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MECHSH SHTE 531E1FEESETY
Jerry D. Griffith

1957

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A GENEML ANALYSIS OF THE FIRST AND SECOi-w

IJLWS OF ’ITELK-IQDYIW-JCS

By

Jerry D. Griffith

AN'ABSTRACT

Submitted to the College of Engineering of Michigan
State University of.Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1957 . 1
13.,

ABSTRACT

Thermodynamics is that branch of science which treats of energy
transformations. As such it is a very general science. The generality
of the First Law offiPhermodynamics is well known. Energy must be conserved.
Even the student of a high school physics course is familiar with this law.
The generality of the Second Law of Thermodynamics, however, is lost in
the obscurity of idealized thermal machines and heat power cycles.

It is unfortunate that the generality of the Second Law should be
lost by such close association with only heat energy transformations. The
Second Law is applicable in all systems Just as is the First Law. The
Second Law must hold for all processes from the irreversible flow of water
down a hill, to the smallest chemical reaction that takes place inside
animal and plant life. If the Second Law is to be placed on a level with
the First Law, the obscurity with which the Second Law is clothed must be
eliminated. This can only be done if the association of the Second Law
with heat engine cycles is removed. This can be done.

The First Law for any system may be written as a Pfaffian equation.
By utilizing the properties of Pfaffian equations and the directional
jprinciples that are observed in natural processes, the Second.Iaw may be
(ierived strictly from mathematical considerations of the First Law. This
is what has been done in this thesis.

The procedure described above has been carried out for adiabatic
systems of constant composition, systems of constant composition in which
no work is done, chemical systems of varying composition, electrical

systems, and mechanical systems.

The analysis of these various systems shows conclusively that the
entropy cannot decrease in an isolated system. Thus the directional.
characteristics of processes is found to be the same as predicted by
classical thermodynamic methods, but it has been found without the
introduction of thermal machines. Thus the Second Law is directly
applicable to any process without first reducing the process to an equiv-
alent heat engine cycle. This removes the association of the Second Law

with heat energy transfers and the generality of the Second Law is increased.

A GENERAL ANALYSIS CF THE FIRST AND SECOND

LAWS OF THERMODYNAMICS

BY
Jerry D. Griffith

A THESIS

Submitted to the College of Engineering of Michigan
State University of Agriculture and.Applied Science
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1957

u?)

ACKNOHIEDGMENTS

The author wishes to express his sincere appreciation
to Professor Ralph M..Rotty whose guidance and encouragement
during the preparation of this thesis added to the quality
of the thesis and the author's understanding of the science

of thermodynamics.

 

fi—fl

' ' "l- I'! PIP-”i—n—n_——.— _. _-_ __

II.

III .

IV.

VI.

VII.

VIII .

TABLE OF CONTENTS

DWTRODUCTICIQ . O O O O O I O O C O 0

PROPERTIES OF PFAFFIAN EQUATIONS. . . . . . .

\

MATHEMATICAL ANALYSIS OF THE FIRST LAW FOR CONSTANT
CCMPOSITION ADIABATIC SYSTEMS. . . . . . . .

MATHEMATICAL ANALYSIS OF THE FIRST LAW FOR CONSTANT
COMPOSITION SYSTEMS IN WHICH NO WORK IS DONE. . .

MATHEMATICAL ANALYSIS OF 'HIE FIRST LAW FOR SYSTEMS
OF VARYING COMPOSITION. . . . . . . . . .

MA‘EEMATICAL ANALYSIS OF THE FIRST LAN FOR ISOLATED
ELECTRICALSYS'JEIVB........ ...

MATHEMATICAL ANALYSIS OF THE FIRST IAN FOR ISOIATED
WCMCAL 8mm 0 O O O O O O O O O 0

CONCLUSION.............

HST OF‘ WNCES O O O O O O O O O O O

21

32

57
67

71

 

.;.-o—

 

INTRODUCTION

Thermodynamics is that branch of science which treats of the laws
which govern transformations of energy. It is based on two general
laws of nature, the first and second laws of thermodynamics.

The First Law of Thermodynamics may be applied to any system. It
states that when any change of state is accomplished, a certain energy
balance must hold. This energy balance may be written as follows:

dQ= dE+dw
Where dQ is the heat transferred to the system during the process, d" is
the work transferred fran the system during the process, and dB is the 1
change in stored energy of the system during the process.

The First Law written in this form is nothing more than a statement
of the law of conservation of energy. Conformance of a given hypothetical
process to the law of conservation of energy, however, is no guarantee
that this process is possible. There are certain directional laws which
have been observed in the physical world. For example, when bodies at
different temperatures are placed in contact, the temperatures of the
bodies always approach a common value; water always runs down hill;
people always grow old; the reverse of these processes is never observed.
It would be possible to reverse many of these processes and not violate the
law of conservation of energy. Thus conformance to the law of conservation
of energy is a necessary but not a sufficient condition to Justify the
possibility of a change of state.

In thermodynamics the fact that a directional law exists results in

a limitation on a change of state other than that imposed by the First Law.

"Inf,"—

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Because the First Law does not completely express the possibility of a
change of state, the Second Law of Thermodynamics has been formulated to
express the directional limitations imposed upon the First Law. The

steam engine had already been invented and.was in Operation at the time

of the first realization of this directional limitation on the First Law.

In truth, it was the conception of idealized thermal machines which
transform heat into work and vice versa or which pump heat from one
resevoir into another that gave the first indications that these directional
limitations existed. These limitations were that heat could not be
transformed completely into work or raised from a lower to a higher state

of temperature without compensation. It was only natural that the

engineer use these same idealized thermal machines that indicated the
existence of a directional limitation to deve10p the Second.1aw, and the
classical development of the Second Law proceeded along these lines. It
was, however, inevitable that someone would object to the introduction of
idealized thermal machines to establish the Second.law, and these Objections
are not without foundation. It would seem much more desirable to establish
the Second Law with sound.mathematics thus making the law directly appli-
cable to a general system rather than simply thermal machines.

A mathematical deve10pment of the Second.law'has been made possible
by the principle of Carathéodory. Carathéodory analysed this problem and
showed that it was sufficient to know the existence of some impossible
processes to derive the Second Law. It must be shown mathematically that
there are states in the vicinity of any given state which are inaccessible
because of the restrictions placed upon the change of state. This problem

-can be solved by the differential equations studied by Pfaff.

PROPERTIES OF PFAFFIAN EQUATIONS

Pfaffian equations are expressions of various functions in space.
There are two types of Pfaffian equations. One type of Pfaffian equation
results in an infinite number of solutions ¢(x,y,z) = constant. This
describes parametrically an infinite number of surfaces in space. Other
types of Pfaffian equations do not have this property. There are physical
examples which describe space distributions of this type. Consider the
space distribution which would result if 8 x 11 sheets of paper were
placed in an 8 x 11 box. Compare this space distribution to the distri-
bution which would result if 1 x 1 squares of paper were placed at random
in the same box. These two space distributions are analogous to the
space distributions of two types of Pfaffian equations. The 8 x 11 sheets
of paper form surfaces in space. The l x 1 squares do not.

Pfaffian equations containing solutions ¢(x,y,z) = constant are
mathematical expressions of some of the most common physical processes.
It will be shown later that the First Law oftrhermodynamics may be written
in the form of a Pfaffian equation. Then from considerations of the
pr0perties of Pfaffian equations and the use of Caratheodory's principle,
which will be stated later, it is possible to derive the Second Law. A
knowledge of the properties of Pfaffian equations is necessary'before‘
this procedure can be followed.

Consider first a Pfaffian equation of two variables, x and y.

as = de+rc1y (1)

where X and Y'are each functions of both x and y.

If the restriction that A be constant is placed on this equation,

the ordinary differential equation

4.2:-3.
dx 1 (2)

is obtained.

Equation (2) may'be integrated and y may be solved for in terms of
x or ‘vice versa. Thus if A is constant only one of the variables may be
independent. 'me differential equation (2) has an infinite number of
solutions O(x,y) equals a constant, representing a set of curves defined
implicitly by 0 in the x-y plane. Now, since the restriction that A be
constant has been placed on any change of position in the x—y plane, all
those points which do not lie on the curve A equals a constant are not
accessible from a point which does lie on this curve. What happens if
this restriction is removed? How is the solution of this equation

affected by a change in A? It can be shown that if “19?) equals a

constant, then

d0 = get” 1%,ng o . (3)
This equation must satisfy the same conditions expressed by the original
Pfaffian equation that
dA= de+Ydy : 0

Thus if the two equations are to be equal
.Y

ax ’
or
.. 19 a
dA-xaxdx+x53dy
dA=id¢ (h)

It has been shown then that each Pfaffian equation of two variables has

an integrating denominator such that Qf is a perfect differential.

Examine a Pfaffian equation of three variables,

dA‘XdX+Ydy+Zdz (5)

where X, Y, and.Z are all functions of both x and y. For.A equal to a
constant, equation (5) defines surfaces in X-yhz space. 2 is not an

independent variable and.may be expressed as follows.

z=z(x,y)
_d_z_ dz 6
dZ-dxdx+a-§dy ( )

Equation (5) may then be eXpressed as

dA= [x+ 2%de +[r+z§-§;]dy
or

ds=x’dx+r’dy 1 (7)
where X’and I’are now each functions of only x and y.

