t o d e t e r m i n e e m p i r i c a l t e m p e r a t u r e s . T y p i c a l l y

i t i s a b o d y w h i c h e x p a n d s a p p r e c i a b l y w h e n i t i s h e a t e d , a n d i t s v o l u m e s a r e

r e g a r d e d a s t h e v a l u e s o f a n e m p i r i c a l - t e m p e r a t u r e f u n c t i o n . In M A X W E L L 's

w o r d s , t h e y m a k e i t " p o s s i b l e n o t o n l y t o f ee l, b u t t o m e a s u r e , h o w h o t a b o d yis ." M o r e t h a n t h a t, j u s t b e c a u s e v o l u m e s a r e m e a s u r a b l e t h e y a r e t a k e n a s

replacing t h e s o m e t i m e s i n c o n s i s te n t e v i d e n c e o f o u r s e n sa t io n s .

T o t h i s e n d w e c a n n o t u s e a n a r b i t r a r y b o d y . L e t u s s u p p o s e , f o r e x a m p l e ,

t h a t o u r f i rs t t h e r m o m e t r i c b o d y c o n s i s t s o f a ir , a n d l e t u s m e a s u r e v o l u m e s o f a



366 C. TRUESDELL

b o d y o f w a t e r a t a t m o s p h e r i c p r e s s u r e w h e n t h e a i r t e m p e r a t u r e is n e a r t o 4 ° C .

F i g u r e 1 s h o w s w h a t w e sh a l l f in d :

p - - 1 a r m

I i

Fig . 1 .0 (a i r the rmomete r ) in deg rees C

A s h a s b e e n k n o w n s i n ce th e d a y s o f t h e A c c a d e m i a d e l C i m e n t o , t h e b o d y o f

w a t e r a t th e p re s s u r e i a t m . o c c u p i e s t he s a m e v o l u m e a t t w o d i ff e re n t t e m p e r a -

t u r es a c c o r d i n g t o t h e a i r t h e r m o m e t e r , o n e a b o v e 4 ° a n d o n e b e l o w it. T h u s a

w a t e r t h e r m o m e t e r o p e r a t i n g i n th is r a n g e at c o n s t a n t p r e s s u r e w o u l d i n d i c a t e

t h e s a m e t e m p e r a t u r e a t h o t n e s se s w h i c h a c c o r d i n g t o t h e a i r t h e r m o m e t e r a r e

d i ff er e nt . T h e m a p p i n g f r o m t h e n u m b e r s d e t e r m i n e d b y v o l u m e s o f w a t e r o n t o

t h e n u m b e r s d e t e r m i n e d b y v o l u m e s o f a i r is n o t o n e - t o - o n e i n a n y o p e n

i n t e r v a l t h a t c o n t a i n s 4 ° C .

T h a t is n o t a ll. S u p p o s e w e t ry t o u s e as a t h e r m o m e t e r a b o d y o f w a t e r a t

a t m o s p h e r i c p r e s s u re i n th e r a n g e f r o m 0 ° t o 4 ° C . T h e t e m p e r a t u r e a c c o r d i n g t o

t h e p r e s u m e d w a t e r t h e r m o m e t e r is t h e n a n i n v e r t ib l e f u n c t i o n o f t h e t e m p e r a -

t u r e a c c o r d i n g t o th e ai r t h e r m o m e t e r , b u t t h e te m p e r a t u r e t h a t is h i gher

a c c o r d i n g t o t h e w a t e r t h e r m o m e t e r w i l l b e l ower a c c o r d i n g t o t h e a i r t h e r m o -

m e t e r . T h u s i t i s i m p o s s i b l e t h a t bo t h o f t h e se a l le g e d s c a le s o f t e m p e r a t u r e c a n

c o n f o r m w i t h ( 1 7 ) . O n e m u s t b e r e j e c t e d .

I n t h i s r e g a r d M A C H 12 w r o t e

77~ose hotnesses wi l l be ca l led h igher in which bodies g ive r i se to gre ater vo lum e

i nd i ca ti ons on t he t he r mos cope . . . . T h e r e f o r e w e avo i d u s e o f w a t e r a st h e r m o s c o p i c s u b s t a n c e . . .

M A C H w e n t t o o f ar h e re . T h e r e i s n o i n h e r e n t r e a s o n f o r r e g a r d i n g g r e a t e r

v o l u m e o f a i r t o b e m o r e f ai th f ul t h a n g r e a t e r v o l u m e o f w a t e r. T r u e , m o s t

o t h e r s u b s t a n c e s a g r e e w i t h a ir a n d d i s a g r e e w i t h w a t e r i n t h e i r b e h a v i o r , b u t

r e s o r t t o m a j o r i t y r u l e w o u l d s e e m d u b i o u s i f n o t d a n g e r o u s p r a c t i c e i n p h y s ic s .

x2 MACH, Temperaturbegriff § 4 . M o d e r n a u t h o r s u s e t h e t e r m " ' a n o m a l o u s " i n t h i srega rd . E .g . WILSON, § 1 .21 o f the boo k quo ted in F oo tno te 6 :

" B y c o m p a r i n g t h e r es u lt s o b t a i n e d b y u s i n g d if f er e n t th e r m o m e t e r s w e c a n r e j e ct a sunsu i tab le those subs tances, such as wa te r , who se beh av io r is ano m alou s . . . . Al l tha t wer e q u i re o f a s u b s t a n c e t o b e u s e d in c o n s t r u c t i n g a t h e r m o m e t e r i s t h a t t h e p r o p e r t y t o b em easu re d .. . sho u ld be a s tr ic t ly inc reas ing func t ion o f the tem pera tu re [i .e . h o t n e s s ] . "

Th e " a n o m a l o u s " s u b s t an c e s a r e f ew i n d e e d i n t h e r a n g e o f h o t n e s s es p r e s e n t l yava i lab le to exper imen t , bu t we a re le f t wonder ing how to te l l wh ich subs tances a re" a n o m a l o u s " a n d w h i c h a r e n o t w h e n n e w r a n g e s o f h o t n e s s s h a l l b e c o m e a cc es si bl e.



Absolute Tem peratures from Carnot's Axiom 367

U n d e r o u r i n t e r p r e ta t i o n o f C A R N O T ' S G e n e r a l A x i o m , i f a n y c y cl e t h a t

a c c o r d i n g t o t h e a i r t h e r m o m e t e r i s a C a r n o t c y c le w i t h o p e r a t i n g t e m p e r a t u r e s

i n t h e i n t e r v a l b e t w e e n 0 ° C a n d 4 ° C d o e s p o s i t i v e w o r k , t h e n t h e v o l u m e s o f a

f i x e d m a s s o f w a t e r a t a t m o s p h e r i c p r e s s u r e a r e n o t t h e v a l u e s o f a n y e m p i r i c a l-

t e m p e r a t u r e s c a l e f o r th e c o r r e s p o n d i n g r a n g e o f h o tn e s s e s . E x p e r i e n c e t e a c h e s

u s t h a t t h e e n g i n e w h i c h a b s o r b s h e a t a t t h e h i g h e r t e m p e r a t u r e a c c o r d i n g t o

t h e a i r t h e r m o m e t e r d o e s p o s i t i v e w o r k . T h u s i t i s t h e s u p p o s e d w a t e r t h e r m o m -

e t e r t h a t w e r e je c t i n t h e r an g e i n c l u d i n g 4 ° C a n d a i r t e m p e r a t u r e s l o w e r t h a n

tha t . Th e fo re go ing , I be l ie ve , is a p re c i se r e p la c e m e n t fo r MA C H ' S va gu e c l a im .

I t a ls o j u s t if i e s u s e o f t h e t e r m " a n o m a l o u s b e h a v i o r " h e re . O f c o u rs e , in o t h e r

c i r c u m s t a n c e s t h e r e i s n o r e a s o n a t a l l t o r e j e c t w a t e r a s a t h e r m o s c o p i c

s u b s t a n c e .

5 . H o t n e s s a n d E m p i r i c a l T e m p e r a t u r e , I I . D i f f e r e n t i a b i l i t y

S o f ar , w e h a v e a p p e al e d o n l y t o t h e h o m e o m o r p h i s m a n d i n tr i n si c o r d e r i n g

r e q u i r e d o f e m p i r i c a l - t e m p e r a t u r e s c al es . I f w e a r e t o u s e t h e m i n c a l o r i m e t r y ,

w e d e m a n d s o m e t h i n g m o r e . T h e v e r y s t a t e m e n t (3) o f t h e D o c t r i n e o f L a t e n t

a n d S p e c if ic H e a t s r e fe rs t o t h e d e r i v a t i v e o f a f u n c t i o n o f t i m e w h o s e v a l u e is

t h e t e m p e r a t u r e , a n d t h e u s e s t o w h i c h t h e c o n s t i t u t i v e f u n c t io n s w , A v , a n d K v

a r e p u t i n t h e d e v e l o p m e n t o f t h e r m o d y n a m i c s a ls o a p p e a l t o d i f f e r e n ti a ti o n .

