AIOLECULAR ATTRACTION. X I . NE&: RELATIONS REVEALED BY DIAGR,4T\IIXG IKTERNAL PRESSURE AS A XEGATIVE PRESSURE BY J. E. MILLS
S?,nzbols.-Subscripts to symbols designate special or particular values. Pressures are expressed in millimeters of mercury, temperatures in degrees Centigrade, volumes in cubic centimeters, and energy in small calories. A, a, 01, P , b, C, c are constants. d denotes density of a liquid. D denotes density of saturated vapor. E denotes energy. E, denotes energy spent in overcoming external pressure. .t denotes force. I, denotes total latent heat of 1-aporization. X denotes internal heat of XTaporization = L - E,. 11 denotes total mass taken. 172 denotes molecular weight, i n / the mass of a molecule. y’ is the constant of molecular attraction given by the L-EL expression !-- \ d13-\D y =
cy/3\171.
is the number of molecules in the mass taken. P denotes the external pressure. p denotes the internal pressure caused b\- the molecular attractive forces. 71
R is the gas constant
=
62 192
.
s denotes the distance apart of the molecules.
T denotes absolute temperature. 2 denotes temperature in degrees Centigrade.
denotes the volume of a gram of liquid. V denotes the x-olume of a gram of \Tapor. i41though the significance of our thermodynamical formulae, and their mechanical explanation in terms of our usual ~8
molecular-kinetic conceptions, leave much t o be desired, yet these formulae stand today apparently entirely deserving of the universal trust which they receive. For this reason I choose t o begin the proof of the conception to be presented by using a thermodynamical equation, although as a matter of fact the conception arose from a consideration of very different equations arid ideas.
The ?'hcl.?nod~'jlanziccllEpatiojl, x
=
JI (dP dT 7 - P )
(V-tc).
The Clausius-Clapyeron thermodynamical equation for the heat of vaporization of a liquid,
can be combined with the thermodynamical equation for the energy spent in overcoming the external pressure during the change of volume,
E,,
2.
1'
=
J
(I7-
;'I
= 0.0~31833P
(Iv-:,) calories,
and we h a w ,
The measurements desired have been accuratelp made for isope~itane~ by Dr. Sydney Young and that liquid is, therefore. chosen for stud?.. The data used are shown in Table I. The
d I'
were obtaiiied irorn the Biot equation, 5. log P = (1 A ha cp, used by lroung for smoothing the x-apor pressures. On differentiating and changing to Saperian logarithms we obtain
~
+
dP
6. .~ ..
(i,l,
dl' ~
1
=
5.3019 P ( b log &.at
+ c log p.p').
~
Sci. Proc. Roy. Dublin Soc., 12, 374 (1910); Proc. Phys. SOC. London,
Session 1894-95, p. 602.
-1Ioleczdar rittractioiz The constants are given in the references cited.
259 The values
dP
of the dT so obtained are given in Table I. Drawing the usual pressure-volume curve, Fig. I , for isopentane a t 120' C, we represent P(V - L ' ) of Equation 4 by the rectangle ABFE, and this area represents the work done
Fig. I The rectangles .1BCD. .IIBICIDI,etc., reprrsent the heats of vaporization given b y the thermodynamical Equation 4, each rectangle being up011 a ditierent temperature plane, and the internal heat of vaporization being represented hy t h a t portion of the rectangle b r l o ~the line of zero pressure. .&C5 rtpresents the critical volume line. The line G-H represents equation I O ol llills. T h e line 11x0 is the projection of the end points of the lines represented by Equation 17 from Dieterici's Equation 15, a complete curve for 1 2 0 ' C being given by the dotted line S O . T h e curx-e K-P is the locus of the points u of Equation 2 2 of Ramsay and Young. The IinFs - \ ~ C SG-H, , R-E'.and Also, intersect a t Cj. The diagram shorn the possibility of plotting ant1 studying the region of negative internal pressure.