I

Thus equation (5) reduces to the case previously studied and it can
be shown that an integrating denominatorfi. , exists such that «6er is a
perfect differential d¢. Therefore, Pfaffian equations of three variables,
only two of which are independent, have this integrating denominator and
dA3‘1dT- These Pfaffian equations define surfaces in space. The
resulting Pfaffian equation of two variables which defines a curve in
the x-y plane is the curve of intersection of the surface in space and
a plane of constant z.

For Pfaffian equations of three or more independent variables this
integrating denominator does not exist. Pfaffian equations of this type
do not form surfaces in space. That these Pfaffians do not form surfaces
in space is easily demonstrated by choosing x,y, and 2 as the three
independent variables in x-y-z space. It is Obvious that once x and y
are chosen ¢(x,y,z) cannot describe a surface in space if z is allowed

'to take on any arbitrary value. That these Pfaffians do not have an

integrating denominator may be shown mathematically.

Consider the Pfaffian equation

dA= -ydx +Xdy +Kdz

(8)
where k is a constant. If it were possible to express GA in the form
Add ,

where A and ¢ are functions of x, y, and z, the following
equation would result-

= 9.9 5.9! 5
(M >13de “Maya—x5342

(9)
By equating coefficients in equations (8) and (9) it may be shown that
23“: iii—as Ail

A, by A 32-}:

Taking partial derivitives of the above equalities gives
fl=2_)-.é(i)
byaz dz A by).

(10)
62 a 186 K
oxozg’a‘é (x1 5:1(1') (11)
fizz-g. I -S. E
dxby ay(>.1 ex (A) (12)
Equation (10) shows that
3225(X)1=Kb%('>?) (13)
but dfi: -—2 5:. Hence xg-z = Kg“; 1
In a similar manner equations (11) and (12) give respectively
y_3__A_. -KQA
yoz bx
(1h)
2X= x3; --+yg—:‘,

(15)

By substituting % and 93‘— from equations (13) and (it) into equation (15)

but

it is found that A= O . Thus Pfaffian equations of three independent
variables do not have an integrating denominator such that dA=Ad¢.

This knowledge of Pfaffian equations gives an insight into what
needs to be done in any physical application of a Pfaffian equation. If
a quantity can be defined by'a Pfaffian equation of two independent vari-
ables, then any differential change in this quantity may be expressed as
Add» It remains then only to express Ado in terms of known, or

if need be, new prOperties.

MA'HIEMATICAL ANALYSIS OF 'EE FIRST LAW
FOR CONSTANT COMPOSITION ADIABATIC SYSTEMS

Since the Second Law of Thermodynamics places limitations on the
First law, it would seem logical to derive the Second Law from
mathematical considerations of the First Law. Caratheodory's showed that
it was sufficient to know of the existence of some impossible processes
to derive the second Law. Moreover, the impossible processes are readily I
obtainable by examining Joule's experiments. They consist of bringing a
system in an adiabatic enclosure from one equilibrium state to another by
doing external work. It is an elementary eXperience that the work cannot
be regained by reversing the process. It can therefore be inferred that
there exist adiabatically inaccessible states in the vicinity of a given
state. This is Caratheodory's principle.

The obvious step now is to construct an adiabatic system in which
Q may be expressed as a Pfaffian equation of two independent variables.
To do this it is necessary to consider a system in which complete control
of the variables of the system is possible. This system must also be
isolated from all external effects thereby eliminating any changes of
state in the system other than those produced by controlled manipulation
of the variables of the system.

To maintain control over all the variables of a system it is
necessary to eliminate the use of rigid bodies and incompressible fluids
in the system. This leads to the selection of gaseous fluids as the
working substance in the desired system.

To isolate this system it would be possible to enclose the gaseous
fluids in a container such as a thermos flask and thereby eliminate any

external effect on the fluid to a high degree of approximation.

The use of such containers, however, eliminates the desired control
over the variables of the system. A new kind of enclosure must be
found, and it is necessary to introduce the idea of walls. These walls
are to be so thin that they play no part in the physical behavior of
the system other than to define the interactionlbetween two neighboring
fluids. These walls may also be moved at will to effect changes in the
pressure and volume of the fluid. It is necessary to define two types
of walls.

An adiabatic wall is defined by the property that equilibrium of
a fluid enclosed by it is not disturbed by any external process as long
as no part of the wall is moved. This type of wall isolates the system
but does not limit the desired control over the variables of the system.

It will be necessary later to allow for the transfer of heat energy
from one fluid to another. This cannot be done across an adiabatic wall.
Therefore, it is necessary to define another type of wall. This second
type of wall is the diathermanous wall, defined by the following
prOperty: fluids separated by a diathermanous wall are in equilibrium
only if the temperature of the two fluids is the same. file: equation
of state of a perfect gas then yields the desired relationship between
the pressures and volumes of these two fluids. The equation of state
of a perfect gas states that:

'7". = ”I R17:

gv2=n2RzTa
where P, V, n, R, and T have their usual meaning. Now if Tl equals T2

and both fluids are composed of the same amount of the same substance then

Plvl = PZVP- ‘ (16)

10

This is the eXpression of thermal contact. The wall is introduced only
to symbolize the impossibility of exchange of material.
It is now possible to apply the properties of Pfaffian equations
along with Caratheodory's principle to derive the Second Law of
Thermodynamics. Consider two equal masses of the same fluid enclosed by

adiabatic walls and separated by a diathermanous wall.

 

Fluid 1 Fluid 2

MlP 1T MPQT

 

 

 

Diathermanous wall

Adiabatic wall

Figure 1.

Either of these two fluids constitute a system and the First law
may be applied.

dQl = dE1+ dwl

ng = d32+dw2 (17)
If movement of the adiabatic walls is restricted to extremely slow
”quasi-static” changes the work done on the fluid enclosed.by the
adiabatic walls is

aw = PdV (18)

11

Thus

dQ, = dE, +P, dV,

an =dsa+r,dv2 (19)

These relationships may be stated in a more convenient form to
describe Pfaffian equations of two independent variables in a temperature-
volume plane.

Since the internal energy is a function of any two independent

variables, v and T may be selected and dE may be expressed as follows:

de-(g—E)T dV+ g—E-LT

Equations (19) then take the following form:

as
dQ.=(5L1§)V:1T.+[3—v‘ +q]cv,

65‘ '
4Q {352) T+ —-¢ p a
z 37“!“ av, 5+ 8 .W (20)
Now the total heat. energy added to the combined system must be
the sum of the heat energies added to the two fluids, and this total
heat energy must be zero since the combined system is entirely surrounded

by adiabatic walls. Therefore,

dQ 3 dQ,+dQ2= 0

do {(1, a.) + {lam + [cf—522%... 5]“, + [(g—E'kfi-‘figano (21)

In equation (21) , T1 and T2 no longer need be distinguished since the
diathemanous wall between the two fluids makes T1= T2= '1'.

Equation (21) describes a Pfaffian equation of two independent
variables in Vl-Ve-T space. This may not be immediately obvious, but

consider what happens if arbitrary values are assigned to any two of

the variables of the two fluids, say V1 and T1. The equation of state

of a perfect gas establishes P1. T2 equals T1, and the equation of

state also establishes the product P2 V2. Since the system is adiabatic
these two properties cannot vary independently in the second fluid

(i.e., if P2 and V2 can only be varied by manipulation of the adiabatic
walls, then a definite value of P2 corresponds to a value of V2). Thus
only one value of V2 and a corresponding value of P2 can possibly maintain
the temperature equilibrium required by the diathermanous wall and
thermal isolation required‘by the adiabatic walls, and V2 is not an
independent variable.

Since equation (21) is a Pfaffian equation of two independent
variables, it should be possible to construct surfaces in Vl-Vé-T space
for dQ = O.

The shape of these surfaces may be determined.by examining the
lines of intersection of the adiabatic surfaces and the Vl-Vé, Vi-Tg
and VQ-T planes.

If V2 is held constant and V1 is increased, T will decrease. .As
V2 approaches infinity, T approaches zero. .As Vé approaches zero, T
approaches infinity.

Similarly for V1 and T. As Vl approaches infinity, T approaches
zero. .As V1 approaches zero, Trapproaches infinity.

It is then Obvious that for a constant temperature plane V1 approaches
zero as V2 approaches infinity and V1 approaches infinity as V2 approaches
zero. Thus these surfaces vanish to infinity in the vicinity of the
Vl-Vé, Vl-T, and V2-T planes, but it is possible to construct these
surfaces for points not in the vicinity of these three planes. These

surfaces have the shape shown in Figure 2.

1-3

 

 

 

Figure 2.

Adiabatic Surface in Vl-V2-T Space

1h

Since dQl, dQe, and dQ have been expressed as Pfaffian equations
of two independent variables, it is possible to express each of these
quantities in terms of m.

dQ1= £16.01

dQe= we

as: M0
But

dQ = dQ1+ dQQ == 0
Hence

xd¢=xld¢i+>~2d¢2= 0 (22)

Equation (22) can hold only if 0 is a function of ¢l and $2.