D i f f e r e n t i a b i l i t y i t se l f r e q u i r es a s s u m p t i o n s r e g a r d i n g ._ fg . I f w e a p p r o a c h h o t -

n e s s t h r o u g h u s e o f e m p i r i c a l t e m p e r a t u r e - s c a l e s , w e m a y r e f er to t h o s e a s s u m p -

t i o n s i n d i r e c t l y b y r e q u i r i n g t h a t t h e t r a n s f o r m a t i o n o f o n e s c a le t o a n o t h e r b e

di f f erent iable . Th e n f i n (19 ) is a d i f f e re n t i a b le func t ion , a nd w e sha l l w r i t e

d0* = f ' ( 0 ) d0 . (20)

B e c a u s e f is a n i n c r e a s in g f u n c t i o n , f ' ( O ) > O . B e c a u s e t h e i n v e r s e t r a n s f o r m a t i o n

f - i is a l so d i ff e r en t i ab l e , t h e p o s s i b i l i t y t h a t f ' ( O ) = 0 fo r some 0 i s e xc lude d .

T h u s

f ' > O . ( 2 1 )

T h e r e q u i r e m e n t t h a t 0 e x i st a l m o s t a l w a y s is i n d e p e n d e n t o f t h e c h o i c e o f

scale .

S t i l l r e s t r i c t i n g a t t e n t i o n t o a s i n g l e b o u n d e d s e g m e n t J r 0 , l e t u s n o w

i n v e s t i g a t e t h e r u l e s o f t r a n s f o r m a t i o n f o r t h e q u a n t i t i e s t h a t e n t e r ( 2) a n d ( 3) .

F i r s t, t h e c h a i n r u l e a n d ( 2 0 ) s h o w t h a t

~ 0 " d O* = - ~ - d O . (22)

H e r e , a s u s u a l i n w o r k s o n p h y s i c s ,

~ a ~ . ( V , f _ 1 ( 0 , ) ) .,~ 0 " - 0 0 "

( 2 3 )



368 C. TRUESDELL

F r o m (2 2) a n d s i m i la r f o r m u l a e f o r th e d e r i v a t i v e s o f th e o t h e r c o n s t i t u t i v e

f u n c t io n s w e s ee th a t t o r e n d e r i n v a r i a n t u n d e r c h a n g e o f e m p i r i c a l - t e m p e r a t u r e

sca l e t he a s sum pt i on t ha t vv be con t i nu ou s l y d i f f e r en t i ab l e , it is neces sa ry t o

a s s u m e t h a t f ' i s cont inuous . W e m a y s u m t h e e f fe c t o f a d j o i n i n g t h is a s s u m p -

t i o n t o t h e p r e c e d i n g o n e s b y s t a t i n g t h a t S o i s a d i f f eomorph o f an open in terval .

L i k e w i se , if a s e g m e n t is th e u n i o n Q ) o ~ o f a fi n it e n u m b e r o f b o u n d e d

s e g m e n t s ~ ,~ ,, o n e a c h o f w h i c h a n e m p i r i c a l - t e m p e r a t u r e s c a le i s d e f i n e d , a n d i f

e a c h i n t e r s e c t i o n o f t h o s e s e g m e n t s s a ti sf ie s t h e c o n d i t i o n s j u s t d e m a n d e d o f . H ,

t hen Q )~¢~, is a d i f f e om orp h o f an op en i n t e rva l and has an i n t r i n s i c o rd e r i ng .

M o r e g e n e r a l c o n c lu s i o n s ca n b e o b t a i n e d f r o m t h e m o d e r n t h e o r y o f m a n i f o l d s

b u t a r e n o t n e e d e d i f o u r o b j e c t iv e is o n l y t o c l e a r a n d s p e c i fy t h e p i o n e e r s ' i d e a s

a b o u t s c a l e s o f e m p i r i c a l t e m p e r a t u r e .

In his t reatment o f 1975 Mr. SERRIN assumed ~ ' to be a diffeom orph of the real

line, equipped with an intrinsic ordering. Such an assumption seems to be implicit in

some earlier studies by others. In his late r w ork NERRIN requires of ~¢' me rely that i t

be a cont inuous manifold. For detai ls the reader should consul t his lecture of 1977,ci ted ab ove in F oo tno te 2. Mr. C .-S. MAN in a master 's thesis accep ted by the

University of Ho ng K ong in 1975 prov ided a far less general b ut c onstruct ive rather

than postulat ional introduct ion of the hotness manifold.

W e r e v e r t t o t h e c o n s i d e r a t i o n o f a s in g le b o u n d e d s e g m e n t o f J r , o n w h i c h

a n e m p i r i c a l - t e m p e r a t u r e s c a l e i s d e f i n e d . T h e q u a n t i t i e s A v a n d K v in (3) wil lg e n e r a l ly d e p e n d u p o n t h e c h o i c e o f e m p i r i c a l - t e m p e r a t u r e s c a le , b u t Q w i ll n o t.

T h e r e f o r e , i n a n e v i d e n t n o t a t i o n ,

H e n c e

A v.o ,(V ,O * ) V + K v , o , ( V , O * ) O * = A v , o (V ,O ) V + K v , o (V ,O )O . (24)

Av ,o , = Av ,o ,

K v. o* dO* = K v, o dO. (25)

I t f o l l ow s f rom (7 ) t ha t

Po* dO * = I1o dO. (26)

I n t h e s e f o r m u l a e t h e s u b s c r i p t 0 m a y b e r e a d " w h e n t h e s c a l e 0 i s u s e d " , a n d

l i k e w i s e f o r 0 " , a n d t h e a r g u m e n t s 0 * o n t h e l e f t - h a n d s i d e s a r e r e l a t e d t o t h e

a r g u m e n t s 0 o n t h e r i g h t - h a n d s i d e s t h r o u g h ( 1 9 ) . I n p a s s a g e s w h e r e o n l y a

s i n g l e e m p i r i c a l - t e m p e r a t u r e s c a l e i s c o n s i d e r e d , w e d r o p t h e s u b s c r i p t .

T h e t r a n s f o r m a t i o n r u l e s ( 2 2 ) a n d ( 2 5 ) s h o w t h a t t he s moo t hnes s a s s umed f o r

w, A v , a nd K v and the s igns o f ~v~/~ ,V, ~w/gO, A v , an d K v are invar iant under

change o f empir ica l - t emperature scale . I n p a r t i c u l a r , t h e c o n s t i t u t i v e i n e q u a l i t i e s

(2)2 and (3) z ho l d fo r a l l sca l e s i f an d on l y i f t hey ho l d fo r on e s ca le .

I n s u m m a r y , t he bas i c ax i oms (2 ) a nd (3 ) hold in t erm s o f one em pir ica l -t emp erature scale i f and o nly i f they hold in t erms o f a l l sca les.

Si nce a d i f f e r en t i a l equa t i on sa t i s f i ed by t he ad i aba t s i s d O /d V = - A v / K v . th e

q u a l i t a t i v e b e h a v i o r o f th e a d i a b a t s n e a r a p o i n t i s t h e s a m e f o r a ll sc a le s . T h a t

m e a n s t h a t C a r n o t c y c le s d e f in e d i n t e r m s o f o n e e m p i r i c a l - t e m p e r a t u r e s c a le



Absolute Tem peratures from Ca rno t's Axiom 369

ex i s t loca l ly i f and on ly i f they ex i s t loca l ly accord ing to a l l such s ca le s ,

t e m p e r a t u r e s c o r r e s p o n d i n g t o t h e s a m e h o t n e s s b e i n g u n d e r s t o o d i n t h e t e r m

" lo ca l ly " . M ore ove r , ~10,(0*)= 0 i f and on ly i f ~L0(0 = 0 , so the s e t o f t em pera tu res

a t w h i c h IL v a n i s h e s o n a n y o n e s c a le c o r r e s p o n d t o h o t n e s s e s t h a t m a k e #

v a n i s h o n a l l s c al es . W e r e c a l l t h a t t h e s e t o f s u c h t e m p e r a t u r e s h a s e m p t y

in te r io r .

6. Kelv in 's Ab solute Tem perature s

K E L V I N u s e d " a b s o l u t e t e m p e r a t u r e " i n t h r e e s e n s e s :

1. A s c a le o f e m p i r i c a l t e m p e r a t u r e i n d e p e n d e n t o f t h e c h o ic e o f th e r m o m e -

ter.2 . A s c a l e o f e m p i r i c a l t e m p e r a t u r e s u c h t h a t t h e w o r k d o n e p e r u n i t h e a t

a b s o r b e d i n a C a r n o t c y c le d e p e n d s u p o n t h e d i f f e r e n c e o f o p e r a ti n g t e m p e r a -

tu re s on ly .

3. A s c a l e o f e m p i r i c a l t e m p e r a t u r e s u c h t h a t t h e r a t i o o f h e a t e m i t t e d t o

h e a t a b s o r b e d i n a C a r n o t c y c l e s h a l l e q u a l t h e r a t i o o f t h e l o w e r o p e r a t i n g

t e m p e r a t u r e t o t h e u p p e r .