J . E . Mills
2 60
during expansion from volume Y to volume V against the external pressure P. If now we undertake similarly to represent dP
the term dT T (V - 2) by an area, it clearly has the same base as before, but the pressure ordinate i s eizormously greater than P a?zd vzust, therefore, be exteiided ilzto the regiou o j izegat+Lie pressure o v must be extcFided as a positize pressure f a r be>o?id P (that i s beyolid i i z the diagram). If extended from
P E in a positive direction, the line dTT
passing beyond A will clearly represent no pressure of which we have any direct evidence.
dP
IVhile if zrT be regarded as
representing a total pressure, made up of the external pressure, and an internal pressure, it may be extended from A so as t o pass into a region of negative pressure and this negatire pressure may pro;,e to represent the pressure arising jrow the molecular attractiot?. One knows certainly after a most elementary acquaintance with mechanical conceptions that the internal pressure arising from the molecular attraction does act in a direction opposite t o the usual pressure due t o the kinetic motion of the molecules, and can be regarded, in a sense a t least, as a negative pressure. dP
Extending dT T, therefore, from A in the negative direction it becomes represented in the diagram by the line AD.
dP
The area XBCD, therefore, represents dT T (V - ;I), which is from Equation 3 the total heat of vaporization. Since ABFE represents the external work done in pushing back the external pressure during the vaporization, the area EFCD will represent the internal heat of vaporization, denoted in Equation 4 by the term 31414A. Therefore, according to our method of representation the internal heat of vaporization will appear always as a pressure-volume area below the line of zero pressure. Similar diagrams are drawn for isopentane a t 140°, 160°, 180' 187' and a t the critical temperature 187.8".
A1401ecular A t t r a c t i m
26 I
It is important t o note that a t the critical temperature no area results because the factor V - Y becomes zero, the facdP
tor dTT still retaining a significant value.
I have here introduced and discussed the thermodynamical Equation 6 to show t h a t it suggests very naturally the idea that the internal heat of vaporization should be represented as an area lying wholly below the line of zero pressure in an isothermal plane. If this idea is correct it should lead to conclusions not hitherto recognized. S o further conclusions of value are a t once apparent from Figure I and this probably accounts for the failure of other investigators to study the facts from the point of view here adopted. Moreover the method of representation adopted would lead one to suppose t h a t the negatize iizternal pressure remained a consta9it during a cha.r?ge iiz ttolume. This certainly cannot be true. The true situation can be stated as follows: The 7iegati;e pressiircs j o u n d a l e the a;!erage iiiterual presstares during tlze g i x i z clzaiige iiz volume except at oiie point, tlze critical temperatirie. =It the critical temperature tlzerc is n o chalige iia c o l m z c aizd tlzc ' ' a x r a g e ' ' $zegatite pressure abotle f o u n d should represeiit tlze trice i7iteriial pressure at this ilolitme. This we find below to be the case when a correct value for the
:2
a t the critical temperature is used.
(We have before
clearly provedl that Equation j a t and very near the critical temperature does not correctly represent the observed vaporpressure curve, and t h a t Equation 6 in this region will in dP
consequence give values for the dT much too low.
