T H E ABSOLUTE ZERO O F T H E EXTERKALLY COSTROLLABLE ENTROPY AND IKTERKAL EXERGY O F A SUBSTANCE OR M I X T U R E BY R. D. KLEEMAS
The properties of the internal energy and entropy of a substance at the absolute zero of temperature and the connection of these properties with those a t higher temperatures has recently attracted a considerable amount of attention, and given rise to a good deal of experimental research. The importance of the subject was probably first recognized by Nernst who enunciated his well known heat theorem in this connection. The development of the subject ultimately resulted in a theorem which is often called the third law of thermodynamics. The object of this paper is to show that these results are only cases of a much more general result, which can be deduced directly from results of an axiomatic character. I.
T h e Diuision of the Internal Energy and E n t r o p y into Two Parts.
The internal energy of a substance or mixture of given mass may be divided into two parts, one of which is a function of the variables temperature and volume only, while the remaining part is independent of these quantities. The first part will accordingly be externally controllable while this will not be the case with the other part, and hence these two parts will be called the externally controllable and externally uncontrollable internal energy of the system. The heat absorbed when the temperature of a substance is increased. or its volume is isothermally increased, are examples of changes in the controllable internal energy of a system. Changes in the uncontrollable internal energy would evidently be spontaneous. An example of such a change is the ejection of 01 and /3 rays during radio-active changes, for these changes are independent of temperature and volume, and therefore take place spontaneously. The liberated energy in such a case, however, becomes controllable internal energy. It would be interesting to know if the reverse may happen. Similarly the entropy of a substance or mixture may be divided into two parts, one of which is externally controllable while the other is not. In speaking of the thermodynamical equilibrium of a system, such as a mixture in a number of phases, it, is tacitly assumed that the nature of the equilibrium will not change unless made to do so from the outside by a change in temperature, volume, or masses of some of the constituents. The equilibrium theref ore depends upon the controllable internal energ- and entropy of the system. Similarly, any reversible process in thermodynamics is externally controllable, and therefore involves changes only in the controllable internal cnergy and entropy. These particular quantities are therefore of great thermodynamical importance.
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R. D. KLEEYAN
If u denote the controllable internal energy of a mass of matter we may accordingly write u = $,(T, V, Ma, Mb, . . . . ) (1) where 2' denotes the volume of the matter at the absolute temperature T , and Ma, M b r . . , denote the masses in gram atoms of the constituent a, b, . . . . Similarly if S denotes the controllable entropy we may write
S
=
$e(T,V: M,.
Mb,
(2)
Since u by definition is the controllable internal energy it will have a zero value for certain values of u and T. For if this were not so let ui denote the least possible value u can have. Now the internal energy u1 could not be externally controlled, or its value reduced by changes in the values of v and T, and therefore it does not consist of controllable internal energy. The values of v and T corresponding to a minimum of u would therefore give a zero -value to the controllable internal energy. Similarly it can be shown that the controllable entropy is zero corresponding to its minimum value for certain values of v and T. If a surface is plotted with v, T, and u as axes, u being taken in its general sense, it will be evident that the zero of the controllable internal energy corresponds to a point where a plane parallel to the v and T axes touches bhe surface. Similarly in the case of a surface with v, T, and S as axes, the zero of controllable entropy corresponds to the point where a plane parallel to the v and T axes touches the surface. The zero of the controllable internal energy and entropy may definitely be determined, t,he reasoning depending in the main on the following theorem :2.
The Specijic Heat ( $ ) ; t
Constant Volume of a ,Yubstance or Mixture
can have only Positive Values. This theorem is a direct consequence of our fundamental conceptions of heat and temperature, according to which an increase of temperature of a mass of matter kept at constant volume is associated with an increase in the heat content. It can therefore be shown to be intimately connected with the laws of thermodynamics. As is well known the thermodynamical scale of temperature is founded on Carnot's cycle. This depends on heat flowing to a substance or mixt,ure on expanding it isothermally and on its temperature decreasing on expanding it adiabatically. It may be remarked that it is one of our fundamental experiences with matter that its temperature may be decreased to any desired extent by adiabatically decreasing its volume sufficiently. Under these conditions only can the heat taken in at the higher temperature during the cycle be larger than the heat given out at the lower temperature, the difference representing the external work done, as is demanded by the laws of thermodynamics. It follows also from these laws that the amount of work obtained is independent of the nature of the substance or mixture used, provided we work between the same temperature limits and the same amount of heat disappears as such, on which considerations the absolute temperature scale is based.
