MOLECULAR ATTRACTIOS. BY J.
XI1
E. MILLS
In a recent article by the author in This Journal,' attention was called to the possibility of diagraming the internal pressure produced by molecular attractive forces as a negative pressure. Several well known equations were studied and new relationships between these equations were pointed out. The facts and forces operating to produce these relations, were not, and are not yet, fully understood. But it seems worth while to point out other relations as they are recognized, in order to simplify the problem as much as possible to successive workers in this field. In order to avoid a rather useless repetition, figures, references, explanations of symbols, etc., used in the previous article are not here reprinted. I . In the previous article it was shown that the negative pressure a t the critical temperature and volume could be found in four ways, namely: dP 31414P' CRTc = a. -Tc-P c = I. dT For isopentane the values obtained were respectively 161,800 = 1j9,500 = 157,700 = 159,924, giving as the most probable negative pressure the average value 159,731. If to this negative pressure the actually observed vapor pressure 25,ooj is added, a total critical pressure of 184'736 is obtained. Now if the theoretical vapor pressure for isopentane a t the critical temperature and pressure is calculated from the equation,
7 ' 7 v'-
2.
P =
62g:T , millimeters
of mercury,
the value obtained is 93,452j . Now since 2 X 93,427 = 186,854 it will be seen that the total external and internal pressure at the critical temperature is equal, within the limit of exJour. Phys. Chem., 19, 2 5 7 ( 1 9 1 j )
X o l e c z d a r rittractiov
6j1
perimental error, to two times the theoretical vapor pressure for the substance under the same conditions, if the substance was a perfect gas. Similar results are obtained for other nonassociated substances. Why such a relationship exists is not yet clear to the writer. 2 . We have pointed out repeatedly since 1905 when the fact was deduced by the author as a consequence of an equation given by Crompton that at the critical termperature, dP
2R
E = v;,
3.
that is to say, at the critical temperature the rate of change of the pressure with the temperature is just twice what it is for a perfect gas under the same conditions. Since it now appears from the section above, that a t the critical temperature the total positive and negative pressure is also just twice what it is for a perfect gas, it is possible to write, dP - ---dT 4. T T’ where T is used to represent the sum of the external and internal pressure, that is the total pressure. 3. A similar equation to the one just given seems also to hold under many other conditions. From the usual equation for a perfect gas PV = RT, it is easy to deduce the relationship a t constant volume, dP dT - _ P T’ Moreover, the equation of Ramsay and Young deduced for constant volume and nearly true for both liquids and gases, 6.
can be thrown into the form,
dP _ -dTT. P+a As shown in the article of which this paper is a continuation a is to be regarded as representing the negative internal pressure. Hence, P a represents the total pressure T and i t i.
+
dT - is a t least very apT’ proximately true a t constant volume over a very wide range of volume, pressure and temperature. But the equation seems also applicable, with a suitably modified interpretation, under circumstances where a constant volume is not maintained. Thus in the Clausius-Clapeyron dP
is clear that the relationship, - = P
L being the heat absorbed (rejected) during the change in volume is equal to I P d : and, hence,
dP - T must represent the dT
azerage total pressure during the given change in volume. This interpretation is enforced in the previous article. Therefore, allowing the former symbols to apply to these changed dP dP dT is average conditions, the relation - T = P or - = T’ dT ir again true. In fact, if the thermodynamical Equation S is combined with the equation given by the author, 9.
= ,u’(~,
- 3 \ D) Calories,
the relationship under discussion can be made much more specific. For we then get, dP T - p - 3’4’4 F ’ ( 3 % d IO. y, dT whence I_?
~ 11.
+
dP _ _ 31414 ,u’(~\2 D) - d T -~ yT’
’\
_-
I_,
As shown in the previous paper,
31414 p ’ ( 3 1 ’ d -
v-
-1
’\E)
is the
mathematical average of the ordinates of the internal pressure curve between the limits V and 5‘. Moreover, Equation 11 has already been proved true. ,4t the critical temperature it reduces to, dP dT 12.