By di-
rectly smoothing the observations of vapor pressure and their dP
rate of change, a more nearly correct value of the dT a t the critical temperature, 379, is obtained. This value is also somewhat too low owing to the fact that the percentage errors of observation are multiplied more than 60 times in obtaining dP
the dT directly from the observations.) 1
Jour. Phys. Chem., 9, 402 (190j)
262
J . E . -Wills
The Equatiou oj’ X i l l s , X = ~ ’ ( ~ t ’ d 3VD). In work upon molecular attraction and closely related phenomena I have shown,’ I think conclusively, that the equation,
is true, and that this equation can be derived easily and follows naturally, if the law governing the molecular force, j, is
The truth of Equation 7 has been admitted finally by nearly all of those who have examined the work critically and who have published the results of their investigations, but a good many of these investigators have never been convinced t h a t Equation 7 is really caused by the law of molecular force given in Equation 8. The objections advanced t o Equation 8 for the law of force have been from time to time answered in the papers cited, and all of this work is now undergoing revision in order that it may be published in a convenient form for critical inspection. We will proceed next upon the assumption that the internal molecular pressure does arise from an attraction obeying the law given in Equation 8 and proceeding from the individual molecules. To determine the effect of this attraction per square centimeter of surface we note that the attraction of a molecule varies inxrersely as the square of the distance apart of the molecules. S o w the number of molecules in any square centimeter of surface also varies inversely as the square of their distance apart. Consequently the internal molecular pressure per square centimeter varies as the fourth power of the distance apart of the molecules, and as a matter of fact Jour. Phys. Chem., 6, 209 i19oz);8, 383,593 (1904):9, 402 (1905i; IO, (1906);11, 132,594 (1907);13,512 (1909);15,417 (1911); 18, 1 0 1 i~1914); Jour. Am. Chem. Soc., 31, 1099 (1909):Phil. Mag., Oct. (1910); July (1911); Oct. (1912);Trans. Am. Electrochem. Soc., 14, 35 (1908);Chem. S e w s , 102, ij (1910); and related papers, X l l s and N a c K a e : Jour. Am. Chem. SOC.,32, 1162 (1910); Jour. Phys. Chem., 14,79j (1910);15,54 (1911). I
Allolecular A4ttraction
263
is given in millimeters of mercury, using the symbols and constants adopted, by the expression, IO.
For isopentane p
=
1,I O l d O O -
~, since
p’ =
105.36.
To those readers who are not familiar with the conception of molecular attractive forces advanced in previous papers, I would explain t h a t the essciztial d.fue71.enc-e between my conception of the behavior of the attractive forces and the conception usually held, lies in the fact t h a t I consider a molecule to ha\-e a certain total power of attraction which remains a constant a t a given average distance apart of the molecules, independently of how t h a t attraction may be distributed t o the surrounding molecules and independently of the number of molecules. (1h a w never found any method of determining certainly how this total attraction is distributed, but it is quite possible, and in my opinion it is very likely, that its distribution in a sense follows Sewton’s law of gravitation.) Consequently in calculating the effect of the attraction upon the pressure per square centimeter I have only to take the attraction of one molecule a t the gil-en distance and multiply this attraction by the number of molecules in the square centimeter of surface. I n other words, the attraction of the molecules is a constant property of the molecule precisely as is its molecular weight ionl?- the attraction varies with the distance apart of the molecules). The above idea has been shown in the papers cited to be in accord with the facts and it is important t h a t it be clearly understood. I will illustrate, therefore, bl;a concrete example. Since the direction and distribution of the total attraction is mathematically a matter of no importance for most purposes, I can consider the total attraction of a molecule A to be exerted upon a molecule B a t distance s . If I then place beside B another similar molecule C the total attraction of A for B and C remains the same as it formerly was for B alone. According to the usual conception the attraction of -4 for B
and C, two similar molecules, a t distance s, is twice what it is for B alone a t distance s. According to m y conception, while the total attraction of A may be shared by both B and C it remains the same for B and C together a t distance s as it was for B alone a t distance s. The fact t h a t the total attraction is now divided between the two molecules B and C has not increased its amount in the least However, I am not here trying to prove that my conception of the attractive force is the correct one. The conception has in the past proved to be in accord with the facts considered and if I can here deduce from this conception new relations in accord with the facts its use will be sufficiently justified. That Equation I O is correctly deduced from the premises, and t h a t it is in accord with the fundamental law stated in Equation 7, will be clear upon considering t h a t
E
11.