K E L VI N'S a b s o l u t e t e m p e r a t u r e o f 1 8 4 8 s a ti sf ie s t h e f ir st r e q u i r e m e n t ; s o

d o e s a n y d i f f e r e n ti a b l e f u n c t i o n o f t h a t t e m p e r a t u r e h a v i n g a p o s i t i v e d e r i v a t iv e ;

one such func t ion de f ines KELVIN ' s abso lu te t empera tu re o f 1854 . KELVIN 'S

s e c o n d r e q u i r e m e n t i s s a t i s f i e d b y h i s f i r s t a b s o l u t e t e m p e r a t u r e b o t h i n t h e

C a l o r i c T h e o r y a n d i n C L A US IU S ' t h e r m o d y n a m i c s . K EL V IN 's t h i r d r e q u i r e m e n t

is s a ti s fi e d i n C L A U S IU S ' t h e r m o d y n a m i c s b y h i s a b s o l u t e t e m p e r a t u r e o f 1 8 54 .

W h e n r e n d e r e d c o n c r e t e b y r e f e r e n c e t o t h e c l a s s i c a l a x i o m s ( 2 ) a n d ( 3 ) ,

CLAUSIUS' a b s o l u t e t e m p e r a t u r e o f 1 8 5 4 r e d u c e s t o KELVIN'S of t h e s a m e y e a r .

T h e h i s t o r i c a l d e t a i l s w i l l b e f o u n d i n m y f o r t h c o m i n g T r a g i c o m i c a l H i s t o r y .

H e r e b y u s e o f th e a p p a r a t u s j u s t n o w a s s e m b l e d I w i ll p r o v e a l l o f t h e f o r e g o i n g

s t a t e m e n t s t h a t r e f er t o K E LV IN 'S w o r k a n d c o m p a r e t h e r e su l ts a p p r o p r i a t e t o

different poss ibi l i t ies .

KELVIN's f i r s t s c a le o f " a b s o l u t e t e m p e r a t u r e " is d e f in e d a s fo l l o w s :

- - S # d O . (27)

S ince d r / d O = I t , t h e r e q u i r e m e n t ( 2 1 ) is s a ti s fi e d i f a n d o n l y i f

# > 0 fo r a l l 0 in i ts do m ain . ( 2 8 )

H e n c e f o r t h w e s ha l l a s s u m e p r o v i s i o n a l l y tha t (28) ho lds , de fe r r ing un t i l the nex t

s e c t io n t h e a n a l y s i s o f j u s t w h a t t h a t m e a n s a n d w h y a s s u m p t i o n s s u f fi c ie n t f o r

i t s h o u l d b e l a i d d o w n . S i n c e / ~ i s c o n t i n u o u s , o u r p r o v i s i o n a l a s s u m p t i o n ( 2 8 )

m a k e s r a n e m p i r ic a l - te m p e r a t u r e f i m c t i o n .M o r e o v e r , f r o m ( 2 6) w e s ee t h a t w e c a n a d j u s t t h e c o n s t a n t s o f in t e g r a t i o n i n

the d e f in i t ions o f ~0, an d z 0 so a s to insu re tha t

ro . (O*)=zo(O) . (29)



370 C. TRUESDELL

T h u s r i s a n a b s o l u t e t e m p e r a t u r e i n t h e s e n s e t h a t i t s v a lue i s i nde pe nde n t o f t he

empir ica l - ten+perature sca le u s e d t o o b t a i n it. T h e p h y s i c a l d i m e n s i o n s o f r a r e

w o r k + h e a t . T h e z e r o o f r is a r b i t r a r y .

A n y c o n t i n u o u s l y d i f f e re n t i a b le f u n c t i o n o f z w i t h p o s i t i v e d e r i v a t i v e s a ti sf ie s

a l l t h e s a m e r e q u i r e m e n t s . T h u s t h e r e a re i n f in i t e ly m a n y a b s o l u t e t e m p e r a t u r e s in

" KELVIN 'S f i rs t s ense . ~ is on e o f t he m . KE LVIN ' s s e c o n d s c a l e o f + ' a b s o l u t e

t e m p e r a t u r e " T i s d e f i n e d a s fo l lo w s :

T- -- e ~/'r. (30 )

J b e i n g a c o n s t a n t h a v i n g t h e d i m e n s i o n s o f w o r k + h e a t . V a l u e s o f T a r e

d im e ns ion l e s s a n d pos i t i v e ; T i s a b s o l u t e i n K E L V I N ' S f i r s t s e n s e . T h e v a l u e T = 1

c o r r e s p o n d s t o t h e a r b i t r a r y 0 o f ~.

A l l o u r c o n s i d e r a t i o n s s o fa r a r e r e st r ic t e d t o o n e b o d y a n d t o a n i n t e r v a l o f

e m p i r i c a l t e m p e r a t u r e s in . .~ , f o r t h a t b o d y o n s o m e o n e e m p i r i c a l - t e m p e r a t u r e

s c al e. T h e i n t e rv a l m a y b e s m a l l , b u t i t n e e d n o t b e . W i t h o u t k n o w i n g s o m e -

t h i n g a b o u t / ~ , w e c a n n o t d e t e r m i n e t h e r a n g e o f ~. I f / ~ is o f th e r i g h t k i n d , t h e

r a n g e o f ~ is t h e w h o l e r e a l li ne . T h e n T = 0 c o r r e s p o n d s t o r = - o e, a n d T = ~ c

c o r r e s p o n d s t o z = oc. T h u s t h e z e r o o f T h a s a s p e c i al s t at u s . W h e t h e r o r n o t r

a s d e f i n ed b y (2 7) c a n h a v e t h e v a l u e - o c , d e p e n d s u p o n t h e n a t u r e o f /~ . I t is

e a s y t o g iv e e x a m p l e s o f f u n c t i o n s # t h a t m a k e z h a v e a f in i te l o w e r b o u n d .

T h e n T n e v e r a t t a i n s t h e v a l u e 0. T h e p i o n e e r s r e g a r d e d # a s a f u n c t i o n t o b e

d e t e r m i n e d b y e x p e r i m e n t . K E LV IN s e e m s to h a v e t a k e n f o r g r a n t e d t h a t t h e

r a n g e o f T w o u l d c o m e o u t t o b e ] 0 , 0 ¢[ .

S i n c e z a n d T a r e e m p i r i c a l - t e m p e r a t u r e s c a le s , w e m a y e x p r e s s t h e f u n c t io n ~ i n

t e r m s o f t h e m . C a l l i n g t h e r e s u l t s / 4 a n d # r , w e s ee f r o m ( 26 ) t h a t

dO

d r J (31)

T h u s t he G e n e r a l C a r n o t - C t a p e y r o n T h e o r e m ( 7) a s s u m e s t h e fo l lo w i n g f o r m s

w h e n z a n d T a r e u s e d :

c~p= _ OPTA v , = = - - , J A v (32)

' o z ' r = l O T "O f c o u r s e

p , ( V ,z) - - w(V, 0(z)) , pr (V , r ) - - - w ( V , O ( T ) ) .

U s e o f a n a b s o l u te t e m p e r a t u r e e n a b l e s u s t o s t a t e t h e C a r n o t - C l a p e y r o n T h e o r e m

wi th ou t r e f e r e nc e t o C ar no t ' s f un c t i on I~.

A l l t h e f o r e g o i n g r e s u l t s f o l l o w f r o m C A R N O T 'S w o r k a l o n e . U s e o f t h e r e s u l t s i n

C o n c e p t s a n d L o g i c e n a b l e s u s t o g o f u r t h e r . F i r s t w e n o t e t h a t

go. = go, h o, = h o. (33)

I f g ' n e v e r v a n i sh e s , g is a n e m p i r i c a l - t e m p e r a t u r e f u n c t io n . I n g e n e r a l h i s n o t .

Be cau se o f (15 ) and (31 ),

h , = g ; , d hr = - r g ' r . (34)




H e n c e ( 1 0 ) b e c o m e s

( 1 1 ) b e c o m e s

a n d ( 1 3 ) b e c o m e s

C - ( if) _ h , ( z - ) _ h r ( T - ) .

C+(¢g) h~(z+ ) h r ( T + ) '

,r+ T+

L ( ~ ) S h~ (x ) dx J S [ hr ( x ) / ' x ] dx- - - T -

C + (~) h~(r + ) h r ( T + )

( 3 5 )

; (36)

(37)

T h u s u s e o f a n a b s o l u t e t e m p e r a t u r e e n a b l e s u s t o d i s p e n s e a l t o g e t h e r w i t h t h e

f u n c t i o n g .

I n t h e C a l o r i c T h e o r y o f h e a t w e m a y a l w a y s t a k e h a s 1. T h e n ( 34 )1 s h o w s t h a tg , = z + co ns t . , w h i l e ( 36 ) r edu ces t o

L ( ~ ) r ÷-- z + - . (38)+(C ~) z - = J l o g T -

T h e f o r m e r o f th e s e e x p r e s s io n s s h o w s t h a t b y i n t r o d u c i n g z K E LV IN o b t a in e d , f o r

t h e C a l o r i c T h e o r y , a t e m p e r a t u r e " a b s o l u t e " i n h is s e c o n d s e n s e a s w e ll a s hi s f ir st .