Therefore,

dO= 3—5 40. +- 3O ”51¢:

Equating'coefficients¢ of the exact differential dO gives:

22... _)\. LT X3
()9,- )\ ’ a¢e=x

Now 0 is an independent property of the combined system and 01 is
an independent property of the first fluid. Since ’3’). is the partial
derivitive of ¢ with respect to O1, and since 9 is a function only of 01
and ¢2,€5xlis dependent of the other pr0perties of the system and cannot
therefore be a function of the temperature, from which

.—$(%')=o . 39ft?) =° <23)
Carrying out the differentiation of equation (23), the following results

are Obtained:

1 a». _ 1 __1_3x
X1354" 55.2%“Xfi (2h)

15

Now )1 is a variable of the first fluid only and is therefore only
dependent on the properties of that fluid. Likewise ’52 is a function
only of the properties of the second fluid. The equality of equation (21:)

holds only if both J- é-él and -'—- 9A3 depend only on T. Hence

A. 3T )2 31'
u a). __ l Ax _ l a)...
xfi-zfi-T‘fi-‘m (25’

It is possible to rewrite equation (25) in the following form:

3 .
7:9,}. = 13119;: gangl‘ =4-‘(1') , (26)

It is now possible to express AdO in terms of known properties of the
system. From equation (26)
w. =/+‘(1')d'r +c (27)
)x has now been expressed as a function of temperature and some
constant of integration, but it is possible to determine the nature of
this constant. Examine a change of state in the A-¢ plane. If dQ=Ad¢,

then the area under the curve represents the heat transferred to or from

the system .

A

 

It is possible to represent constant temperature lines as any
general lines in the )s-Q plane. The actual sloPe and orientation of
the constant temperature lines will prove unimportant . It is already
known that )s is a function of temperature and the above diagram shows
that ). can then be at most a function of one other independent variable.
This variable is arbitrary, but it will prove desirable to use O. Thus
equation (27) may be expressed as a function of T and O. By re-examining
the diagram it is possible to determine what this function of O is. It
is evident that x can be made to vary as a function of 0. Then In x must
be proportional to me“). Therefore equation (27) may be written:
mas-flu)“ + mo)
X?§e/‘ptr)d'r (28)
where §=+‘(¢) .
It is now possible to define a thermodynamic temperature as that

'part of A which is a function of T.

T’(T) see/HT)” (29)

where the constant C may be fixed by prescribing the value of T142 for
two reproducible states of some normal substance (e.g., Tl—TQ = 212°F if
T1 corresponds to the boiling point and T2 the freezing point of water
at one atmosphere of pressure).

A new property, entrapy, may be defined as follows:

s(¢)= é/Idd (30)
This new property is defined such that

dQ=AdO =TdS
Similarly

dQ. emu, =1;d$.

4Q; =Xad¢2=TadSz

 

17

The above equations refer to "quasi-static" processes which may
_be represented as sequences of equilibrium states. Idealhyall heat
transfers can be made by'a succession of equilibrium states, and so
the above results are quite general and refer to all such reversible
heat transfers.

dQR=TdS (31)

To obtain a knowledge of real irreversible phenomena, Carathéodory's
principle must be applied considering a finite transition from an initial
state (v0, so) to a final state (v,s) where v equals v1+v2. It is
possible to reach the latter state in two steps: first changing the
volume "quasi-statically" and adiabatically from V0 to V the entrapy
remaining constant, equal to So, and then changing the state adiabatically
by the addition of external work at constant volume, so that Sogoes over
to S. This second change of state could be made by stirring the fluid
with a paddle wheel. It should be noted here that work must be added to
the system.since it is impossible to extract energy from a system during
a constant volume adiabatic process. The application of Caratheodory's
principle may be more easily understood if these two changes of state are

pictured on temperature-volume diagrams.

 

 

 

Figure 3

18

Figure h represents step one of the two steps discussed previously.
It is obvious that any state (V,So) is accessible from any initial state
(V3, So) since the volumes of the two fluids are arbitrarily changeable
adiabatically by the definition of adiabatic walls. But Carathéodory's
principle states that there are adiabatically inaccessible states in the
vicinity of any given state. Therefore, step two must provide the desired
restriction on the First Law.

Examine step two on a temperature-volume diagram.

 

 

 

Figure 5.

Figure 5 shows that once the state (V,Sa) has been reached from
any initial state by step one, there are two possible relationships
between S and 80, that is Sisoor 8:80. One of these conditions is
then not possible or else all states would be accessible to any given

date by adiabatic processes.

 

 

 

19

The actual choice of sign 3: or 6 depends on the choice of the
constant C in equation (29). If this constant is chosen so that T is
positive then the relationship dQ=TdS will show that the entrOpy never
decreases in an adiabatic system.

This fact may be seen by considering the process in which 3, was
changed to S at constant volume by the addition of paddle wheel work.
There is experienced during this process a rise of temperature in the system.
Thus the initial temperature is raised to some final tailiperature ‘1'. This
temperature T and the volume of the system fix all the other preperties
of the system including the entrapy, S. this value of entropy will be
the same at this given volume and temperature regardless of how this
state is reached. But it is possible to produce this rise in temperature
by the reversible addition of heat to the system at constant volume until
the temperature, T, is reached. Since this heat is added to the system,
it is a positive quantity. T has already been made a positive quantity
by the coice of the constant C in equation (29). 'nderefore, the relation-
ship dQ=TdS shows that as must also be a positive quantity, and it has
been shown that the entropy never decreases in an adiabatic system.

Consider the relationship dQ=TdS in a different way. Since T is
always positive, then the only way the entropy can decrease is to r-ove
heat from a system. Also since the entropy can increase but never decrease
in an adiabatic system, dQé ms for any process. The equality of course
holding in reversible processes such as in the hypothetical system used
to derive the relationship dQ=TdS. Hence for any process“?Q $115. For

any cycle, integrating around the eyelei

I“?
f? 5 (13
or

dQ
f?.so

since entrOpy is a prOperty and has zero change in a cycle. Hence the
inequality of Clausius has also been proven without the use of idealized
thermal machines.

Thus by the introduction of Pfaffian equations and Carathéodory's
principle it is possible to derive the Second Law strictly from

mathematical considerations of the First Law. A develOpment similar to

the one presented here has been presented by Born.1

1. Born, Max, Natural Philosophy 9;; Cause and Chance,
(New York:0xford University Press, 1919), pp. 31-16.

21

MATHEMATICAL.ANALKSIS OF THE FIRST LAW
FOR CONSTANT COMPOSITION SYSTEMS IN WHICH N0 WORK IS_DONE

It is surprising that the use of the prOperties of Pfaffian equations
has ended with the previous deve10pment since the introduction of Pfaffian
principles introduces new areas of thought in other phases of the First
Law. The Second Law is directly concerned with irreversibility. Therefore
it was natural to examine irreversible phenomena to find the restrictions
placed on the First Law. Joule's experiments indicated that there were
adiabatically inaccessible states in the vicinity of any given state. This
phenomenon-ms then used to derive the Second Law. But Joule's experiments
do not uniquely describe irreversibility. There exist other irreversible
phenomena (e.i., a free expansion). It is obvious that if a fluid is
allowed to expand freely, it cannot be returned to its initial state
solely by the addition or removal of heat. werk must be done to compress
the fluid back to its initial volume. Therefore it can be stated that there
exist states in the vicinity of any given state which are inaccessible if
no work is to be done on the system.

From the First Law

dfls=dQ -dE
or if all heat transfers are done reversibly and "quasi-statically"

dH==TdS~dE.

The work, dfl, may be expressed in the form of a Pfaffian equation

of two independent variables.

w: E-(g—gg dS— (3%)54 P (32)

22

where P denotes pressure. If the correct system is selected, it
should be possible to express 6." in terms of Ad) and make a development
similar to the previous one.

To construct the desired system it is necessary to define two new
types of walls. The first type of wall is rigid so that no work can be
done by deforming the walls, and it is conducting so that heat energy
may be added reversibly to the system. The second type of wall requires
that the pressures of the two fluids be the same. Once again the equation
of state for a perfect gas written for the same amount of the same fluid

at the same pressure yields the desired relationship between these variables

11:22 J
T1 T2 (33)

Since there is a possibility that the temperatures of two fluids separated
by this wall will be different, it is also necessary that this wall be
adiabatic (non-conducting, non-absorbing).

Now examine two fluids enclosed by the first type of wall and

separated by the second type of wall.

' III/II/III/II/II/I /IIIIIII//////I///I////f

 

 

 

 

I
I
Low temperature é :MlTlP MQTEP 3 Low temperature
sink g ; sink
.1 . a f .
a '- 'z
5 9
9 Q1 Q1 5
’ I
fil/I/I/I/I l/l/l/II/I // ////////. 7/////////’/

 

 

 

 

I ' I ' . Pressure
H g emperature ou equilibrium

Rigid membrane

Walls

Figure 6.

Either fluid constitutes a system and the First Law may be applied

in the form'of equation (32)

=[ (tins.- (:2?

These are Pfaffi(an equations of two3 3independent variables.
“'1 3 E191
dW2=hzd¢2

The work of the combined system must be equal to the work of the two

(3h)

Thus

separate systems, and this sum must be equal to zero since the combined

system is entirely surrounded by rigid walls. Hence

dwsdwl‘: due: 0

”'- [“‘(§I)p]d5u +[Te’(§§:)p]‘s‘ " [(3§'(31-( 3' 1]de (35’
In equation (35) P1 and P2 need no longer be distinguished since the wall
between the two fluids makes P1: P2=P.