=J”d\-,
and substituting for P its value from Equation calling the internal energy given out X,
IO,
we have,
’V
I
3
1-
31414P ( t d - 3 t n ) ,
an expression which can be reduced to calories by dividing b y 31414 and is then in accord with the law given in Equation 7, and already proved in the numerous papers cited. It is really a source of much gratification to be able to deduce Equation 7 in this way, because Equation j was originally (after correction of an .earlier error) deduced’ from a consideration of the individual action of the attractive forces upon the individual molecules. And yet it is clear if Equation 7 really resulted from the law of molecular force given in Equation 8, and if the effect of this force upon the total Jour. Phys. Chem.,
of the law.
11,
143, 147 (1907)
Note the two derivations
Molecular Attractio.tl internal pressure is correctly given in Equation IO, that one should be able t o arrive a t the same result either by considering the individual forces between the molecules or by considering their resultant. Consequently the demonstration above given really completes a cycle of operations, and to arrive exactly a t the starting point is gratifying evidence that no errors were made in the individual operations and arguments. Moreover my idea as to the independence of the kinetic and attractive energies and of the connected external and internal pressures urider the coi?ditioiis studied were, to put it mildly, viewed with suspicion," by some investigators. Yet this same independence is here again in evidence. I have plotted in Fig. I , line GH, the internal pressures calculated by Equation I O against the volumes, the data being given in Table I. The ordinates of this curve represent the internal pressure in millimeters of mercury per square centimeter and it will be noted that at the critical teniperatzire this iiqtertial pressitre does equal the iizterqial pressure obtaiiqed f r o m the tizeruzodyizamical equatioii uithiii the 1 iwit o j experiFneizial error. (149,63j = 159,500, attention has already been
TABLE I-ISOPEKTANE Volume of
Temperature
Presstire
t "C
P
0 20
40
60 80 IO0 I20
140 I60 I 80
18j
187 187.8
Liquid "
2j7 jj2
74 j9
1131 I
203j 6 3400 8 5354 j 8039 9 11620 1628j 22262 23992 24713 2500j
tn
Vapor
Densitv of
Liquid d
Vapor D
o (1392 o 001090 5644 917 4 6141 426 6 o 6196 o 002344 6700 224 4 o j988 o 004456 I 7329 12; 6 o 5769 o 007837 o jj40 o 01287 I 8Ojj 7; 7 o 5278 o 02020 I 8940 49 2 003; 32 2 0 o 4991 o 03106 2 153 21 I j 0 4642 0 04728 2 378 13 72 o 4206 o 07289 2 8j8 7 95 o 3498 o 12j8 3 183 6 355 0 3142 0 1573 3 500 j 455 0 2857 0 I833 4 268 4 268 0 2343 0 2343 = 72 IO,p' = 10j.46, c = 1.688 I I I
0.8614
0.852j 0.8429 0.832j 0.8213 0.8081 0.7932 0.7743 0.7493
0.7046 0.6799 0.6586 o.616j
266
TABLE I (Continued) ~
Temperature
VD
t "C
I-
I
'3
Y4/3
o o o o
1029 1328 1646 1986
0 0001121 o 0003112 0 0007339 0 0 0 1 jjLj
2344 2724
o 003018
IO0
0 0
I20
0
0 20
40 60 80
140 I 60 I80 185 I87 187 8 187 8
dT
11.16
0 4177 0 5011 0 54.00 0 5681
o 616j -
2789 5592 I 003I 16522 2 5492
3047
6165 I I 162
35.66
18558
55.73
o 005506 o 009771 1 5 5 . 2 o 01~09 204.4 264.0 0 03045 0 0630j 335.9 0 OS503 356.I 0 1041 364,4 367.8 0 I445 Ob. 379.0
3144 o 3616
dT
21.04
81.85 114.7
dP ,,T-P
CpT
dP -
25893 42783 60986 84425 114.312 1j2163 163100 167600 169480 174640
37429
52946 7250j 98027 129900 139100 I42890 1444i.5 149635
TABLE I (Contznzied)
x Temperature 1 "C
0
3 1414 !.' 3 i'li 3
80 96
69620 93900
76 0 2 71 37 66 59 61 80 56 63 50 77 43 89 35 31 21 55 14 74
207800
I I 4900
9 49
I j9j00
1
5574C0 530300 j02400
140
I60 I 80