A l s o ( 3 7 ) b e c o m e sOKv, ~ = 8Av , ~ O K v , T = OAv, r (39)

~ V 8 z ' 8 V ~ T '

c o n d i t i o n s n e c e s s a r y a n d s u f fi c ie n t th a t a h e a t f u n c t i o n s h a l l ex i s t a t l e a st l o c a ll y .CLAUSIUS l a i d d o w n a s h i s f u n d a m e n t a l a x i o m t h e r e q u i r e m e n t t h a t i n a l l

cycles c~L ( f f ) = J C(Cg), (40)

i n w h i c h J is a u n i v e r s a l p o s i t i v e c o n s t a n t h a v i n g t h e d i m e n s i o n s o f h e a t + w o r k ,

w h i l e C ( f f ) is t he ne t ga i n o f hea t i n c~. C o ro l l a ry 10 .3 i n § 10 o f C o n c e p t s a n d L o g i c ,

r e s t a t e d a c c o r d i n g t o t h e d i r e c t i o n s o n p . 9 8, m a k e s C L A US IU S ' a x i o m e q u i v a l e n t ,

f o r t e m p e r a t u r e s t h a t c o r r e s p o n d t o p o i n t s o f N , , t o

g = J h + cons t . (41)

H e n c e i f g i s a n e m p i r i c a l - t e m p e r a t u r e f u n c t i o n , s o is h.

W e r e c a l l t h a t i f g ' d o e s n o t v a n i s h , g is a n e m p i r i c a l - t e m p e r a t u r e f u n c t i o n in a ll

t h e o r i e s c o n s i s t e n t w i t h C A RN O T 's G e n e r a l A x i o m , b u t h g e n e r a l l y is n o t . O n t h e

o t h e r h a n d , h i s a l w a y s a p o s i t i v e f u n c t i o n . B y m a k i n g h b e c o m e a n e m p i r i c a l -

t e m p e r a t u r e f u n c ti o n , CLAUSIUS' h e r m o d y n a m i c s p r o v i d e s a n e m p i ri c a l t e m p e r a -

t u r e w h i c h i s a l w a y s pos i t i ve .

I n t h e d e f i n i t i o n (3 0) o f T t h e c o n s t a n t c a l l e d J w a s a r b i t r a r y . I f w e c h o o s e i t a s

be i n g t he J o f C LA U SIU S ' ax i o m (40), f r om (34) w e see t ha t

h = K T , (4 2)

K b e i n g a n a r b i t r a r y p o s i ti v e c o n s ta n t . T h e n (3 5) b e c o m e s

C - (~) _ e - ( '+ -~ - )/a - T -C + ( , ~ ) - - T + . ( 4 3 )



372 C. TRUESDELL

K E LV IN r e g a r d e d t h e s e c o n d o f t h e s e s t a t e m e n t s a s e x p r e s si n g a d e s i r a b l e p r o p e r t y

o f a b s o l u t e t e m p e r a t u r e : f o r CLAUSIUS' h e o r y T is " a b s o l u t e " in K E L V I N 's t h i rd

s e n se o f t h e t e r m , a n d f o r th a t r e a s o n KELVINad op te d i t i n 1854 . i n CLAUSIUS"

t h e o r y ( 3 6 ) r e d u c e s t o

L ( ~) - ( r + - r - ) T -

J C + fig) = 1 - e x p J = 1 - T--- . (44)

T h e f o r m e r e v a l u a t i o n o f e f fi c ie n c y is d u e t o K E L V I N . I t s h o w s t h a t e v e n i n

CLAUSIUS' h e o r y r is a t e m p e r a t u r e " a b s o l u t e " i n KELVIN's e c o n d s e n s e a s w e ll a s

h i s f i r s t . The l a t t e r eva lua t ion in (44 ) i s due to RANKINE and CLAUSIUS.m o r e o r

l es s. D e t a i l s m a y b e f o u n d i n m y Tragicomical History. Fina l ly , i n CLAUSIUS' h e o r y

( 1 3 ) b e c o m e s

c K v , ~ = c A v . ~ Av .~ c K v . r = T c_~_[A v,r ~

(45)?,V ~ r J ' ~ V g T \ T ] "

I n ( 32 ) 2 a n d ( 4 5 ) , t h e r e a d e r w i ll r e c o g n i z e t h e b a s i c c o n s t i t u t i v e r e s t r i c t i o n s

o f c l as s ic a l t h e r m o d y n a m i c s . W e h a v e o b t a i n e d t h e m h e r e o n l y f o r t h e f u n c t io n s w ,

A v , K v , a n d / ~ b e l o n g i n g t o s o m e o n e b o d y , a n d o n l y f o r a n i n t er v a l o f e m p i r i c a l

t e m p e r a t u r e w h i c h m a y b e v e r y sm a l l. T h u s , s o f ar , t h e r a n g e o f T m a y b e s m a l l, a n d

w e m i g h t o b t a i n d i f f er e n t s c a le s T f o r d i ff e r en t b o d ie s . W e p r o c e e d n o w t o r e m o v e

t h e l i m i t a t i o n a n d t h e v a ri e t y . A t t h e s a m e t i m e w e return to the generali ty of

CARNOT'S theory.

7. Consequences of Part II of Carnot's General A xio m

CARNOT a s s e r t e d a l s o t h a t t h e mot ive pow er o f a Carn ot cycle was independent

of the body that underwent i t . T h i s is P a r t I I o f h is G e n e r a l A x i o m . A x i o m I V o f

Concepts and Logic e x p r e s s e s i t i n o n e w a y . F o r o u r p u r p o s e s h e r e a s o m e w h a t

w e a k e r a x i o m w ill do : Carnot 'sfunction l~ is a universal function i n t h e s e n s e t h a t

w h a t e v e r b e t h e c o n s t i tu t i v e f u n c t i o n s rv a n d A v u s e d t o d e t e r m i n e i t t h r o u g h (7 ),

t h e s a m e f u n c t i o n p r e s u l t s . H e n c e t h e s a m e a b s o l u t e t e m p e r a t u r e r i s o b t a i n e d

t h r o u g h ( 2 7 ) . The value of K E L V I N ' s absolute temperature r is independent not

only o f the empirical-temperature scale used but also of the body em ploye d to

determine #. I n o t h e r w o r d s , z m a p s h o t n e s s es o n t o n u m b e r s w i th n o r e fe r e nc e

t o a n y p a r t i c u l a r b o d y .

O u r c o n s i d e r a t i o n s a r e s t i l l l o c a l . T h e r e a r e m a n y w a y s t o e x t e n d t h e m t o t h e

w h o l e h o t n e s s m a n i f o l d ~ o r a s l a r g e a p a r t a s m a y s e e m d e s ir a b l e. I w i ll c h o o s e

o n e .

I n p a r t ic u l a r, l et ~ a n d ~ , b e b o u n d e d s e g m e n t s o f W w i t h a n o n - e m p t y

i n te r s e ct io n ~ 2 . L e t e m p i r i c a l - t e m p e r a t u r e fu n c t io n s 01 a n d 0 : b e d e fi n e d o n ~ 1

an d ) f ' : , r e spec t ive ly . Th en on o'¢g~2w e m a y i n t e r c o n v e r t 01 a n d 0 : b y a r u l e l i k e (1 9),

r e s t r ic t e d b y ( 21 ). L e t b o d i e s B 1 a n d B z h a v e n o r m a l s e ts w h o s e s e ts o f e m p i r i c a lt e m p e r a t u r e s a c c o r d i n g t o 01 a n d 0 2, r e s p e ct i v e ly , i n c l u d e t h e r a n g e s o f t h o s e

f u n c t i o n s . T h e c o r r e s p o n d i n g f u n c t i o n s p a r e tL 1 a n d p a , s a y . O n ) f 12 w e m a y

exp res s P2 a s a fun c t ion o f 0~ . S ince /~ fo r a g iv en sca l e i s a universal fun ct io n , IL_,s o

e x p r e s s e d mu st be the same fu nc tion as It ~. B y c h o i c e o f t h e c o n s t a n t o f i n t e g r a t i o n i n




(27), we ge t o n ~f~ 2 the s a m e a b s o l u t e -t e m p e r a t u r e f im c t i o n r f o r b o t h b o d i e s . N o w

con s i de r i ng o n l y B z , s ti ll on J{~ , , w e m ay u se 02 i ns t ead o f 0~ and bec ause r is an

a b s o l u t e - t e m p e r a t u r e s c a l e a g a i n c a l c u l a t e r t h r o u g h ( 27 ). I n t e rm s o f 0 , t h e

d e f i n i t i o n o f z s ti ll m a k e s s e n s e o n a ll th e i m a g e o f ~ , . W e h a v e p r o v e d t h e

f o l lo w i n g l e m m a . O n b o u n d e d s e g m e n t s ~ 1 a n d ~ 2 h a v in g a n o n - e m p t y i n t e r s e c t io n

le t t here be e m pi r i ca l - t em pera ture f im c t io n 0 t and O z ; l e t t he ranges o f O1 and 0 e be

in t e rva l s i nc luded in t he t em pera tu re se t s o f t he norm al s e t s o f bod ies B ~ and B 2 . Then

the absoh t t e t em pera ture - func t ion z , un ique to w i th in an arb i t rary cons tan t , ex i s t s on

T h e l e m m a s h o w s t h a t i f a s e g m e n t o f H is t h e u n i o n o f a fi n it e n u m b e r o f

b o u n d e d s e g m e n t s o n e a c h o f w h i c h # e x i st s a n d i s p o s i ti v e fo r s o m e b o d y , th e

a b s o l u t e - t e m p e r a t u r e f u n c t i o n r e x i s t s o n t h e w h o l e s e g m e n t a n d p r o v id e s a c o -

ord ina te sy s t em upon i t .