iVIolecular Attractio91
653
giving the relationship discussed in Paragraph 2 , more specifically, for isopentane. V Dieterici's equation, X = C R T In - would give similarly 21,
dP
dT -V T CRT In 1
As a matter of fact, the relationship, dP dT =Total pressure T' i s so persistentlj* true O X Y s z d a wide range o j coiiditious as to suggest Fery Jorciblj' that perhaps the relatioii expresses a uviversal law. hloreover, since both the temperature and the pressure are functions of the velocity of the molecules a general consideration of the meaning of Equation 14 rather strengthens the hope that it does contain a universal truth. Equation 6, and hence Equation 7, are not exactly true, and other objections to accepting Equation 14 as a universal law will appear to the critic familiar with the field here studied. After considerable consideration of the question I am inclined to the belief that Equation 14 does represent a very fundamental truth, and that properly interpreted it may prove to be a universal and exact law applying to changes a t constant volume and to phase change equilibria. For that reason I have suggested it here. Probably the pressure of the numerator should be taken also as referring to the total pressure. I am a believer in thermodynamical formulae, but I have never been able to understand, and I have sought to understand, how such a relationship as the Clausius-Clapeyron Equation 8 can be applied to fusion, vaporization, and other phase change phenomena. Some explanations ignore infinitesimals of the first order at will, though usually this is done in a non-mathematical sort of way to lessen the crime. Some apply the perfect gas, reversible cycle, reasoning without troubling much as to how the reasoning happens to be applicable Others are at great pains to explain that there really
is a reversible process in phase change phenomena, because the process really can be reversed by infinitesimal changes, etc., etc. As a matter of fact the various arguments are made to give correct equations but I have always thought that this might be due to some undiscovered relationship. If Equation 14 is true of phase change equilibrium conditions extensive explanations of reversible cycles will not be required. 4. I n the previous article the negative pressure due t o molecular attraction was expressed by the equation, 15. p
=
'w. 3v
Writing the equation
p2'3
=
10471,~'~
and comparing with the equation for an adiabatic of a perfect gas, PVy = constant, their similarity is noted. The so-called intrinsic energy of the adiabatic given by the equation,
corresponds with the energy given out by the molecules on coming together from an infinite distance apart to volume V, as derived in previous papers. In a perfect gas, pressure is caused by a given amount of kinetic energy, EK, according to the equation,
while from Equation I j , the internal pressure, caused by a giren amount of energy derived from molecular attractive forces following the inverse square law, is given by the equation,
Is.
P
31414,~'~td
= ---~
3v
The same amount of energy produces twice the pressure in Equation 17 that it does in Equation 18. There is no error here, mathematical or otherwise. Since the pressure is produced by the operation of different laws the same amount of energy corresponds to twice the pressure in one case as in the other. Since there is always present in a substance both temperature energy and energy derived from attractive forces i t is important to recognize the above fact. It suggests a
655
JIolccular Atiractio?i
new meaning for the relationship pointed out some years ago' 19.
p'?%d c _ c = ___ 3RTc
-
energy necessary to overcome molecular attraction -~__ , 2 x kinetic energy of molecules, ~
C is the important constant of Dieterici's equation, 1-
X = CRTIn;.
20.
L
Transforming Equation 19 from terms of energy to pressure, it will be seen that C should equal the ratio of the internal negative pressure a t the critical temperature to the theoretical critical pressure. This is, as a matter of fact, true, for taking isopentane, and using values given, C = 159j731 = 93427
.
I .'7 IO
The actually observed average value of C for isopentane is I .688. For the 26 normal non-associated substances measured by Dr. Young and his co-workers the average difference between the value of C calculated as above and the observed average value of C was 2.6 percent. The greatest divergence was 4.j percent. The relation, 21.
c
=
Internal negative pressure a t critical temperature Theoretical critical pressure
is, therefore, certainly very nearly true. the cause of the relationship. .i. It has been shown? that,
I do not understand
where D, = theoretical, and d, = actual, density at the critical temperature. This relation could now be expressed as follows for the critical conditions, 23.
Xegative pressure Observed pressure = Theoretical pressure + Theoretical pressure ~~
2,
which reduces to the relation already given in Section Total pressure = Theoretical pressure ~
Jour. Am. Chem. SOC., 31, 1 1 1 7 (1909). Ibid., 31, 1 1 2 1 (1909).
2.