185 187 187.8
~
Ther SIlllS D ieterici Equation 3 Equation 7 Equation I j
124
40
80
3
3 44 810 171s 3330 6050 10790
583200
I00 I20
3T4
60S000
20
60
314I4P'
470800
437200 396800 g48000 272200
236100
IS880 33620
59500
0
called t o the fact that the error in obtaining the value 149,635 is very large. That the poor agreement shown is only due to errors of observation will appear more clearly later.) Moreover the internal pressures given by the
Molecular Attraction
267
thermodynamical equation a t other ternperatkres are the (1
mean’’ or
average” values of the integral, , [ j ( ; l ) d ~ i , that
is, they are the mean values of the ordinates between the proper limits, V and 2 of the curve GH. This follows because,
The first equality was shown above. The second equality was experimentally proved for numerous substances in former papers. After reduction to calories the values for isopentane are given in Table I under the headings A-Ther., Mills, for convenient comparison here. In other words, the nieati x l z i e s o j the ordiiiates to the c u r x G” betweeiz the liwiits 1- atid i are dP
haTe pro;!ed experiineiitallj3 i s eqzial to d~ T
-
P obtaiiicd
JOY
the correspoitding tewtperaturc, piesstire, colzctw, ioiiditioiis. I t should be constantly remembered that we really have t o deal with a tri-dimensional pressure, \volume, temperature, diagram, and that the curves ABCD, AIBICIDl, etc., shown on Fig. I are really on different temperature planes, and that Fig. I is really made by the projection of points and lines that exist on different temperature planes. The fact that no temperature function enters into Equation 1 2 , and the fact that it is true a t all temperatures studied, cause the projection of all of the points and curves considered to fall into one line GH. However much we may be inclined to an opinion on the subject, we have as yet no experimental evidence to show that limits on the curve G H can be taken arbitrarily a t any temperature. Before leaving Equation 1 2 it is important to note that its definite integral can be obtained as follows on the supposition that it is true for all values of V.
J . E. &fills
268
For when the volume is infinite, E is zero and d is zero. quently C is also zero. We can write X,
‘4.
= pI3.u’d calories,
or sX,
Conse-
= p‘34;,
where X, indicates the heat given out when the molecules come from an infinite distance apart to a distance apart represented by s, t h a t is to a density d. This result I have previously derived in the papers cited from aquation 7, but this new deduction is worth while because it shows that the internal attractice energy given out by a substance o n contractio+z from an h z j n i t e colume to colzme ij, or density d , can be represe?.zted gyaphically by the area between the V axis, the curee GH, and the ordinate at the a o l u ~ tei.
The Equation OJ Dieterici, X
=
d C R T l n D.
Dieterici found an empirical equation for the internal heat of vaporization, d T d log - calories = I j . X = CRTIn - = 4.573 C D nz D CRTlnV - CRTlnu. This equation has been carefully studied1 by the author, by Dieterici himself, by Steinhaus, and by others. It is a remarkably accurate equation with a tendency to give slightly too high results a t very low vapor pressures. The constant varies slightly for various substances but is usually around 1.7. For isopentane C has the value 1.688. The values of x calculated from Equation 15 are given in Table I for comparison with the values of X obtained from Equations I and 6 (see under the heading X-Ther., Mills, Dieterici). Equation 15 can be expressed, 16.