T h e r e a d e r f a m i l ia r w i t h t h e t h e o r y o f d i f f e r e n ti a b l e m a n i f o l d s w il l s e e t h a t i t is

u n n e c e s s a r y t o r e s t r ic t a t t e n t i o n t o a s e g m e n t w h i c h is t h e u n i o n o f a f in i te n u m b e r

o f b o u n d e d s e g m e n t s o f H . I h a v e d o n e s o b e c a u s e I d e s ir e to p r e s e n t a n a r g u m e n t

u s i n g o n l y e l e m e n t a r y id e a s ; I h a v e n o w i sh t o m a k e K E L VIN 's c o n c e p t u a l p r o b l e m

s e e m t r i v i a l b y i n v o k i n g a m o d e r n g e n e r a l t h e o r e m . M o r e o v e r , b e c a u s e e x p e r i -

m e n t s b y t h e i r n a t u r e c a n d e t e r m i n e o n l y f i n i t e l y m a n y n u m b e r s , t h e r e i s n o

p h y s i c a l g a i n in w e a k e n i n g t h e a s s u m p t i o n , a n d i f t h e p a r t o f ) f a c c e s s ib l e t o u s

t h r o u g h u s e o f e m p i r i c a l - t e m p e r a t u r e s c al es c o u l d n o t b e c o v e r e d b y a f i n it e

n u m b e r o f t h e m , n o e x p e r i m e n t c o u l d d e t e c t t h a t f a c t. P e r h a p s Y ?' i ts e l f c o u l d b e

c o v e r e d s i m i l a r l y , b u t n o e x p e r i m e n t c o u l d e s t a b l i s h t h a t .

8. T h e r m o m e t r i c A x i o m

I t r e m a i n s t o c o n s i d e r t h e c o n d i t i o n (2 8), w h i c h a ll t h e f o r e g o i n g c o n s t r u c t i o n s

h a v e p r e s u m e d t o h o l d . I n o r d e r t o e s t a b l i s h ( 2 8 ) , w e n e e d a f u r t h e r a x i o m . M r .

SERRIN, w o r k i n g u p o n a d i ff e r e n t f r a m e w o r k o f i d ea s , h a s e m p h a s i z e d t h e

i m p o r t a n c e o f s u c h a n a x i o m , w h i c h h e c a ll s t herm om et r i c . H e r e I a d o p t a s p e c i a l

case o f h i s ax i o m of 1975 . ph rase d a l it t le d i f f e r en t l y :

T h e r m o m e t r i c A x i o m . J Y' c o n t a i n s a s e g m e n t -H th w h i ch i s t h e u n io n o f a . fi n i te

num ber oJ bound ed segm en t s , on each oJ w h ich an em pi r i ca l - t em pera ture sca le i s

de f ined , l f Oo i s an em p i r i ca l t em p era ture o f a ho tnes s i n ~ th , i t li e s in t he t em p era tur e

in t e rva l o f t he cons t i t u t i ve dom ain ~ o f som e body such tha t a t one po in t on the

i so therm 0 = 0 oK v ~ K v . (46)

L e t u s c al l P t h e p o i n t w h i c h t h e T h e r m o m e t r i c A x i o m p o s it s. T h e n b e c a u s e o f (9 )

w e k n o w t h a tK p > K v at P . (47)

U t 2 7A v - ~ - > O at P . (48)

H e n c e A v # 0 . H e n c e P b e l o n g s to t h e n o r m a l s et o f t h e b o d y t h a t t h e

B y (4 ) , t hen ,



374 C. TRUESDELL

T h e r m o m e t r i c A x i o m p o s i t s. H e n c e (7 ) h o l d s a t P . B e c a u s e /L is a u n i v e r s a l f u n c t i o n ,

i t is n o w d e t e r m i n e d o n c e a n d f o r a ll a t 0 o ; b e c a u s e o f ( 4 8 ) ./ z (0 o ) > 0. B u t a c c o r d i n g

t o t h e T h e r m o m e t r i c A x i o m , f o r e a c h h i n ~ h t h e r e is a 0 0 o f t h is k i n d . B y a p p e a l t o

( 29 ) w e e s t a b l i s h t h e f o l l o w i n g

T h e o r e m . z as defined by (27) i s an empirical - temperature scale not only local ly but

also ol7 all of .)f~h ; fo r each h th e v alue r(O(h)) doe s n ot d epe nd upon the empirical-

temperature scale 0 used ro calculate i t .

T h u s r i s a n absohae- temperarure scale o n a l l o f ' . Y ~ , .

I h a v e c h o s e n ( 46 ) a s th e c r it i ca l c o n d i t i o n o n w h i c h t o b a s e t h e T h e r m o m e t r i c

A x i o m b e c a u s e t h e i n e q u a l i t y (4 7) r ef le c ts c o m m o n e x p e r i m e n t a l k n o w l e d g e .

A c c o r d i n g t o t a b l e s o f p h y s i c a l q u a n t i t i e s , n e a r l y a l l r e a l m a t e r i a l s s a t is f y ( 47 )

a l m o s t a l w a y s . T h e r e a r e s o m e e x c e p t i o n s ; i n d e e d , w e c a n s ee d i r e c t l y f r o m ( 4)3 t h a t

K p = K v i f a n d o n l y i f A N ~ w / ~ 0 = 0 . S o m e f ew m a t e r i a l s s e e m t o o b e y v e r y c l o s e l ya n e q u a t i o n o f s ta te o f th e f o r m p = w ( p ) , a n d f o r t h e m K p = K v a l w a y s . A f e w

o t h er s , li k e w a t e r, e x h ib i t c u r v e s o n w h i c h ~ v / ? 0 = 0 ; F i g u r e 1 s h o w s o n e p o i n t o n

s u c h a c u rv e . B e c a u se / ~ > 0 i n v i r tu e o f t h e T h e r m o m e t r i c A x i om ~ w e c o n c l u d e t h a t

A v = O i f a n d o n l y i f ~ w / ~ 0 = 0 , s o th e c u r v e s j u s t m e n t i o n e d c o u l d b e d e f in e d

a l t e r n a t i v e l y a s c u r v e s o n w h i c h A v = 0. C o u n t e r e x a m p l e s s h o w t h a t w i t h o u t s o m e

a s s u m p t i o n s u c h a s t h e T h e r m o m e t r i c A x i o m i t w o u l d b e po s s i b le f o r kt t o v a n i sh

f o r c e r t ai n p a r t i c u l a r t e m p e r a t u r e s . I f p ( 0 0 ) = 0, (7 ) s h o w s t h a t ~ v / ? 0 = O f o r a l l V

and fo r all bodies o n th e i s o th e r m 0 = 0 0 . S u c h a s t a t e m e n t w o u l d a b u n d a n t l y

c o n t r a d i c t e x p e ri e n ce . T h i s fa c t p r o v i d e s f u r t h e r s u p p o r t f o r t h e T h e r m o m e t r i c

A x i o m 13I n t e n t i o n a l l y I h a v e l ef t , ~t h u n s p e c i f ie d . M a t h e m a t i c a l a u t h o r s u s u a l l y w i s h t o

h a v e a s in g le s c al e o f t e m p e r a t u r e o n a ll o f af t. A n o b v i o u s m o d i f i c a t i o n o f t h e

t h e r m o m e t r i c a x i o m c a n a d j u st t h e f o r eg o i n g t h e o r e m t o th a t r e q u i r em e n t . T h e n

~3 Th e func t ion o f the the rm om et r ic ax iom i s to m ake su re tha t /~(0o ) ex i s t s and#(0o) > 0 for all 0o in range 0 . As sum pt ion s su ff ic ien t to th i s end have been m ade , exp ress ly o rtac i tly , s ince the beg inn ings o f the rm ody nam ics .

CARNOT res t r ic ted h i s a t t en t ion to idea l gases, fo r wh ich o f cou rse ~ v / g O - p / O > 0 . He

s ta ted exp ress ly tha t Av>O. Since he a l so de r ived the co r re spo nd in g spec ia l ca se o f theGe nera l Ca rno t -C lape y ron Th eo rem (7) , i t was unn ecessa ry fo r h im to s ta te th a t ~t > 0. H is

successo rs ex tended the scope o f the theo ry , bu t a s they ag reed th a t # w as a universalfunc t ion , the sam e fo r al l bod ie s, and a s they u sed f ree ly the conc ep t o f idea l gases, they ha dno need to de te rmin e the s ign o f # af re sh , fo r CARNOT had a l ready don e so . Th ey to ok fo rg ran ted tha t ~ > 0 , a s is shown by the i r d iv id ing b y i t wheneve r co nven ien t .