ZERO O F EXTERNALLY COh'TROLLABLE ESTROPY
749
X o w suppose that a mass of matter has its temperature increased from
TI to Ta at constant volume, giving rise to a algebraical increase in entropy
r( T25
aT, where C, denotes the specific heat a t constant TI T volume. S e s t suppose that the volume is adiabatically increased till the temperature has fallen to TI. Lastly suppose that the volume of the matter is isothermally decreased till it regains its initial value, during which a quantity of heat Q is given out, and a corresponding decrease Q Ti in entropy takes place. This decrease is equal to the foregoing increase, and hence
equal to
According to this equation C, or
(E),
is positive.
It may also be noted that since the theorem is based on the laws of thermodynamics, which are the outcome of experience, physically absurd results (in the light of our esperience) mould be obtained if we suppose that the theorem does not hold. Thus it is evident that if the temperature of the surrounding medium is increased by d T an initial flow of heat to the substance will take place, whether its specific heat is positive or negative, otherwise nothing would ultimately happen with the substance. This will be attended by an absorption of heat. It will then be found that if we suppose that the specific heat is negative it may have an indeterminate value, or the lowering or raising of the kmperature of the substance over a finite range can only be carried out by periodically reversing the flow of heat. These are of course absurd results in the light of our experience. Besides the above theorem some other results will be necessary in order to determine the zero of the controllable internal energy and entropy whose t,ruth according to our experience is even more obvious, and which will therefore be designated as postulates. 3.
T h e Coej'icients
(2);(g)"
cannot hace Infinite Values.
Since the pressure of a substance is due to the motion of translation of the molecules, a n increase in the magnitude of this motion through an increase of temperature is not likely to be attended by an infinite increase in pressure, or
(
(g)
cannot have an infinite value.
For the same reason
cannot have an infinite value.
We shall now proceed to apply the results of this and the preceding Section.
T h e Absolute Zero of the Controllable Internal Energy of a M a s s of Matter. Let us consider a mass of matter in the condensed state under the pressure of its vapour at the absolute zero of temperature. It can be shown that the controllable internal energy always increases in passing from this state 4.
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R. D . KLEEhfAS
to another physical possible state. Since such a change is brought about by a variation of 2: and T, it may be noted at the outset that the controllable internal energy only !Till undergo a change. Thus suppose that the matter is isothermally compressed. This can be shown to he attended by an increase in internal energy. On multiplying the thermodynamical equation
by d l and integrating it between the limits z’ and v’, u and u‘ we obtain
A t the absolute zero of temperature this equation becomes u - uf since T =
0,
and
=
-1; p .
av
(4)
(g)”
is not infinite according to the postulates in the
preceding Section. If v is taken equal to vo, the volume of the mixture in the condensed state at the absolute zero of temperature under the pressure of its vapour, and v’ is less than vo>p is positive for values of v lying between vo and v’ inclusive. Hence u’ - u is positive, or if the volume is isothermally decreased from v, to v’ this is attended by an increase in u. This result follows also from the fact that the atoms of a substance in the condensed state at the absolute zero of temperature are in equilibrium under their forces of attraction and repulsion. Hence when the volume is decreased repulsion is the outstanding force, and the internal energy is accordingly increased during a process of compression. If the volume of the substance is plotted against its temperature the foregoing operation corresponds to passing from the point vo to the point v’ in Fig. I . S e s t suppose that the substance kept at the constant volume v ’ has its temperature increased to T, which in Fig. I corresponds to passing from the point v’ to &hepoint T. This is attended by an increase in internal energy according to the theorem in Section 2 . Thus the whole process, since it involves variations of 2’ and T, gives rise to an increase in the controllable internal energy. This result will evidently apply to any point lying in the compartment in Fig. I made by the line v, a and the positive axes of ti and T. Kext suppose that the mass of matter in the same initial state as before, namely in the condensed state in contact with its vapour at the absolute zero of temperature, has its volume isothermally increased to v”, which corresponds to passing from the point vo to the point v ” in Fig. I . The matter
ZERO O F EXTERNALLY COSTROLLABLE ESTROPY
751