V ,IU
X l l s : Jour. Am. Chem. SOC.,31, 1099 (1909); Dieterici: Drude’s 569 ( 1 9 0 8 ) ; 35, 2 2 0 (1911); Richter: Dissertation Rostock, 1908; Steinhaus: Dissertation, &el, 1910. 1
Ann.,
25,
Molecular Attraction
269
When IL' = I , In li' = o and X = C1. The value of the constant of integration can, therefore, be found if the value of X can be found when I gram of liquid occupies a volume of I cubic centimeter. The indefinite integration gives the area bounded by the ordinates 'L' = I and it = i', plus a constant. It is clear that we here deal with a curve in every respect similar t o the usual PV curve for a perfect gas except t h a t equals CR instead of R.
The curve is a parabola referred
to its asymptotes as axes. The internal pressure is given by the expression, CRT 1.688 X 62392 'I' T ' 7 . 1, = = __ - = 1460.7 for isopentane. ~
v
Values of
72.10
';T
v
L'
for liquid and saturated vapor are given
in Table 11. The curves represented by Equation 17 are plotted in Fig. I , the end points of the curves alone being shown on the line 34x0 except for 1 2 0 ' when the dotted line KO indicates the complete curve. These curves are, of course, projections from the corresponding temperature planes. The area bounded by these curves, the pressure ordinates, and the volume axis represents the internal heat of vaporization. These areas are equal to the areas between similar limits of volume as obtained from Equations 3 and 12 as is proven by the equality of the values of the internal heat of vaporization X given in Table I, under the headings, Ther., Mills, Dieterici. A t the critical temperature we have
and we get 149,635 = 159,joo = I 57,700, the values agreeing t o within the limit of experimental error. Combining Equations 7 and 15 we have 18. X = p'(3vd-331D) = CRTIYZV-CRTIYZV, or, po3311'd CRTInn = ~ ' ~ . \ i j j CRTZnV. '9. Steinhaus (106. cit.) first examined the equations in this form
+
+
but seems t o have been unable to understand the result. The values of the various terms for isopentane are given in Table 11. Plotting the values of the various terms of Equation 19 against the temperature me have the curves shown in Fig. 2 . Now these curves become identical if the diagram is 100
89 60 U
0
$0 20
C 2EGREES
CEkTlGRADE
Fig
2
The curves sholi the iiiergy chnngc> indicated 115 Equation I of AIilis and 15 of Dieterici The curves are cliarly the 5ame curves b u t mith a different origin and direction One curve becomes i,ractically identical mith the other if s 180' rotated around the a x ~ AB
In other words the curves rotated around the axis AB 180'. are the same curves but with a different origin and direction (sign). IYe find experimentally for isopentane (see Table 11) that Equation 19 takes the form, p'j\;
20
+ CR'l'l~z, = 96.38
= ,LL'{\D
+ CR'l'InT-,
or. 21
P'
'
=
96 j8-CRTInL,
or CRTin;8 = g G 3 5 - p " i \ d ,
\\There D is the x-olume and d is the density either of the liquid or of the saturated vapor a t an>- temperature. This equation for isopentane is remarkably accurate Considering isopentane a t IZO', Figure I , it now becomes clear that y l J t d represents the area EGH-V when V represents an infinite x-olume, C R T h represents the area PQNOF, the latter area extending from the line PQ drawn a t volume L~ = I , in the direction indicated by the arrow. If one is called upon to deal with liquids whose density is
-17lolecular rittractioiz
X
5 h
L
._
great so that i’ becomes less than I then C R T h i’ will give negative values. This is true with stannic chloride. Also when L’ is I and li? I = 0 , d = I , and Equation 2 1 becomes p’ = 96.35. As a matter of fact p’ for isopentane is 10j.46. This discrepancy indicates nothing except that either Equation 7 , or Equation I j > or both, break down if extrapolated to points outside of the saturated vapor-liquid equilibrium for which they were proven. That this view of the question is correct follows from a consideration of stannic chloride. The data for this liquid are not detailed here, but i = I lies in the saturated \rapor-liquid region attainable a t about 310’ C , and a t this point , L L ’ ~ \ > C R T h i’ does equal y’. It may be remarked in passing that for stannic chloride CRTIIIL’ does not always equal a constant, but varies slightly, and apparently linearly, with the temperature, passing through the value y ’ a t about the temperature indicated. The peculiar form of the curve M N O for isopentane shown on Figure I should be noted, particularly in connection with the fact that Dieterici’s Equation 15 gives too high results a t low temperatures, and with the fact just shown that Equation 2 1 cannot be extrapolated for isopentane to volume
+
+
._, L
- I. -
As yet too little is known regarding this region of negative internal pressure that I am attempting to explore t o indulge in speculation regarding the meaning of the relations found. But it must not be supposed that we are here studying merely mathematical relationships. The equations here under investigation have been extensively studied for many substances (see papers cited), and they certainly do closely represent physical relationships. Any study which will throw additional light upon these relationships is well worth while. The Equation oj Ramsay a.ild You?zg, P = bT-a. Sir Wm. Ramsay and Dr. Young so long ago as 1887 called attention to the fact that van der Waals’ equation, 2 10,.