Th e f ir st t ime tha t th i s a rg um en t fa i led to rem a in va l id was in KELVIN's second theo ry o fa b s o l u t e t e m p e r a t u r e . 1 8 5 4. H e c o u l d n o l o n g e r h a v e r e c o u r s e t o e v a l u a t i n g # b y u s e o f a nidea l gas. and h i s ana ly s i s o f the behav io r o f wa te r in 1 853 had show n h im tha t fo r somes u b s t a n c e s i n s o m e c o n d i t i o n s Av<O. A s h e c o n c l u d e d t h a t I~=J/T, T be ing h i s secondabso lu te t empera tu re , he ce r ta in ly conc lud ed a l so tha t # > 0 , bu t h i s wor k in these yea rs was

ske tchy , We cann o t be su re ju s t w ha t he a s sum ed o r how ca re fu l ly he thou gh t o u t the de tai ls .H e w as the f ir st pe r son to pub l i sh (4) 3 in ful l gene ra l i ty , and ce r ta in ly ne i the r he n o r any o the rea r ly s tuden t wou ld have hes i ta ted to a s sume tha t K p > K v i f ~ r a / ~ 0 # 0 , f o r th a t w a sp rec i sely wha t a l l ava i lab le expe r imen ta l da ta ind ica ted . I t is fo r th i s ve ry reason tha t I havec h o s e n t o s t at e a T h e r m o m e t r i c A x i o m i n te r m s o f K p a n d K v . O f c o u r s e a n a x i o m m u s t g ob e y o n d e x p e r i m e n t a l d a t a , m u s t e x t r a p o l a t e f r o m t h e k n o w n i n t o t h e u n k n o w n , f o ro therwise i t would be use less in predic t ion .

L a t e r a u t h o r s h a v e a b a n d o n e d CARNOT'Sa p p r o a c h . N o t h a v i n g t h e G e n e r a l C a r n o t -C l a p e y r o n T h e o r e m ( 7 ) o n w h i c h t o b a s e t h e i r c o n s t r u c t i o n s , t h e y h a v e h a d t o u s e m o r e



A b s o l u t e T e m p e r a t u r e s f r o m C a r n o t ' s A x i o m 3 75

e a c h h o t n e s s lie s i n t h e d o m a i n o f s o m e e m p i r i c a l - t e m p e r a t u r e f u n c ti o n . W h i l e I s e e

n o o b j e c t i o n t o s u c h a n a s s u m p t i o n , it s e e m s b o t h d a r i n g a n d u n n e c e s s a r y . W e

m i g h t b e c o n t e n t t o t h i n k o f .3f~h a s t h e s e t o f a l l h o t n e s s e s s o f a r a c c e s s i b l e t o

e x p e r i m e n t . T h a t s e t m a y w e ll g r o w a s ti m e g o e s o n .

9 . T h e P h y s i c a l D i m e n s i o n o f A b s o l u t e T e m p e r a t u r e

A v a l u e o f K E L V IN 'S s e c o n d a b s o l u t e - t e m p e r a t u r e s c al e is a d i m e n s i o n l e ss

n u m b e r . W e ar e a c c u s t o m e d t o t e m p e r a t u r e s t h a t b e a r a d i m e n s i o n c a l l e d " t h e

d i m e n s i o n o f t e m p e r a t u r e " . L o o k i n g b a c k a t a n e m p i r i c a l - t e m p e r a t u r e s c a le 0 ,

w e m a y i f w e l ik e t h i n k o f it as b e a r i n g t h e p h y s i c a l d i m e n s i o n w h i c h w e m a y

c al l " t h e d i m e n s i o n o f t h e sc a le 0 " . P h y s i c a l d i m e n s i o n s s u c h a s t h o s e o f m a s sa n d l e n g t h a n d t i m e a r e a s s i g n e d o n c e a n d f o r al l, a n d o n l y t h e m a s s , l e n g t h ,

a n d t i m e - i n t e r v a l t o w h i c h t h e v a l u e 1 is a s s i g n e d m a y b e c h o s e n a t w i ll . A l l

s ca le s o f m a s s a r e p r o p o r t i o n a l . S u c h is n o t t r u e o f e m p i r i c a l - t e m p e r a t u r e s c al es .

W e m a y i n d e e d d i v id e t h e m i n t o e q u i v a l e n c e c l as se s, al l m e m b e r s o f a n y o n e o f

w h i c h a r e p r o p o r t i o n a l . T h e n a p h y s i c a l u n i t a p p e r t a i n s t o a g i v e n c l a s s b u t h a s

n o m e a n i n g w i t h r e s p e c t t o a n y o t h e r c l a s s . T h u s t h e r e a r e i n f i n i t e l y m a n y

d i st i nc t p h y s i c a l d i m e n s i o n s o f e m p i r i c a l t e m p e r a t u r e .

K E L V I N ' S a b s o l u t e t e m p e r a t u r e s ~ a n d T a r e c o n c e i v e d i n s u c h a w a y a s t o

c o r r e c t t h is u n s a t i s f a c t o r y s t a te o f a f fa i rs . They annul the differences b e t w e e n t h e

p h y s i c a l d i m e n s i o n s o f a ll p o s s i b l e e m p i r i c a l s c a l e s. T h e v a l u e s o f a n y o n e o f th es c a l e s T a r e dimensionless numbers w h i c h d o n o t d e p e n d u p o n t h e ch o i ce o f t h e

e m p i r i c a l - t e m p e r a t u r e s c a l e 0 u s e d t o o b t a i n ,u a n d t h e n t o c a l c u l a t e T a n d T . I f

w e c a n p r o v e a ll s c a le s T t h a t c o r r e s p o n d t o a g i v e n J t o b e p r o p o r t i o n a l , w e

c a n t h i n k o f t h es e p u r e n u m b e r s a s c o o r d i n a t e s o n o n e a n d t h e s a m e l -

d i m e n s i o n a l v e c t o r s p ac e , t he s p a c e o f absolute-temperature vectors. I t is j u s t t h e

s a m e t h i n g t o s a y t h a t a b s o l u t e t e m p e r a t u r e h a s it s o w n p h y s i ca l d im e n s i o n , t h e

dimension o f absolute tem perature. O n l y t h e c h o i c e o f u n i t o f a b s o l u t e t e m p e r a -

t u r e d is t i n g u i s h e s o n e a b s o l u t e s c a l e f r o m a n o t h e r .

e labo ra te reason ing . T hey a l so have recou rse f ree ly to a "Fi r s t Law " , wh ich I wish exp ressly

to avo id .The ea r l i e s t exp l ic it the rm om et r ic ax io m I have found i s Pos tu la te V ( the rmom ete r s ) in

t h e p a p e r o f J . B . B O Y L IN G " A n a x i o m a t i c a p p r o a c h t o c l a ss ic a l t h e r m o d y n a m i c s " ,Proceedings of the R oyal Society (Lo ndon ) A 329 (1972), 35 -70 . Tha t pos tu la te r eads in pa r t :

"T he r e ex i st s a c la s s o f s imp le sys tems ca l led the rm om ete r s w i th the fo l lowing 'p r o p e r t i e s : . . .

(e ) Fo r e v e r y th e r m o m e t e r M . . . t h e re s t r ic t i o n o f [ t h e h e a t f o r m ] ~ , t o a n a r b i t r a r yi s o t h e r m o f M is e v e r y w h e r e n o n - z e r o ( o n t h a t i s o t h e rm ) . "

In m y no ta t io n the ax iom j.us t quo ted a s se r t s tha t o n each su ff ic ien tly sho r t i so the rm a l

s e g m e n t A v * O f o r s o m e b o d y .

T o Mr . SE R R IN w e o w e a d e e p e r u n d e r s t a n d i n g o f t h e r o le o f a T h e r m o m e t r i c A x i o m ;m o s t o f w h a t I k n o w a b o u t t h e m a t t e r I h a v e l e a r n t f r o m h im . I n h i s n o te s o f 1 9 75 h e a s su m e stha t fo r each h the re i s one body such tha t AvOw/~O#O loca l ly on some empi r ica l -t e m p e r a t u r e s c a le c o v e r i n g h . I n t h e f r a m e w o r k o f CARNOT's deas , h i s ax iom a nd the one Ip rop ose he re a re equ iva len t . In h i s l ec tu re o f 1977 he a s sumes o n ly tha t fo r each h the re i s oneb o d y s u c h t h a t A v +-0 loca lly . To wi th in t echn ica l de ta i l s th is ax iom seem s to be the sam e a sp a r t o f BOYLING's.