Equation (35) describes a Pfaffian equation of two independent
variab les in 81-82-P Space. That there are only two independent variables

in this combined system may be demonstrated by prescribing any two of the

Variab leg of this system, say P and 81. Then all other properties of

fluid one are fixed including V1. 'lhis fixes V2 also since V2 is equal to
the total volume of the combined system minus V1, and the total volume of
the combined system is a constant. P2 13 also fixed by P1 and the wall
separating the two fluids. Therefore, all the properties of fluid. two are
fixed, and 82 cannot be an independent variable.

since equation (35) is a Pfaffian equation of two independent variables

it
Should be possible to construct surfaces in P-Sl-Se for space dH 3 O.

The
se Surfaces take the form shown in Figure 7°

 

. Fv\F.

2h

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 7.

Surfaces of no work in 31-82-P Space

25

Note that the variation of 51 with respect to 82 in a constant
pressure plane may be found by examining the change of 31 with respect

to $2 at constant pressure.

as C
(Q) ___ (star. __3 tr“ ...
“$2 " (Shea tare W

Since constant pressure may be maintained only-if heat is added
to one system when it is removed from the other and vice versa, dTl and
T
dTé must have Opposite signs and the quantity'§§%%% is always a negative

quantity. thus in any P: constant plane the curve of intersection of

the surface of no work and this plane has a slape %§%%%’and this is always
negative. [At a point in a constant pressure plane where S1 is large and
82 is small %§ will be small since heat must be removed from system 2 and
added to system 1 to accomplish this state. When 81 is small and $2 is
large %§ will be large by similar reasoning. This determines the slopes
at these points.

To find the slope of the curve of intersection of the surface of no
work and a constant entropy plane, examine the change of $1 with respect
to P when 82 constant. Consider the case when V2 is small enough to be
negligible compared to V1. Then V1 will approximately equal the total
volume of the system which is constant. .As heat is added to this system

dS will increase and may be given by

. 41’ (iv
dSV - CV :r-V +%_VV
Where 3 is the coefficient of cubical expansion $6.9,” K is the
coefficient of compressibility— $6: Tend CP is the specific heat at
I

constant volume.

Since dV is zero andd—t= dP

-—"v 'r P
a
dsv=c

(23:9)... ”

The slope of the curve of intersection of the surface of no work and

—at constant volume for a perfect gas

the constant entropy planes decreases with increasing pressure. The surface
may now be constructed and takes the form of Figure 7.

Since equation (35) is a Pfaffian equation of two independent
variables it is also possible to express d“ in terms of )tdO.

dw=xd0 (36)
But

aw: owl: dw2 = 0
Thus

M0 ”pm-rang: 0 (37)
Equation (37) can hold Only if O is a function of ¢l and (32. Therefore,

(MW-3104“ ‘1'“30 d¢2

m2

Equating coefficients of the exact differential d0 gives

5 (..XI 3 - X2

6%;- ‘x, 3%; x

0 is an independent pr0perty of the combined system and 01 is an
independent prOperty of the first fluid. Since % is the partial derivitive
0f 0 with respect to 61, and since 0 is a function only of 01 and 02, 7‘”

cannot be a function of any other property of the system and is therefore

“‘4

independent of the pressure,P. 'me partial derivitives of AJIA and

with respect to P must then be equal to zero.

a x. b 5:
6P 3: " 0, a? " 0
BY cfiu‘li‘ying out the differentiation the following equality is obtained:

a». tax .3).
13.4: “Kat—PL- T375 (38)

27

This equality can hold only if—'- Q34 and -!- 3%; depend only on P.
2

A. as A
Therefore,
1.3" -13_"2.J.3’*-
~35" xzap ' 2.33"”) (39)
Equation(39) may be rewritten as follows:
951%.)! = égéaJ—slg" = m) (1+0)

It is now possible to express AdO in terms of known prOperties.

lax: f(P)dP+C

This constant can be shown to be a function of at most one other
independent variable just as was done in the previous develOpment. 'Jhis
variable is arbitrary but selecting Q will give the most desirable results.
Then

in). = firm? + my
where §= f0”.
It is now possible to define a thermodynamic pressure as that part of )k
which is a function of P.

P’m = comma? (to)
The value of C may be fixed by prescribing the value of Pg—Pl for two
reproducible states of some normal substance.

By defining another prOperty as follows:

V(¢)=5'- M (1+2)
it is seen that

dW=AdO=PdV= 0
Similarly

531: Ndolzfidvl

dug = Adez : [’2de

28

These results are valid for all reversible changes of state and in general
awR= PdV (1+3)

Tb Obtain knowledge of real phenomena it is necessary to apply a
unidirectional principle similar to that of‘Caratheodory's. 'lhis
principle is the previously mentioned fact that there exist states in the
vicinity of any given state which are inaccessible to the given state if
no work is to be done on or by the system.

Examine a finite transition from an initial state (To, v“) to a
final state (m). It is possible to reach the final state in two steps:
first changing the temperature of the system by adding heat to or removing
heat from the system reversibly, the total volume of the system remaining
constant equal to V5 and then changing the state with no work being done
by allowing a free expansion from We to V, the temperature of the system
remaining constant. The first step of this transition may be pictured on

a temperature-volume diagram as follows:

T W? V.)

 

a (To y.)

 

 

Figure 8.

29

Figure 8 shows that from any initial state (T,, V, ), any other
state having the same volume, V, , is accessible by the controlled addition
or removal of heat to or from the system. But there are states which are
inaccessible in the vicinity of any given state. merefore step two must
provide the restriction on the First Law.

Examine step two on a temperature-volume diagram.

T (T91) (mt) (13V)

 

 

V

Figure 9 .

Figure 9 shows that once the state (T,V,) is reached by step one,
there are two possible relationships between Va and V; that is Vaiv or
mg V. One of these conditions is not possible or all states would be
accessible to an initial state with no work being done on the system.

The actual sign a oré depends on,the choice of the constant C in equation
(1&1). If this constant is chosen so that P is positive, then the relation-
ship aw = PdV may be used to show which sign holds Just as was done with

5Q ="- 'l‘dS in the previous deve10pment.

30

In systems in which no work is done the only volume change possible
is a volume change done against no external forces (i.e., a free expansion).

Consider the process in which V, was changed to V by allowing a free
expansion. There is experienced during this process a decrease of
pressure in the system. Thus the initial pressure is decreased to some
final pressure P. This pressure, P, and the temperature of the system
fix all the other preperties of the system, including the volume, V, and
the entrOpy, S. his value of volume will be the same at the given
entr0py and pressure regardless of how this state is reached. It is
possible to reach this state by adding heat to the system reversibly at
constant volume until the value of entropy, S, is reached, and then
expanding reversibly at constant entropy (thus doing work and decreasing
the pressure) until the state (P,S) is reached. This work is a positive
quantity. P has already been made a positive quantity by the selection
of c in equation (hi). Thus the relationship dw=1>dv shows that the
volume change must also be a positive quantity. 'lhus V0.4: V and the
volume can never decrease in an isolated system.

Consider the relationship dw=PdV in the same manner as was done for
dq=TdS. Since P is always positive, the only way the volume can decrease
is to add work to the system in the form of PdV work. Also since the
volume can increase but never decrease in an isolated system, dVéPdV
for any process. The equality of course holding in reversible processes.

Hence, for any process flédv. For any cycle, integrating around the cycle

P
d"
or

since volume is a state prOperty and has zero change in a cycle.

31

The above equation is more often stated that the work of a cycle is less
than or equal to the cyclic integral of PdV, but it is interesting to
note the ana10gous form that this equation takes to the previously derived

equation

ng-‘leo

The above results indicated nothing new about systems in general.
However, it is Observed that application of the preperties of Pfaffian
equations when applied to one term of the First Law gives different results
than the same principles when they are applied to another term. The one
result in no way contradicts the other since the relationship between the
properties of a system,

Tdsl==dE‘+-PdV
results in the following relationship between entropy and volume:

B
82-51: Cvln %+E (Va-V1)

This relationship shows that when volume increases the entropy increases
also. However, an increase in entropy is possible without a corresponding
increase in volume. Thus the increase in entropy is a more general
criterion for irreversibility than the increase in volume, and the state-
ment that the entropy change must be a: O in any isolated system is

not violated.

32

MATHEMATICAL ANALYSIS OF {II-IE FIRST IAN IN SYS'EMS
OF VARYING COMPOSITION WHERE NO WORK IS DONE

Until now only systems involving constant quantities of mass have
been considered. There exist, however, systems in which the mass is not
constant. It might be possible to examine these systems in the same
manner as has been done in the previous systems. It was shown in the
previous developments where all the masses were constant

Tds=dE+PdV
or

dE =TdS-PdV. (uh)

In these systems the internal energy may be expressed as a
function of S and V. In a system where the masses of various substances
are not constant, E may be expressed as a function of S, V, and the

various masses, m; :

E = f(S,V,m]_,ne, """ mn)

Hence for small changes of entrOpy, volume, and composition:
_, E 5E1 3E1
at) (—) —) u
S vaime'" + OV 59ml:m£°" + 3m, V,S,m2--- ( 5)

By comparing coefficients in equations (ML) and (1+5), it is seen that

(a)

bs v)",nM£"' (’46)

(23-)

5V S’m',mZ ""- (h?)

as)
If (— i f' :
Om, S,V,II s de mad as follows

(‘3) '
$1 S,V,m2--- p1
then equation (#5) takes the form

as = TdS-Pdv + 1%. ”151.1 (t8)

 

—
33

The p in the above equation is called the chemical potential. Its
similarity to the potentials temperature and pressure in equation (#8)
is obvious.