would divide itself into a solid and vaporous phase. The amount of matter in the latter phase would however be zero, since the pressure is zero. The process is therefore not attended by a change in internal energy. Since, when v” is infinite the amount of evaporated matter may be finite, we may proceed more strictly thus: The heat absorbed, if any is absorbed, can only have a positive value according to the fundamental properties of Carnot‘s cyclc pointed out in Section 2 . Kow the external work done is equal to ART, where A is a constant, and therefore zero, since T = 0. Therefore a positive increase in internal energy only can take place. la
I
vi-
~
FIG.I
FIG.2
S e s t suppose that the temperature of the matter is raised to T keeping the volume v ” constant. which corresponds to passing from the point v” to the point T in Fig. I . This is attended by an increase in internal energy according t o the theorem in Section 2 . Thus the controllable internal energy of the matter has been increased during the process. This result will evidently also apply to any point in the space in Fig. I to the right of the line vo a. It follows then that the total internal energy of the mass of matter in its initial state has a smaller value than for any other physically possible state. Therefore according to the definition of and properties of the controllable internal energy given in Section I , its value is zero, when the matter is in the condensed state under the pressure of its vapour a t the absolute zero of temperature. We may now proceed to the consideration of the entropy. 5 . T h e Absolute Zero oj the Controllable Entropy oj” a M a s c o j Matter The absolute zero of the controllable entropy according to Section I corresponds t o the state for which it can be shown that the controllable entropy has a smaller positive value than for all other physically possible states. Such a state should exist. For suppose that the entropy is measured from a given state chosen arbitrarily. Let the state of greatest negative entropy be obtained corresponding to this state. We may then transfer the zero to the latter state and measure the entropy from this state. It would eviderltly correspond to the foregoing definition of the zero of controllable entropy. Let the point uo on the 11 axis in Fig. z correspond to the zero of the controllable internal energy. Let the curve ab be an adiabatic corresponding to zero entropy. S o w as we pass from the zero point of internal energy uo to the point a on the adiabatic which corresponds to the same volume, the internal energy is increased, and hence the entropy is increased. Therefore on passing from the point a to the point uo the entropy is decreased. But
752
R. D. KLEEMAN
since the adiabatic corresponds to zero entropy, the entropy should increase under these conditions. It follows therefore that the adiabatic must pass through uo, or a substance or mixture in the condensed state in contact with its vapor at the absolute zero of temperature corresponds to a state on the adiabatic of zero entropy. We may therefore take this state to correspond to zero entropy. This zero, which is the same as that of the internal energy, will for convenience be called the absolute zero of control. The possibility that the substance may exist in several different (crystalline) forms a t the absolute zero of control which possess different internal energies and entropies, remains to be considered. Suppose that two such forms are mixed a t the absolute zero of temperature. The change Au in internal energy is given by the thermodynamical equation dW Au=T--W dT where Vvr denotes the external work done during the process of mixing. This may be carried out by allowing both forms to evaporate, mixing the resultant gases, and condensing the mixture. I t will now be clear that the work done may be expressed in the form AAT, where A is a constant. Hence when T = o it follows from the above equation that Au = o j or no change in internal energy takes place on mixing. Thus the two forms have the same internal energy, which is zero. I t can be shown that a similar result holds for the entropy. The change in internal energy during an adiabatic change is given by
which may be written
by means of equation (3). At the absolute zero of temperature for any state this equation becomes
I t expresses that as we pass isothermally from one form of the substance to another no change in entropy takes place.' Since the results of this Section are based on thermodynamics, deductionsmade from them will not apply to substances which cannot be passed through any conceivable thermodynamical cycle. They would therefore probably not apply to such a substance as glass, which, if evaporated and condensed again, is hardly likely to be in the same state as before, i.e., glass. A supercooled liquid probably does not admit of being passed through a cycle, a t least in some cases, and to them these restrictions will therefore also apply.
' This proof is not quit? satisfactory, hut will have to suffice here. be given in a subsequent paper.
A better proof will
ZERO OF EXTERNALLY CONTROLLABLE ENTROPY
753
A number of relations corresponding to the zero of control may be found by the help of the postulates of the next Section. 6.
($)>,(g).are
T h e Diflerential Coejicients
and continuous for all Possible States of a Homogeneous iMuss o j Matter.