376 C. TRUESDELL

C o m i n g t o t h e d e t a il s , w e fi rs t r e n d e r e x p l i c it t h e r o l e o f t h e c o n s t a n t o f

i n t eg ra t i on i n (27 ) and so expres s ( 30 ) i n t he fo rm

1 0 =_ exp/79_> 0,T - A e x p ( -J J o oP O (X )d x , A (49)

r o b e i n g t h e v a l u e a s si g n ed t o r w h e n 0 = 0 0 . I f ~ o = 0 , t h e n A = I , a n d t h e

a b s o l u t e t e m p e r a t u r e 1 is a s s i g n e d t o t h e a r b i t r a r y e m p i r ic a l t e m p e r a t u r e 0 o . If.

o n t h e o t h e r h a n d , w e m a k e t h e c h o i c e o f 0 o a n d r o d e p e n d u p o n s o m e p h y s i c al

p h e n o m e n o n , w e m a y o b t a i n a v a l u e o t h e r t h a n 1 f o r A . B y v a r y i n g th e p o s i t iv e

n u m b e r A w e o b t a i n a l l p o s s ib l e a b s o l u t e - t e m p e r a t u r e s c al es T c o r r e s p o n d i n g t o

a g i v e n c o n s t a n t J . I t i s t o t h e e q u i v a l e n c e c l a s s s o o b t a i n e d t h a t w e a s s i g n t h e

p h y s i c a l d i m e n s io n o f a b s o l u t e t e m p e r a t u r e . C h o i c e o f A d e f i n es t h e u n i t o f a

p a r t i c u l a r sc a le o f a b s o l u t e t e m p e r a t u r e Z I n t e r n a t i o n a l c o n v e n t i o n n o w t a k e s

0 o a s a n e m p i r i c a l t e m p e r a t u r e o f t h e t r ip l e p o i n t o f w a t e r a n d a s s ig n s t h e v a l u e

2 7 3 . 1 6 K t o A .

K E L VIN h i m s e l f p r e f e r r e d t o a s s ig n a p a r t i c u l a r d i f f e re n c e o f t e m p e r a t u r e s

r a t h e r t h a n p a r t i c u l a r t e m p e r a t u r e s . I f T 1 a n d T2 a r e t h e v a l u e s o f T t h a t

c o r r e s p o n d t o t h e e m p i r i c a l t e m p e r a t u r e s 0 1 a n d 0 2 , t h e n ( 4 9 ) s h o w s u s t h a t

T = ( T 2 _ T 1 T / T 1T z / T - 1 '

( 5 0 )

e x p ( ~ i l ~ o ( x ) d x )

= ( T 2 - T l )( , O s =

e x p \ a o , ~ ° ( x ) d x ] 1

When 01 and 02 a r e g i ven , t o a s s i gn A i s one and t he s ame t h i ng a s t o a s s i gn 7"_ ,

- T 1F o r e x a m p l e , w e m a y c h o o s e f o r t h e e m p i r i c a l - t e m p e r a t u r e s c a l e 0 t h e

C e l s iu s s c a l e : t h a t w h i c h is p r o v i d e d b y t h e a i r t h e r m o m e t e r w i t h 0 ° a n d 1 00 °,

r e s p e c t iv e l y , a s s ig n e d t o t h e h o t n e s s e s a t w h i c h w a t e r a t s t a n d a r d a t m o s p h e r i c

p r e s s u r e f r ee z e s a n d b o i ls . I f w e w i s h a n a b s o l u t e s c a le w h i c h p r e s e r v e s t h e

d i f f e re n c e o f 1 0 0 ° in t h e t e m p e r a t u r e s a s s i g n e d t o t h e s e h o t n e s s e s , w e s i m p l y p u tT2 - T a = 100 i n (50 ) and so ob t a i n

l O O e x p ( l i p o ( x )d x )

T = l O O • (51)

T h i s f o r m u l a c o n v e r t s d e g r e e s C e l s i u s o n t h e a i r t h e r m o m e t e r t o d e g r e e s

a b s o l u t e w i t h t h e s a m e d i f f e r e n c e o f t e m p e r a t u r e s , n a m e l y I 0 0 °, b e t w e e n t h e

b o i l i n g p o i n t a n d t h e f r ee z i n g p o i n t o f w a t e r .

I t w as sugges t ed by H ELMH O LTZ and JO U LE t ha t 11 v a r i e d i n v e r s e l y a s t h et e m p e r a t u r e a b o v e a b s o l u t e c o l d . T h a t i s , o n t h e C e l s i u s s c a l e

J~ 0 - 0 + 0 a , ( 5 2 )




- 0 ~ b e i n g t h e C e l si u s t e m p e r a t u r e o f a b s o l u t e c o ld . I f t hi s f o r m u l a w e r e

r i g o r o u s l y c o r r e c t , ( 5 1 ) w o u l d r e d u c e t o

r = O + O ~ . (53)

T h e e x p e r i m e n t s o f J OU L E & K E L VIN o n t h e p o r o u s p l u g a r e i n t e r p r e t e d a s

s h o w i n g t h a t ( 5 2) is v e r y n e a r l y b u t n o t e x a c t l y tr u e . T h e r e f o r e th e a b s o l u t e s c a l e

de t e rmi ned by (51) i s ve ry nea r l y (53 ) , w h i ch i s t he C e l s i us s ca l e w i t h i t s ze ro

s h i f t e d t o a b s o l u t e c o l d . T h i s c o n c l u s i o n r e q u i r e s e x p e r i m e n t a l d e t e r m i n a t i o n

n o t o n l y o f / z 0 b u t a l s o o f 0 a . I t h a s l o n g b e e n k n o w n t h a t 0o i s ve ry nea r l y

- 2 7 3 ° C .

T h e f o r e g o i n g tr e a t m e n t r e fe r s o n l y t o t e m p e r a t u r e s, n o t t o h o t n es s e s. W e

m a y t h i n k o f e a c h p o s s i b le c h o i c e o f t h e s c a le T a s c o r r e s p o n d i n g t o a p a r t i c u l a r

ho t nes s h o s e l ec t ed by som e ru l e . I f w e w r i t e Tho fo r t he s ca l e t ha t a s s i gns t o h ot h e v a l u e 1 ,

Tho h o )= 1, (54)

t h e n it is e a sy t o d e m o n s t r a t e t h e r u le o f t r a n s f o r m a t i o n u n d e r c h a n g e o f u n i t

h o t n e s s :

L : ( h ) = L , . ( h ~ ) L 1 ( h ) . ( 5 5 )

A n y t w o a b s o l u t e s ca le s T t h a t c o r r e s p o n d t o o n e c h o i c e o f t h e p o s it iv e

c o n s t a n t J a r e p r o p o r t i o n a l t o e a c h o t h e r , a n d i f w e s e le c t t w o p a r t i c u l a r

h o t n e s s e s h : a n d h 1, t h e c o n s t a n t T h2 (h l) p l a y s t h e r o l e o f a u n i t o f t e m p e r a t u r e :

Th (.~, , "Fa2 h ) - Th2 (h'), v ' 9 = ~ . (56)

T h e r e s u l ts p r e s e n t e d a t t h e b e g i n n i n g o f t h is a p p e n d i x c a n b e d e r i v e d f r o m ( 55 )

a n d (5 6), p r o v i d e d / ~ b e o f t h e r i g h t k i n d . I t h i n k t h is a p p r o a c h i s c l e a r e r a s w e l l

a s nea t e r . F o r ex am pl e , i t show s t ha t i f ( 52 ) he l d s t r i c t l y , t hen t he ab so l u t e s ca l e

d e f i n e d b y t h e h o t n e s s h I a t w h i c h w a t e r f r e e z e s w o u l d b e g i v e n a s f o l lo w s i n

t e r m s o f th e C e l s iu s s c a l e 0 :

O(h) + 0 aTh, = 0a (57)

10. C o n c lu s io n

T h e c o n c e p t o f a b s o l u t e t e m p e r a t u r e h a s a l w a y s b ee n i n h e r e n t i n CARNOT'S

g e n e r a l t h e o r y . T o c o n s t r u c t t h e a b s o l u te t e m p e r a t u r e s i n t r o d u c e d b y KELVIN, we

r e q u ir e n e i th e r th e " F i r s t L a w " n o r t h e " S e c o n d L a w " o f t h e r m o d y n a m i c s .

1 1 . A p p e n d i x . A x i o m s o f C l a s s i c a l T h e r m o d y n a m i c s

In C o n c e p t s a n d L o g i c , A p p e n d i x t o C h a p t e r 1 5, M r . B H A RA T H A & i h a v e

p o i n t e d o u t o n e w a y t o m o d i f y o u r p r e s e n t a t i o n s o a s t o d e l i v e r c l a s s i c a l



378 C. TRUESDELL

thermodynamics quickly: Adopt CLAUSIUS' "'First Law" (40). In §6 of this note I

have shown that the corresponding special choice of J in (30) then gives to KELVIN's

second abso lute temperature T exactly the proper ty he desired for it. namely (43)2.

a property which it most certainly does not have for other choices of J.However. I think the basic program of Concepts and Logic is more elegant:

Prove both the "First Law" and the "Second Law " as theorems based upon axioms

which render formal CARNOT'Sown assumptions, of course not restricted by his

subsidiary and undesirable adherence to the Caloric Theo ry of heat. In Concepts

and Logic we attained this goal by our Axio m l/. There is a body of ideal gas such that

the whole I7-0 quadrant is a thermodynantic part, and both specific heats are

constant. There we used the concept of ideal-gas temperature.