Equation (h8) is then the general relationship between properties
of a system of varying composition. Consider a system of only one kind
of mass. Iffldm for this mass is denoted by dM, then equation (1+8)
takes the form

dM =63 + PdV-TdS

dM=l:@—5)s + P] dV +[(g—§)V-T] d5 (1,9)

It will later be desirable to construct surfaces of constant M.in

OI‘

T-Ver space. ‘mese surfaces may be constructed by examining the variation
of each of the properties with respect to another when the third is held
constant. The above equations do not indicate the variance of one
variable with respect to another in T-V space. Therefore, another
relationship between the variables must be found. This relationship
exists in the form of free and available energies.

By definition, the work done in a reversible isothermal process is
equal to the decrease of the maximum work function of the system.

dW== -dF
From the First Law, for a reversible isothermal process

-dw = dE-Tds
Thus,

dF==dE-THS.
Since the temperature is constant, it follows that

dF=d(E-TS)
or,

F=E-'rs. (50)

31*-

Thus it is seen that the maximum work function, F, is a prOperty of a
system. Equation (50) is sometimes taken to be the definition of the
maximum work function.

By definition, the work obtainable in a reversible isothermal
process at constant pressure is equal to the decrease in the free
energy of the system. It will be shown that the free energy is also a
prOperty of a system. The free energy of a system must necessarily be
less than the maximum work function of a system since in any process at
constant pressure where there is a volume change, part of the total work
available must be used in changing the volume at constant pressure. That
part of the total available work which may be utilized for other purposes
is the free energy, G.

From the definition of free energy

dHfi;P:= -dG.
But

dwnP-awf- dwv
where dwv denotes the work done in changing the volume at constant
pressure. Therefore,

dG=dF+PdV
or since pressure is constant

as = d(F+PV)

as .-.-. a(s-rs+PV)

G s E-1'5+Pv. (51)

Equation (51) is the definition of free energy. This free energy
is defined such that the work for a constant temperature, constant pressure
process is-GG. However, equation (51) shows that G is a property of the
system, and as such it refers not only to constant temperature, constant

pressure processes but all processes in general.

 

__—
35

Then for any general process
as = (dE-TdS+PdV)-SdT+VdP (52)

If once more systems of varying composition are considered where

dE= TdS-PdV + i pidmi
1's!

the expression in parenthesis in equation (52) becomes simply
n
Emmi
Thus
n
dG = -SdT+VdP +2111“; (53)
‘3
Consider equation (53) for a single homogeneous substance.
d6 = -SdT+VdP «mam (51+)
Equation (5h) expresses the desired relationship between the

variables, temperature, pressure, and mass.

(3%).“ z -8 ($313": V ($91, p: I“

if??? = '(g'g)tm'(byr)gm (55)
«73%; = ”(g—Eh?” (glam (56)
672.2%“ 36%).»; = (%§)Em (57)

Equation (56) shows that at constant pressure and mass p varies
inversely and linearly with temperature. Equation (57) shows that at
constant temperature and mass )1 varies directly and linearly with pressure.

This information will be utilized later to construct surfaces of constant

M in T-V-Ve space.

 

36

Consider the following system composed of an initial amount of
gaseous fluid contained inside two volumes which are separated by a

rigid super-conductive semi-permeable membrane.

High temperature source
1 i ll 1
{ Q1 {Q‘i Rigid Super

Control 1/] 2/11,P,T age Conductive heml—
volume #

“2,1351.w Permeable Membrane
Qo
l Q0 :%—-‘ Control '
I ' Volume #2

Low temperature sinks

 

 

 

 

 

 

 

Figure 10.

These two masses of the same fluid are at the same temperature
and pressure and have the same ’1 and therefore also the same Menu
It is possible to add heat reversibly to each of the control volumes
and to move the walls containing the two fluids at will. In this
manner.M may be held constant in the combined system by manipulation
of the temperature and pressure of the two fluids while mass is
allowed to pass from one control volume to the other. For each of

the control volumes,

.... [(2-$:)s.+ P}!v + [(3%) ,-n]ds.

d"2=[(35.)s +Pa] “”2 +[(%§Z)v;'2] “'52

or since dMl and dMé are Pfaffian equations of two independent variables

“‘1" ’9’1
cu“2"‘2‘1‘a

For the combined system,

., (straw. + lea-v.1» +[(:—:;)..—-r.]«s. +

 

 

37

It can be shown that equation (58) is a Pfaffian equation of two
independent variables. If T is fixed in the system by adding or
removing heat and V1 is changed an arbitrary amount only one value of
V2 can possibly maintain a constant M for the system and there can be no
other independent variable. It is possible to construct surfaces in
Vl-Vz-T space by utilizing the information that p varies inversely and
linearly with temperature and directly and linearly with pressure, and
that ’1 must be constant in the system.

M = m

dM=pdm+mdu
Since dm and did are equal to zero dp must equal zero also.

The mass in V1 may be increased by decreasing V2 and increasing V1
at constant temperature. Since the pressure also must remain constant
during this process to maintain p constant, the total volume of the two
control volumes must be constant by the equation of state. Thus V1 varies
inversely and linearly with V2 at constant temperature.

It has already been shown that

(£4913, manegative constant, (gglfim a positive constant.

To determine the slope of the curve of intersection of the surface of
constant M and a constant volume plane, consider the case when V2: constant:
0. The total volume of the system is then V1 and the mass is constant.

The above partials then hold and since ,1 is a function of T and P

at constant mas s ,

due.- (%)PdT + (%)T (IF :0

-AdT+BdP=O

%dedP

 

 

Since for a perfect gas iflp=g dV
dT Lav
0.4T: %__ T , C.=e- and is always a positive constant.
(1.1- __£_
ov" "c,v-R

If 01V>R the slope is negative. If Clv<R the slope will be positive.
When ClV‘=R the s10pe will be infinite. If we consider only those volumes
which are small enough to give a positive lepe, the slape will increase
with increase in temperature and pressure. It is not knomif ClV can

ever be greater than R but C V can definitely be less than R since the

1
volume may be made arbitarily small. Thus the existence of the surfaces
of positive slepe is assured. The surfaces of negative sIOpe might not

exist and are not included in the construction. The surfaces of constant

M then take the form of Figure ll.

 

39

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 16 .

Surfaces of constant M in Vl-Ve-T space

 

#0

Since dM is a Pfaffian equation of two independent variables
our-MO
But
dM.= dM,: dMe==O
Therefore
Xd¢= wagwg =0 (59)
Equation (59) can hold only if ¢ is a function of m and @e. Therefore,

(108 g'%' (1.. + 53‘“:

Equating coefficients of the exact differential d¢ gives

gggél figs—52
u, A ’oo, A

O is a property of the combined system and M and 9: are prOperties
of the amount of the fluid on each side of the membrane. Since 53‘
andxfix are the partial derivitives of Q with respect to 0. and (be they
are independent of the other prOperties of the system and cannot be a

function of the chemical potential, ,1. Therefore,

a x b). _
6)? x‘=° : m”

It is not necessary to distinguish between the chemical potentials
of the two fluids since the semi-permeable membrane maintains pressure
and temperature equilibrium in the entire system thereby making p'zngp

By carrying out the differentiation of the above equation

The equality of the above equation can hold only if both ;— 36—3de
3
_L _3*2
*2 an

Hence,

depend only on p.

 

—~r—f
Ll-l

This equation may be written in the following form:

om). _ blnx _ am-
TE" 7.7"37 "W

from which
lnx=/1°(H) an + m M)

A:§e/flu)dn
where is f“).

The chemical potential may now be defined as that part ofA

which is a function of M-

A») = c e/fli‘W
Then the mass, m, may be defined as follows:

n(¢)=-é-/§d¢
'Jmus,

dM=Ad’ = pd! s 0
Similarly,

d-MF’W‘WO =Fldml

TW‘MF Mad":
and in general for all processes in which ,1 is constant

an: pd-

If a principle existed such as Carathéodory's, to the effect that
there are states in the vicinity of any given state which are inaccessible
if M is to remain constant, then the application of this principle to
define the directional limitation on the First law would be possible.
There exists, however, no such principle, for all states are accessible in

a given control volume simply by adding mass to or removing mass from

the volume .

 

To gain knowledge of the directional limitations imposed on the
First Law in systems where the composition of the system is changing, it
is necessary to examine the irreversible phenomena involved in these
systems. These irreversible phenomena are associated with chemical
interactions between two or moreelements. In isolated systems these
chemical interactions proceed only in one direction.

The introduction of more than one substance in a system, however,
results in the inability to express the First law in the form of a
Pfaffian equation of two independent variables, since the second sub-
stance obviously introduces another independent variable. This problem
may be remedied by lumping the individual pdm's of the system so that
equation (h8) takes the following form:

Izn’pgdn; = dE -TdS + Pdv

.

OI‘

é: mm; = [GB—3V ’7] d5 + “393+ P] «iv (59)

Consider now a system composed initially of hydrogen and oxygen

‘which is allowed to unite chemically to form water.