I,
6. The causes operating to produce the relations cited in this article, and in the previous article, are not fully understood and no extended discussion of the causes will be undertaken here. It is, however, not out of place to point out some additional facts that should be borne in mind when seeking an explanation of the relations derived. First. The total heat added to a monatomic element such as aluminium, copper, silver, lead, from the absolute zero to the molten condition, is approximately not quite three times the energy necessary to effect the temperature change if the elements behaved as perfect gases. I n other words, the entropy of these monatomic elements in the molten condition a t their melting points is roughly a constant, and is for many of these elements from 8 t o 8.5 per gram molecular weight. Secoxd. The indications are that from the melting point to the boiling point of the liquid element the specific heat will be from 2 to 3 times as great as for the element in the condition of a perfect gas. Consequently a t the boiling point of a monatomic element it is certain that there is, in many cases a t least, a very large amount of energy in the liquid element that cannot be accounted for as molecular energy of translational motion (temperature energy). Third. Estimates based on thermodynamical relations indicate that the specific heat a t constant volume of a liquid or solid element is not greatly different from the specific heat a t constant pressure of the same element. Hence no very large proportion of the excess energy of the liquid indicated above is directlj' due to the change in volume of the element under the action of the molecular attractive force. The idea of molecular attractive force proposed in the present series of papers leads to the same conclusion. Fourth. While the experimental data on more complex compounds is not sufficient, nor sufficiently understood, to enable the above remarks t o be extended with certainty so as to include them, yet probably everything that we do know
i2iIolecular Attraction
657
indicates, that in a general way a t least, the same conclusion should be applicable to these compounds. Fijth. In a previous paper‘ the author proved that the energy gi;ien out bit aizy two bodies origiually at rest at a72 iwjinite distaizce apart i i i f o r n z i n g aiiy stable colljiguratio9z g n d e r the action o j gra2itatioizal attractioiz, i s equal to the kinetic eulerg31 which the3 retaiiz aud i s , .tor either bodji, imersely proportional to tlze niea.tz distaizce .from their cowzntoi~ center o j mass. This proposition is a necessary consequence of the law of gravitational attraction and of the laws of motion. The proof rests entirely upon well known, universally accepted, dynamical principles. It is possible to extend the proposition to a system of IZ particles as follows: Sewtonian mechanics depends upon the truth of the principle that “If two or more forces act on a body, each produces its own change of momentum in its own direction independently of the others.” It has been shown that two particles coming from an infinite distance apart to a distance s and forming a stable s!-stem under the action of gravitational attraction retain exactly as much energy as they give out. Kow making use of the fundamental principle above stated, it is obvious, that when I I particles come together to form a stable system, that we can consider the action of any one of these ?z particles upon any other particle of the system independently of the i z ? - 7 1 - I actions of the other particles upon these two particles and upon each other. But the result of the action of any two particles upon each other is that the eizergj, lost = euergj’ retained, and when the mutual action of every particle has been determined we will have i t ? - ii similar equations. Since in every equation the energy lost = energy retained, the total energy lost by the system must be equal t o the total energy retained by the system and we can state that, The avzomt of e?iergy retained bj’ a systenz o j 71 parti‘cles iii jormiiig a stable systenz zuzder ?he actimz qt tlze gravitational Phil. Mag., ( 6 ) 22, 84 (1911).
law of jorce i s equal to the amount o j eiiergy gizega out by these particles i.lz cowing together from aw iqtfiqtite distatzce apart. There seems to be no escape from this conclusion. For the individual particles of a stable system are in dynamic stable equilibrium and it is clearly possible to separate the tangle of individual forces into their components, and these components act independently and must be in equilibrium in accord with the analysis that has been made. Sixth. A similar result must follow if the molecular law of force, f =
p 2 - --2,
S
is substituted for the gravitational law of
force. This result can be obtained by paralleling the previous argument in its entirety. Or it can be seen to follow directly. For the result that, the energy lost by the system = the energy retained by the system, is true independently of the absolute value of the masses of the particles and of the absolute value of the attractive forces. Sc;e.tztlz. The above arguments take no account of the temperature energy. It is interesting to note the actual relations that exist for one gram of isopentane a t the critical ' d 6~5.01 calories, surrendered in coming temperature. ~ ' ~ t = to the critical density, and the same amount of energy should be retained by the liquid isopentane according to the above argument. The kinetic translational energy for isopentane a t the critical temperature E, = 19.04 calories. If this energy is additional energy, the total retained energy of the liquid would be 84.05 calories. As a matter of fact only one-half of the temperature energy also, seems t o be retained, making the total energy retained 74.53 calories. This amount of energy substituted in Equation I O of the previous article gil-es a total pressure of 182,900 millimeters of mercury, which is t o be compared with the value 184,736 given in Section I of this paper. However. it must be noticed that there is no cnpciiiizciztal evidence to indicate that ~ ' ~ v =d ,65.01 calories, and the extrapolation may not be justified. Also I frankly do not understand the relation between the temperature energy (trans-
AIolecular Atfractiopi
65 9
lational molecular motion) and the retained attractive energy (orbital motion) of the molecules. The relation given just above holds true for other substances, and I point it out only for what it is worth as a possible explatzatio.iz of the pressures obtained at the critical temperature.
Summary I.
The persistent exact, or very approximate truth,
under many circumstances, of the relation,
dP
=
d'I'
F, suggests
that the relation ma>- always be true for constant volume and for phase change equilibrium conditions. P refers to the total pressure, i. c.. to the numerical sum of the external and the internal pressure. 2 . The relationship of several deductions previously made is discussed. 3 . The following relation is derived: The amount of energy retained by a system of 71 particles forming a stable system under the action of the molecular attractive force is equal t o the amount of energy given out bl; these particles in coming into the system from an infinite distance apart. Temperature affects this relation in a manner not yet understood. 17?iizeusztj of Sozith Cc.iolinc1 J ~ C6I,J1 9 1 j