While in this note I have excised all use of ideal gases in the sense of the pioneers,

if we start from KELVIN'S second abso lute tempera ture T as defined in terms of

an arbitrary positive cons tant J we can by modify ing a little the axiomat ic structure of

Concepts and Logic uphold its program. By using absolute temperatures only we

can again prove both the First Law and the Second Law as theorems. I now present

the details.

We replace the primitive concept of temperat ure by the primitive concept of

hotness and lay down

Axiom O. The set of all hotnesses is a diffeomorph of a real interval, equipped with

an intrinsic ordering.

Definition O. An empiricaI-temperature fun'ction is an order-preserving chart on a

bounded segment of the hotness manifold.

Axioms I-I II of Concepts and Logic are then to be understood as referring to

some one empirical-temperature scale and hence to all such scales on one bou nde d

segment.

Axiom IV should be modified slightly: Any two fluid bodies which may undergo

Carnot cycles with the same operating temperatures do in those cycles the same

amount of work per unit of heat absorbed.

Next we lay down the 77wrmometric Axiom stated above in § 8. On that basis we

may construct in terms of any positive constan t J the absolute-temperature scale T

on all of YC~,h- We m ay then conve rt all the local rest rictions shown to be valid forsome one empirical- temperature scale into statements valid for all T. Mos t of these

have been listed above in § 6.

Definition. A body of ideal gas is a body such that

for all T. pV=RT, R=c ons t. >0, (58)

The temperature measured by the volumes of such a body at constant pressure provides "anatural measure of temperature . . . . a real temperature . . . . a sort of Platonic idea oftemperature," of which the temperature read on an air thermometer is "only an incomplete.

inexact expression". The results of JOULE & KELVIN'sexperiment with the porous plugshow that the inexactness of the air thermometer is negligible for most purposes. Thus we

are not astonished to learn that the idea "'absolute cold" or "'absolute zero" is 150 years

older than the idea "absolute temperature". The relations (31)a, (32)~, (34), and (35) wereall first obtained by use of the "Platonic idea" of an ideal gas.




A p p e a l i n g t o (3 2) a n d ( 4 ) 3 , w e see t ha t t he spec i f i c hea t s o f an i dea l gas r e f e r r ed

t o t he s ca l e T sa t i s fy t he r e l a t i on

d(K p , r - K v , r ) = R , ( 59 )

Y be i ng t he p os i t i ve c on s t an t u sed t o d e f i ne T . Th i s r e l a t i on , l i ke (58 ) i ts e lf , is

f a m i l i a r b e c a u s e o f i ts c o u n t e r p a r t w h e n i d e a l- g a s t e m p e r a t u r e i s u s ed , b u t t h e

d i m e n s i o n s o f t h e q u a n t i t i e s o c c u r r i n g i n t h e t w o a r e n o t t h e s a m e . A s t h e a b s o l u t e

t e m p e r a t u r e T is d i m e n s i o n l e s s , R h a s t h e d i m e n s i o n s o f w o r k , a n d K v , r h a s t h e

d i m e n s i o n s o f h e a t. F o r e a c h c h o i c e o f t h e a r b i t r a r y c o n s t a n t J w e o b t a i n a n

a b s o l u t e t e m p e r a t u r e T t h r o u g h t h e d e f i n i t io n (3 0), s o t h e d e f i n i t io n (5 8) o f " id e a l

g a s " d e p e n d s u p o n J . A s u b s t a n c e t h a t i s a n i d e a l g as f o r o n e c h o i c e o f J is n o t a n

i d e a l g as f o r a n o t h e r . F o r a n y c h o i c e o f J w e se e t h a t (5 9) h o l d s . T h u s w i t h a n y

cho i ce o f 9" t he t w o spec i f c hea t s a r e d i s t i nc t ; a l l o f t he V -T q u a d r a n t t h a t

c o r r e s p o n d s t o ~ h is n o r m a l ; a n d b o t h s p ec if ic h e a t s a re c o n s t a n t i f a n d o n l y if

o n e o f th e m is . T h e r u l e o f t r a n s f o r m a t i o n ( 2 5 ), w h e n a p p l i e d t o ( 30 ) f o r d i f f e re n t

cho i ces J l a nd 9'2 o f J show s t h a t

9 '2 T ~ K ( 6 0 )Kv , ra = dl T2 V.T,.

T h e e x a m p l e o f th e C a l o r i c T h e o r y o f h e a t s h o w s t h a t t h e r e n e e d b e n o c h o i c e o f 9'

s u c h a s to m a k e K p , T a n d K v , r c o n s t a n t ( C o n c e p t s and Logic, H i s t o r i c a l S c h o l i o n

a f t e r D e f i n i t i o n 1 3 i n C h a p t e r 6 ) .

W e a r e n o w r e a d y t o l ay d o w n a f in a l a x io m s i m il a r in f o r m t o A x i o m V o f

Concepts and Logic. b u t c o n c e p t u a l l y q u i t e d i f f e r e n t :

A x i o m V . There is a J such tha t fo r one o f the corresponding bodies o f ideal gas

K v , r = cons t .

T o s u c h a g as , w h i c h A x i o m V m a k e s c o m p a t i b l e w i t h t h e g e n e ra l t h e o r y , w e m a y

a p p l y t h e r e a s o n i n g t h a t l e a ds t o T h e o r e m 1 5 in C h a p t e r 1 5 o f Concepts and Logic.

H e n c e t h e f u n c t i o n s g r a n d h r o c c u r r i n g i n ( 34 ) 2 h a v e t h e f o r m s

g r = 9"h r + cons t . , ( 61 )

h r = M T ,

a n d (3 7) 2 r e d u c e s t o ( 4 5 ) : , w h i c h is t h e l o c a l " S e c o n d L a w " o f c la s s ic a l

t h e r m o d y n a m i c s . T h e o t h e r c l a s si c al c o n s t i t u t i v e r e s t ri c t io n , n a m e l y (3 2)2 , w e h a v e

a l r e a d y f o r a l l c h o i c e s o f 9", s o i t h o l d s f o r t h e c h o i c e p r o v i d e d b y A x i o m V .

Thus the entire form al structure o f classical thermodynamics results. I n p a r t i c u l a r ,

o n e d e m o n s t r a t e d c o n s e q u e n c e o f t h e a x i o m s h e r e is t h e " F i r s t L a w " i n

C L A U S I U S ' f o r m ( 4 0 ) , a n d t h e c o n s t a n t J t h a t A x i o m V p r o v i d e s t u r n s o u t t o b e

t h e u n i f o r m a n d u n i v e r s a l m e c h a n i c a l e q u i v a l e n t o f a u n i t o f h e a t i n c y c li c

p r o c e s s e s .

T h e T h e r m o m e t r i c A x i o m o f § 8 h a s e n a b l e d u s t o c o n s t r u c t K E LV IN 'S s e c o n da b s o l u t e - t e m p e r a t u r e s c a l e u p o n t h e g e n e r a l f r a m e w o r k o f C A R NO T 'S i d ea s . U s i n g

it, w e h a v e s h o w n t h a t t h o s e i d e a s , d u l y m o d i f i e d b y A x i o m V , su f fi c e t o c o n s t r u c t

c l a s s i c a l t h e r m o d y n a m i c s witho ut use o f ideal-gas temperature, the " 'First Law ",

o r th e " S eco n d L a w " .



380 C, TRUESDELL

Acknowledgment. I am g rateful no t o nly to Mr. SERRIN bu t also to Messrs. C .-S. ~VIAN

and M. PITTERI or long and p atient discussion of the subject and for scrupu lous cri t icism of

this note . T he wo rk i t presents was par t ia l ly sup por ted by grants f rom the U.S. Na t ional

Science F oun dat ion ' s Program s in the H is tory and Phi losoph y of Science , in Appl iedMathematics, and in Solid Mechanics.

The Johns Hopk ins Un ive r s i t yBa l timore , M ary land

(Received Decem ber I5, 1978 )

N o t e a d d e d in p ro o fi T h e d e m o n s t r a t i o n s o f t h e t h e o r e m s in Concep t s andLog ic o n w h i c h t h i s p a p e r d r a w s d o n o t r e q u i r e t h a t OKv/~.O exis t ; (25) ,_ sh ow s

t h a t i t s e x i s te n c e is n o t i n v a r i a n t u n d e r a ll c h a n g e s o f e m p i r i c a l - t e m p e r a t u r e

s c a l e . A c c o r d i n g l y , i n t h i s p a p e r I h a v e s u p p o s e d o f K v o n l y t h a t i t b e

c o n t i n u o u s a n d t h a t ~ K v / g V e x i s t a n d b e c o n t i n u o u s .

absolute temperatures as a consequence

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