' Hugodmuzo +(Pozd"°z+l‘ugd'”"z) = [Gav ’1’] d8 +[(§_Vg)3+ P] dv

where dmozand din" are negative quantities. If this chemical reaction is
carried out at constant volume as in a bomb calorimeter
d dm dn - QE- d5
Map "‘uonF'oz oath“,z a) - as V’T (60)
Equation (59) indicates that twopossible relationships exist for
:EJIII.
dill dm dm 2.-
or

”"30 dmflgo + (madlo‘ + ”Hadmu‘) é- o

 

M3

The actual sign E.- oré depends on the quantity
3%)V‘T
Since the volume is constant, this quantity may be rewritten as follows:
dE-TdS. '
By the First Law
dE-dQ: -dW
But dw=0 since the process was carried out at constant volume. Therefore
dE-dQ=0
Since TdStdQ
dE-Tdsso (61)
Thus the quantity dE-TdS is a negative quantity and
Hugo d”n,o+(flo,d"oe+fluzdmfl,) ‘-‘-= 0 (62)
Examine the quantity mogdflsfipugdmue). This represents a summation.
For summations of this. kind in general
n _ n
3 gain; £3,»ng ..é. pEIdni
Where 2 denotes the least andfi'the maximum of all the possible ”1'3
It is then possible to find a ,i'such that
guiding = p'lgdm;
Where
Aép'éfi
This P’represents an equivalent chemical potential of the system for each
total infinitesimal change in mass.
Formdmog-l-Aewt is possible to construct an equivalent chemical
potential, p(Hz,O:), such that when it is multiplied by the total change

in mass, dma‘i- dez , the amount of energy represented will be the same

as that represented by (”ozd"0¢+}‘ugmflz)°

”Ogd"°¢+”uf'“uz =P("z:°z)d'“n,+o,

 

 

an

Since »
Pagodmnp +fl(”z:°z)d'”u¢+o,-‘-
and

“H20 = -dnflszz
it is seen that

szo "Mflzfld -‘-'O (63)
It has been shown then that the chemical potential of an isolated

system cannot increase.
Once again these results are consistent with the fact that the
entropy cannot decrease in an isolated system since the relationship

dQé TdS was used in equation (61) to derive the result that the

chemical potential could not increase.

MATHEMATICAL ANALYSIS OF THE FIRST’LAH
FOR ISOLATED ELECTRICAL SYSTEMS

Energy transformations in electrical systems are made in accordance
with the First Law. In electrical systems the same forms of energy are
involved that are involved in all other systems, namely heat, work, and
some means of storing energy.

It is known that in isolated electrical systems a unidirectional
principle exists. This may be demonstrated by examining a charged
capacitor in an isolated system with an Open switch. When this switch
is closed, the observed effect is an increase in the temperature of the
system at the expense of the electrical energy stored in the capacitor.
The reverse of this process has never been observed. It is not possible
for the capacitor to be recharged solely by decreasing the temperature
of the system. If this system is to be brought back to its initial state,
heat must be removed to restore the system to the original temperature
and work must be done by an external source to recharge the capacitor.
If this is done, the system is no longer isolated. Therefore, there
exist in isolated electrical systems states which are inaccessible from
any given state. It should then be possible to determine this direc-
tional limitation imposed on the First Law if an isolated system can be
Constructed in which one of the quantities involved may be expressed as
a Pfaffian equation of two independent variables.

Consider first the following electrical system composed of a
Charged capacitor and an inductance connected.by wire of negligible

resistance.

‘hl '-
II

C

__®__

 

 

L

 

Figure 17 .

When the switch is Open for the electrical system represented by
Figure 17 , energy is stored in the capacitor according to the relation
lit-'nécv2 where C denotes the capacitance and-V denotes the voltage across
the capacitor. When the switch is closed a current begins to flow
causing a voltage of self induction to be produced by the induction coil.
This voltage is not a voltage in an actual sense but is only a tendency
to resist the flow of current. The voltage of self induction is equal
to the voltage across the coil and is given by

e s -L 3%8V
where i denotes the current and L the inductance. This voltage of
self inductance will limit the current but will not stop it since the
tendency to oppose the flow of current is only present whengitfio.

Thus the current will increase to a maximum value when the voltage
across the coil and capacitor is zero. At this time gjt- will be zero
also. The energy of the system is stored entirely in the inductance
coil in the form

E = $12

21.7

The current after having reached its maximum value will continue
to flow in the same direction and will begin to recharge the capacitor.
This causes a voltage across the coil and capacitor which tends to
oppose the flow of current and the current decreases until it is zero,
at which time the energy is once again stored in the capacitor. The
only change in the system at this time from the initial state is that
the capacitor has Opposite polarity. This system will continue to
oscilate in the above manner as long as the switch is closed.

The First Law for this system could be written as E: constant or
dE=O since dQ, and dfl are both zero.

E = %m2+ fCV2 = constant

If the capacitance and inductance of the system are Constant,

dE = Lidi+CVdV= 0 (6h)

The above system serves to show how energy may be stored in an
electrical system. It remains to construct an isolated system for
which a mathematical analysis of the First Law is possible. The

following systems of negligible resistance make this possible.

I" _ ... _- _ ___-___
.I 7:": ‘ I
System#l J V

 

 

Adiabatic wall

 

\

Adiabatic wall

 

I

14/-
--I

I

I

I

M—

I

L___.| ‘3
, ..

l
I
l
I.--- ___-.-

Figure 15.

 

y‘-
CU

Both capacitors are originally charged and the voltage across
both capacitors is the same. If the switches of Figure 18 are closed
simultaneously the system will oscilate as described in the previous
example.

If the two capacitors and inductances are initially of equal
magnitude, the energy stored in each capacitor at zero current is the
same and the energy stored in each inductance coil at maximum current
is the same.

Thus there is no change in stored energy in either system. But if
both C1 and L1 are varied, the energy stored in the two systems is not
the same. If Cl is made less than C2 the energy stored in Cl at zero
current is less than that stored in C2 since Ezécw2 and the voltage
across both capacitances is the same. Likewise if L1 is made less than
L2 the current will increase more rapidly in L1 since V:L%and the
voltage across both inductance coils is the same. Thus L1 will have a
larger maximum current and the energy stored in L1 will be greater than
that stored in IQ since E==§§L12.

For adiabatic systems the First Law is -dW==dE. Thus when there
are changes in the stored energy of these systems work must be done
reversibly on one of the systems by the other system.

The change in stored energy for a system such as the above two
systems has already been found to be if L and C are constant

dE = Lidi+CVdV

In the above systems this same relationship will hold if Cl is
varied only when the voltage of the system is zero and L1 is varied only
when the current flow is zero. By carrying out the variationaof Cl

and IQ in the above manner, not only are all energy transformations made

| .e
Lil)

at constant Cl and L1 but also no work is needed from an external source
to vary these quantities.

The change in stored energy of each system is thus a function of
two independent variables, 1 and V, and may be expressed as follows:

E

“’(§7)V“ 3— )1 (65)

Therefore, dwl and awe may be expressed as Pfaffian equations of
two independent variables:

AWE-(gt?) V.611 +(g—El')i dVl

4512:1652)de +(‘ v :12) 'de (so)

The work done by the combined system is equal to the sum of the
work of the two separate systems and this sum is equal to zero since no
work crosses the boundries of the combined system.

dw = dW1+ awe: o
(a) we) a is. (at

The voltage of systems one and two need no longer be distinguished
since V1: V2: V.

The quantities -dW1 and -dW2 have been expressed as Pfaffian
equations of two independent variables. Therefore,

-dwi= anti

-dw2= Agate ’ ‘ (63)

It can be shown that -dw is also expressed as a Pfaffian equation
of two independent variables. This can be shown by constructing the
surface for ~dN equals zero.

Examine first the intersection of the surface of no work with the
V-i planes. Let the voltage be increased and vary L1 such that il is
constant. This could be done by increasing L1. An equivalent capacitance

C for the system may be found. C==Cit02. From energy consideration:

50

$ch = in 1.3 this
But 11 is constant. Therefore,
ice-n: sin
or
..LI2
V2-K2— E i
This describes on V-i planes lines of constant 11 with the

following form:

V

 

 

 

Figure 19.

The shape of the surface is then determined.

8 :mn

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

20.

Figure

Surfaces of no work in 11-12-V space

Since ~dW is a Pfaffian equation of two independent variables

there exists an integrating denominator). such that €31 is an exact

differential, dQ.

-dV = AdQ
But
dw = aw1+ avg
Therefore,
MO 3 Mdhfizfi’g (69)

The equality of equation(59)can hold only if 0 is a function

of .1 and 02. Therefore,

at %%_ i
d a d ( 0)
.3 “1 01+ 2 h ‘
Equating coefficients of the exact differential d0 gives

561:; 9 M23}:

0 is a property of the combined system and Q1 and 02 are properties
of the individual systems. Since- -;‘:' andxi‘ are the partial derivitives of
¢ with respect to $1 and $2, they are independent of the other properties
of the system and cannot be a function of the voltage. 'Jherefore,

9. 5'- 9. he
bVA’O; on” (72)

It is not necessary to distinguish between the voltages of the two

systems since they are equal.

By carrying out the differentiation of equation (72)

l a». .
X 87" fight“ o. (73)

The equality of equation (73)can hold only if both— ' N%%'and— ’{—2 _ch

depend only on V. Hence,

1 &.1 29.2.1». wt
x.a7'-xzsv-xv 1"“ m

IThis equation may be rewritten in the following form:

61M... blnAg_. bln). __
TV "5?" “TV -' N")

from which
lnA= fiIv>av+1nim
A = fie/“V’dv (75)

where §=f(0).
The voltage may now be defined as that part of). which is a “

function of the voltage -

v’m = Ce/f(v)dv (76)
By defining a new pr0perty, charge, as follows, 5.
' L..-
Q( )3- , ..
O C Q“ (m

it is seen that

-dw - MO = m. 0
Similarly,

-dw1= A901 = VldQI

-dW2 .1: AdeQ = VedQZ
and in general for all such reversible processes

-dw=)«d0 a m (78)

To gain a knowledge of the directional limitation imposed on the
First Law in electrical systems it is necessary to examine the irreversible
phenomena associated with these systems.