An atom in a substance is in equilibrium under the influence of the forces of attraction and repulsion of the surrounding atoms. If these forces are not discontinuous for certain distances from their sources the above postulates would evidently hold. There does not seem to be any evidence whatever in nature of the existence of such discontinuities. 7 . Some Relations ut the Absolute Zero of Control. Since the zero of internal energy corresponds to the least value it can have
and
(
(g)
are not discontinuous, it follows from the Differential,
Calculus that the zero satisfies the conditions for a minimum, and hence we will have :
(g),= O
(&)"= 0
A similar set of equations and inequalities hold for the entropy at the absolute zero of control. Since at this state the entropy has the least possible value, and
(g ) T a n d (g)are not discontinuousaccordingto the postulates
in the preceding Section, it follows that the entropy satisfies the conditions of a minimum, and hence we have:
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R. D. KLEEBIAN
Equation (6) expresses that the specific heat of a substance or mixture in t.he condensed state kept at constant volume is zero st the absolute zero of temperature. This result is part of Kernst's theorem. h formula for the specific heat of a substance in the gaseous state at infinite volume may be found which expresses an important property of this quantity. On multiplying equation (3) by a v and integrating it between the limits a and 2' it, becomes
where uE denotes the internal energ?- of the substances at infinite volume. Let us suppose that we are dealing with x mixture whose equation of state may be expanded in the form
where A is a constant, B, C , . . . functions of T. and a , 8. . . . . positive constants whose values are greater than unity. According to our experience this is likely t o be always possible. The integral in equation ( 9 ) corresponding to the upper limit would accordingly lie zero. On differentiating the equation with respect t o T at constant volume we have
where C , denotes the specific heat at the volume v, and CvCcthat at infinite volume with the matter in the gaseous state. This equation applies t o matter in any state. On applying it to a substance at the absolute zero of control the integral will not become infinite since
s2)
(
is not infinite according to
"
the postulates in Section 3, and the volume has the finite value vo. The integral term therefore becomes zero, since T = 0. Since we also have C, = o the equation gives
c,,
= 0
(11)
or the specific heat of a gas at constant infinite volume is zero at the absolute zero of temperature.
ZERO O F EXTERSALLY COSTROLLABLE ESTROPY
-..e
i3 3
This is an interesting and important result, and is especially striking when a non-atomic gas is considered. Evidence of its truth already esi and Heuse' have measured the specific heat of helium at various temperatures and found that it decreased though slightly with decrease of temperature. Thus for example a t the temperatures I S O F and - 1 8 o T the specific heats were found to have the values 3.008 and 2.949 respectively. The esplanation of the result is probahly that though the atoiii absorbs energy on increase of temperature due to an increase in tlie kinetic energy of translation, energy is also given out clue to internal changes in the atom. These two changes balalice each other at and near the absolute zero of temperature, or if -u denotes the heat given out per grain niolecule by the molecules from their internal controllable soiirce of enrrgy per unit inereaye in temperature, untl 3 ' 2 I1 the heat ai>sorhed through the incresse in kinetic energy of motion of translation. Tve ~ r o u l dhave 311 2 - 11 = 0 (121 at the absolute zero of temperature. It should tic noted i n this connection that the internal energy of a number of inolecules in the gaseous state at tlie absolute zero of temperature is not zero according to Section 1. It appears froin this esplanntion of the specific heat of a gas at the absolute zero of temperature, that it is not necessarily associated Tvith a deviation from the gas laws, or the gas laws may hold for all temperatures. It is not impossible of course that the correct esplanation is n different one than the foregoing. though a t present it seems the most plausible. It should he mentioned that Sernst? has deduced a forinula for the decrease of the specific heat of a gas with decrease of temperature from a given theory of the atoiiiic model and quantum considerations. Formulae for the controllable internal energy and entropy of a substance or niisture may immediately be deduced corresponding to the volume 1' and absolute temperature T. Suppose that a substance or niisture at the absolute zero of control has its temperature raised to T keeping its volume vo constant.
The change in internal energy is equal to
.i
C,: dT. where c', denotes the
specific heat, corresponding to the constsnt volume vo. S e s t suppose that the volume is changed to 1' at constant temperature. The change in internal energy is
I'
I
du. The total change in internal energy, which is the controllable
v.
internal energy
ti
of the substance, is therefore given by u
=
i"
C,. dT
+
Jo 1
2
.Inn. Physik, (4) 40, 484 (1913). Der. deutsch. physik. Ges., 18, 8j-116 (1916'.
I"
~ L I
Jvi
756
R. D. KLEEYAN
On substituting for au from equation (3) and for C, from equation ( I O ) applied to the substance at the volume va, in the foregoing equation, it becomes
The expression for the entropy S obtained along the same lines is given by
The value of vo is given by vo =
2.6
2
s,d/aw
(16)
according to Traube', where K,,denotes the number of gram atoms of atoms 6f atomic weight a, relative to the hydrogen atom the substance contains. Thus if the equation of state of the substance or misture and the specific heat C,, be known the controllable internal energy and entropy may immediately be evaluated. Any empirical equation of state may be used provided its constants are determined from the facts. The controllable free energy F and the controllable thermodynamical potential 9 may then be obtained from the equations F=u-TS ( 1 i) 9 = u -
TS + pv
(18)
I t will be found that in all thermodynamical formulae useful for calculation we may substitute the controllable internal energy, entropy, free energy, and potential since the uncontrollable parts would disappear through differentiation or otherwise. A number of other iinportant formulae may be derived which will be given in a subsequent paper. Physik. Z., 1909, 667.