Examine the following irreversible system.

g,/’

'-[c 1. R
T

Figure 21.

Consider a finite transition from an initial state (Vo ,Q.) to a
final “state (V,Q). It is possible to reach state (WQ) in two steps;
first changing the voltage adiabatically by varying the capacitance, the
charge remaining constant, equal to Q, and then changing the state
adiabatically at constant capacitance by closing the switch for a finite
length of time until Q,o goes over to Q.

Examine step one on a capacitance charge diagram.

6

 

 

 

 

Figure 22 .

Figure 22 shows that from any initial state (Va-,Q.) any other state
(V,Q.) having the same charge may be reached since the capacitance is
arbitrarily variable at constant charge. Step two must then give the
restriction on the First Law.

Examine step two on a capacitance charge diagram.

 

 

 

Figure 23 -

 

Figure 23 shows that once state (V,Q) has been reached from any
initial state (V,Q.), there are two possible relationships between Q.
and Q, that is 61.; Q or Q09. Q. One of these conditions cannot be
possible or else all states would be accessible to any given state in
isolated systems.

Consider the process in which Q0 was changed at constant capaci-
tance to Q. {were is experienced during this process a decrease in the
voltage across the capacitor. This value of voltage and the capacitance
determine the charge on the capacitor and this value of charge will be the
same regardless of how this state is reached. This state could be reached 5'
by allowing this capacitor to reversibly charge another capacitor. If _
the capacitor does work reversibly in this manner the work done by the
system has been shown to be aw: -VdQ. This work has been a positive
quantity since it was out of the system. If the constant in equation
is chosen such that V is positive the relationship div! = a-VdQ shows that
the charge cannot increase in an isolated system.

Since V is always positive, the only way the charge in a system can
increase is to add work to the system in the form of VdQ work. Also since
the charge can decrease but never increase in an isolated system,
dw é ~VdQ or dW‘IVdQI for any process. The equality holding in
reversible processes. Hence, for any process #54de . For any cycle
integrating around the cycle,

ft£IMI
as a 0
since charge is a state point and has zero change in a cycle.

This result is not in conflict with the classical statement of the

Second Law that the entropy cannot decrease in an isolated system since

it can be shown that the entrOpy increased also during this process.

Consider first the change of state of the capacitor during this
process. The capacitor performed an amount of work on the resistor.
This work could have been done reversibly and adiabatically on another
capacitor. Therefore there was no change of entrOpy in the capacitor.

Consider next the resistor. If a large source of air at a
temperature equal to the initial temperature of the resistor is blown
over the resistor so that no change of temperature is caused in the
resistor or the air, no change of entropy takes place in the resistor
since it is in a steady state condition.

Consider finally the cooling air. There has been an amount of
heat energy transferred to the air equal to the work done on the

resistor. Therefore the increase in entropy of the air may be calculated

by
as = “31.19
'where T is the temperature of the air and resistor.
Thus the decrease of charge and the increase of entropy in

electrical systems indicate the same directional restriction on the

First Law.

ANALYSIS OF THE FIRST LAW
FOR ISOLATED DECHANICAL SISTENB

In the first two constant mass systems, energy could be stored in
the system only in the form of molecular activity. In systems of varying
composition it was seen that there existed another form of stored energy
in the form of the chemical potential. In mechanical systems there exist
two other methods by which a system may store energy (i.e., kinetic and
potential energy). Thus for mechanical systems in general there exist
four ways of storing energy in the system (electricity, magnetism and
capillarity being neglected).

If chemical interactions were present in a mechanical system, they
could be treated separately by the methods developed for systems of
varying compositions Also heat exchanges may be treated separately by
the methods develOped for adiabatic constant mass systems. This leaves
then for reversible adiabatic mechanical systems two forms of energy
storage within the system, kinetic and potential energy.

It is known that a directional principle exists for isolated
mechanical systems (i.e., if a mass object in an isolated system is
allowed to fall a certain distance while doing work on a friction device,
the system cannot be returned to its initial state by an adiabatic
process). There is experienced during this process a rise in the
temperature of the system at the expense of the potential energy of the
mass object. The process has never been observed when the potential
energy of a mass. object has been increased solely by a decrease in the
temperature of the system. If the above system is to be returned to its

initial state, work must be done by an external source to restore the

mass object to its initial state of potential energy, and heat must be

removed to restore the temperature of the system to its initial value.

When this is done the system is not isolated. Thus in isolated mechanical

systems there exist states which are inaccessible from any initial

state. It should then be possible to construct an isolated system in

which one of the energy quantities envolved may be expressed as a

Ffaffian equation of one or two independent variables to determine what ma
this directional principle is. E

Consider first the following system.

 

 

 

 

Position 1 __J' Initial rest position L ,
_m—J
Xi
- '7 4 Reference Elevation
POSition f I ‘"* 717777777777777777777
I... .J

Figure 21+ .

If the mass in Figure 21+ is initially at rest, energy is stored in
the system in the form of potential energy.
1.".3. .
E: 3. X,

Where 3 is the acceleration due to gravity and 3. is a dimensional

H‘s:

constant 32.17 mg .

If the mass is allowed to fall to position f , it no longer has
any potential energy. The energy is totally stored in the system as
kinetic energy.

£=2£mV2
where V denotes the velocity of the mass obJect.

The First Law for this system could be written as E: constant or
dE=O since dQ and dW are both zero. For any position intermediate to
the two shown in Figure 2’}, E will be given by

E=ux + 1W2 a constant.

35 2

For any infinitesimal change in energy

ds=’;-}dx+dev=o (79)

The above system serves to show how energy may be stored in a
mechanical system. It remains to construct an isolated system for
which a mathematical analysis of the First law is possible. Consider
the following isolated system of negligible friction which is initially

at rest and in which m1 is greater than me.

 

 

m, I System #1

dP—I

 

 

 

 

1" x'
s [35—J—
y #2 X2 Reference Elevation
1, - m

Figure 25 .

 

m1 and mg each constitute separate systems, and the First Law may
be applied. For adiabatic systems the First Law is -dW-‘-"dE. 'Jhus if
there is a change in stored energy of either of these systems, work
must be done reversibly on or by the other system.

The change in stored energy for a system such as the above two
systems has been shown to be

dB: 25;? dx + mvdV 1-..1

The change in stored energy of each system is thus a function of
two variables, x and v, and may be expressed as follows:

dE‘ (gav “95).. ‘N i

Therefore -dW1 and -de may be expressed as Pfaffian equations of

two variables:

“d"1'(bx.v.) “W 3%.)x ‘V'
'd"2=(g_eg)vgdx2+(3'$:)xz av, (80)

The work done by the combined system is equal to the sum of the
work of the two separate systems and this sum must be equal to zero since
no work crosses the boundries of the combined system.

dw=dw 1+dW2=O

-aa—[s)vaai]aa+[(35)..+(352)Jdv=°

In equation(81)dx, and dxaneed no longer be distinguished since
dx, = (IX; and v2 and v1 need not be distinguished since V1: V2 =V.

Equation(81)is a Pfaffian equation with one independent variable
since for the system constructed if one variable is known say V1 all the

others are determined. V2: V1 and the kinetic energy in the system is

determined. Since the total energy in the system is constant the total

potential energy is also determined. Thus
oAKE-APE

where KE denotes the change in kinetic energy and PE the change in
potential energy of the system.

61

-4KE=-’%}X,{+ m§1x23+ 2%!("2’0’0

where xd and x2,“ are the initial rest positions which are known.

The only unknown in equation (Blhs 4x and the equation may be
solved for 53:.

It has been shown that equation<81)is a Pfaffian equation of one
independent variable. It should be possible to construct curves of no

work in a vl-xl plane. These curves take the form of Figure 26.

3

Wk“

 

 

XI
Figure 26.

Curves of No Work in the Vl’xl Plane
The slope of the curve in Figure 26 is determined by
deV,+ WVZdVg =—[’£;de, -- ’2}? 4x2]
deV: - W olx
Jo
211- ..an - .9.
dx‘ mVy. ' V
Thus when V is large the slope is small. When V is small the sIOpe

is large and the slope is always negative.

..A " t‘

62

Since —dWl, -dW2, and -dW have all been expressed as Pfaffian
equations of two variables,

-dWl= Aid“.

«3342 = A3”

-dW : A d¢

Since dW= dWl+ dW2=O,

Ad¢=A1d¢1+Azd¢2 (82)

The equality of equation(82)can only hold if O is a function of O1
and 02. Therefore,

- M N
“1° " 3‘61“” is” (83)
Equating coefficients of the exact differential d¢ gives

bib-X
Est-'1’. Wu F

O is a property of the combined system and O1 is a property of the
first system. Since N/A is the partial derivitive of ¢ with reSpect to
01, and since 0 is a function only of O1 and 02, AI/" cannot be a function
of any other property of the system and is therefore independent of the
force, F, exerted on system one by the coupling between system one and

system two ; similarly forxz. The partial derivitives offi»I and {32 with

respect to F must then be equal to zero.

bx_ ,
o'i-X"° ) an" =°

F1 and F2 need not be distinguished since F1: F2= F. By carrying out

the differentiation, the following equality is obtained.

I AM ... I 5’2... IOA
EFF-- A23?- XdF

This equality can hold only if

33L?

%' and {—2 %3depend only on F. Therefore,
lax. = I Aka: I DA... =‘F(F)

Nfi- xsz A3- (81*)

Equation (81+)may be rewritten as follows:

blnAl = blnA3_ 3111A 3 fur)

TF- "Bi" "Sf-
from which
lnxs f(F)dF+C (85)

This constant can be shown to be a function of at most one other
variable. Selecting Q will give the most desirable results. Equation “-31
(85) may then be written

in). =/f(F)dF + in}
where is f( O),

Asia/“I9“:

It is now possible to define F as that part of A which is a
function of F.

F'(F)=ce km” (86)

The value of C may be fixed by prescribing the value of Fe-Fl for
two reproducible states.

By defining another property x as follows

X(®)=-£-/§d¢

- (87)

it is seen that

-dW s Ad0=Fdx=O
Similarly,

-dw1. AldOlu-Fldxl

-dw2.).¢d¢2crpdx2
and in general for any reversible process

-dWR .-.- Fdx

Equation (87) refers only to the reversible processes from which
it was derived. To obtain a knowledge of real irreversible phenomena it

is necessary to apply the previously mentioned fact that there exist in

isolated mechanical systems states which are inaccessible from any given state.

Consider the following system.

i -

 

   

friction brake

 

 

Figure 27.

Examine a finite transition from an initial state (Va ‘X.) to a
final state (V,x). It is possible to reach the final state in one step.
Consider the system of Figure 27 in which there is now a frictional

device fastened to the pulley. Examine the change on a V-x diagram.

(V, X)

 

 

 

Figure 28.

it:

"3.7-.6‘ a:

hag ‘3.

a 1‘." u‘I’n I I‘—

Figure 28 shows that four possible relationships exist between
(Vo,xo) and (V,x).

V03 V and xyl- x

1:}. V and X’s x

IL: V and age x

\Lé V and x.‘ x

Any value of velocity is obtainable within the limits of the system
simply by varying the friction in the system. Therefore the relationship
between x, and x must give the restriction on the First Law. Consider
what happens when the frictional element is applied.) The pulley
experiences a force which tends to Oppose the direction of rotation. If
this force had been exerted by another suspended mass in a way similar to
Figure 25 , this force could have been utilized to raise this mass through
a distance. If this had been done, work would have been done reversibly
by the system. Since this work would have been taken from the system it
is a positive quantity and the relationship -dW 8 Fdx shows that dx is
a negative quantity. Therefore xg x0 and it can be shown that the
potential energy could not increase in this isolated system.

It is obvious that the maximum kinetic energy of the system is
also decreased by the friction device, and in general in any mechanical
system the sum of the kinetic and potential energies cannot increase.

These results are not in conflict with the classical statement
of the Second Law that the entropy cannot decrease in an isolatedsystem
' since it can be shown that the entropy also increased during this

irreversible process .

66

Consider first the falling mass. There is no change in entrOpy in
the falling mass since the loss in h1netic and potential energies could
have been utilized reversibly'and adiabatically to do work.
Consider next the pulley and brake. If a large source of air at
the original temperature of the pulley and brake is blown over the pulley
and brake there is no change in entropy in the pulley and brake since
they are in a steady state. f‘tmm
Consider finally the cooling air. There has been an amount of '
heat added to the air equal to the change in stored energy of the falling

mass. The change in entrOpy may be calculated by

423.
T

....m L--. A 4.4... _ ‘ i

52-Si=

where T is the temperature of the air.
Thus the increase in entropy and the decrease in potential and
kinetic energies indicate the same directional restriction on the First

Law in mechanical systems.

67

CONCLUSION

It is seen that by establishing a method of performing a
mathematical analysis of the First Law, the directional characteristic
of any process is directly obtainable from the First Law. Thus the
association of the Second Law with idealized thermal machines and heat
power cycles is removed.

The common conception that thermodynamics is concerned with
heat-work relationships in thermal machines is erroneous. The laws of
thermodynamics are concerned with energy transformations; as such they are
very general. The breadth of the field of thermodynamics is Obscured,
and the generality of the laws of thermodynamics is frequently lost
in the maze of thermal machines classically employed in their proof.

The establishment of the Second Law for all systems without the use of
idealized thermal machines is at least one step toward making the
science more general.

The analysis of various systems shows conclusively that the
entrOpy cannot decrease in an isolated system. However, the mathematical
analysis of the various systems did not of itself indicate this direction.
Only in adiabatic systems of constant composition did the mathematical
analysis show that entrOpy could not decrease. This resulted in a
reduction of the temperature potential.

In systems where no work is done the mathematical analysis of the
FirssLaw indicated only that the volume of the system could not decrease.

This resulted in a reduction of the pressure potential.

' i 68

The mathematical analysis of chemical systems showed that in
isolated systems the chemical potential could not increase.

In electrical systems it was shown that the charge in the system
could notincrease if no work were done on the system. This resulted
in a reduction of the voltage potential.

Similarly for isolated mechanical systemS'Umatotal stored energy
in the form of potential and kinetic energy cannot increase.

The results of the mathematical analysis of these various systems
leave some doubt as to whether the increase in entrOpy points the
direction that an isolated process must take. Possibly the increase in
entr0py only measures the displacement in this direction during the
process as a result of the reduction in the potential of the system.
This, however, is irrelevant since both directions are one and the same,
and it makes no difference whether it is stated that the entropy cannot
decrease or that the potential cannot increase in an isolated system.

If the entropy must increase in isolated systems, then let the
Second Law be so stated. It is, however, unnecessary to prove this
directional characteristic by the use of thermal machines. There is an
obvious gain in generality by being able to analyse any system directly
rather than applying a result which was established indirectly by the
use of thermal machines.

Consider for example the classical proof that a free eXpansion is
an irreversible process. The Second Law is first stated in one of its
classical forms: it is impossible for a heat engine to produce net work
in a complete cycle if it exchanges heat only with bodies at a single

fixed temperature.

 

69

To demonstrate the irreversibility of a free expansion the following
procedure is followed. An insulated container, Figure 29 , is divided

into two regions A and B by a thin diaphragm.

7/,”IIIdllllllllllllllllI[III/III]Ill/IIIIII/IIIII/IIII/I
r ‘ .

\\.\\\

\\\\\\\\\\\

\\

f
/
/
5
I
r
/
x
I
z
I
5
x

\\

lllll/lllllllllllllllll IIIIIIIIII/l/IIIIII/

 

 

Figure 29 .

Region A contains a mass of gas and region B is completely evacuated.
If the diaphragm is punctured, the gas will expand into region B until
the pressures in A and B become equal, but there will be no effect on
the surroundings.

Assume that the free expansion is reversible. let the reverse
process occur, by which the gas in B returns into A.with an increase in
pressure, without any change in the surroundings. To construct a cycle
which shall violate the Second Law, install an engine between A and B,
Figurefitand permit the gas to expand through the engine from A to B

instead of using the free expansion.

    

 

{Villfl/[llll/I/l /’(/Ifll r1. VIII”.

   
 

4

  
 
  

Assumed reversed
free expansion

m \\‘.‘fi.\\‘

 

Heat Source

The engine can obtain work from the system at the expense of the
internal energy of the gas as it eXpands. ‘After the expansion through
the engine, the internal energy of the system can be restored to its
initial value by heat transfer from a source at a fixed temperature. Thus
the system can be brought to the same state as if the free expansion
had occurred. New, by use of the reversed free expansion, the system
can be restored to the initial state of high pressure in.A and vacuum gr“
in B. The result is a cycle which violates the classical statement of

the Second.law and is therefore impossible. Since the expansion through i

the engine and the heat transfer from the fixed temperature source are

glen-1 ’51—“-

known to be possible, the reversed free expansion must be impossible; and
a free expansion is irreversible.

This proof leaves much to be desired. The very statement of the
Second Law used in this proof implies that the Second Law is associated
only with heat engines and cycles. This is not the case. The Second Law
applies to all processes, not just heat engine processes and not just
cycles. The free expansion has been analysed independently of heat
engine cycles and the resulting directional characteristic shown.

Other irreversible phenomena may be analysed in a similar way. Only
by this removal of the association of the Second Law with heat engine
cycles may the generality of the Second Law be elevated to a position
consistent with that of the First Law, and the scope of the science of
thermodynamics extended beyond the realm of heatawork relationships in

thermal machines.
\\ \x\ t; _
‘\
i}

\_
‘x'
‘\..
\—
‘V
L.
v

\/ \r

,
w

\

LIST OF REFERENCES . !
Born, Max, Natural Philosophy of Cause and Chance, New Yerk:
Oxford University Press, l9h9, pp. 314KB.
Partington, J. R., Thermodynamigs, Constable 86 Company Ltd.

Schouten, J..A., Pfaff's Problem and Its Generalizations,
Oxford Claundor Press, 19h9.

Sears, F.'W., Thermodynamics, Addison-Wesley Publishing
Company, 1955.

Keenan, J. H., Thermodynamics, New York: John Wiley'& Sons, Inc.,

19h1.

 

Caratheodory, C., hathematische Annalen, 1909, Volume 67, p. 355.

V

n
E

I ' vi: “2!. "'
PM" ’2‘ “(5: Chi-3’.
(Inlay/r; :JJ!‘

2.

 

 

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