at coiistant volume reduced to
,VI o 1ecul a r 24t t ractioM
where a t constant volume b
dP
= d~
273
= constant, and a mas a
constant. This equation mas tested in a careful experimental study' for a good man>- substances both in the condition of liquid and gas and was found to be very nearly true.
dP dT
a t constant volume does not remain exactly and absolutely a constant a t all temperatures. Dr. Toung found for isopentane? that a t volumes lower than 4.6 cubic centimeters per gram, the values of b increase with rise in temperature, while a t greater volumes up to about 400 cc they diminish. A t still larger volumes they appear t o be constant. Values of a obtained by interpolation from Young's results are shown in Table I1 for the corresponding volumes. The values found for a for isopentane by Young are plotted on Fig. I forming the line R-I'. (Young obtained the values of a by two methods. I use his values from drawn isochors throughout.) It will be noted that this line intersects the other lines, AjC,,, GH, hlNO, a t the negative pressure corresponding to the critical volume. I have now obtained the value of the internal negative pressure a t the critical temperature by four methods getting 2.3.
149,635 = 159,500 = 1 j 7 , j o o = I j g , g 2 4
I may add here that the first value, 149,63j, is too low, due dP
solely to the fact that no a c z i i a t e dT can be obtained a t the
critical temperature either directly from the observations or -
-
I Rarnsai a n d X70ung Phil 1 1 a g (5)23, 195, 435 ( I 8 8 j l Phil l r a n s , 180A, 13; 18891 183A, 1 0 7 11892 Young Proc Phys SOC London 13, 6 0 2 (189 j 1 "Stoichiornetrv' pp 2 0 3 - 2 13 RowInnes and Young Phil llag , ( 5 ) 48, 213 ( 1 8 9 9 ) 43, 1 2 6 '18971, 47, 35.4 (1899 ( 6 ) 2, 208 ( I g O I ) , -1rndgat Cornptes rendus 94, 8 4 j 11882), A-lnn c h m phys , (6) 29, (18931, Barus Ph11 l I a g l ( 5 ) 30, 358 (1890) Proc Phys Soc , 13, 648 (189j)
J . E. M i l l s
2 74
indirectly through the Biot formula, due to the great multiplication of the errors of the observations in obtaining the dP dP The correct value of the dT a t the critical temperature is dT. nearly always given by the expression 24.
first discovered as an empirical equation by Dieterici and later independently derived by the author from an equation given by Crompton.'
L-siiig the \-due 4oj.J of the
Jour I'h? i C h c m
9, 4 0 2
f
rcog
i ~ ~ iother d pipci-5
dP
so derived
cltcti
dP
we would obtain for d,l, 1' ~- P the value 161,Soo. The importance of the relations given in Equation 23 and their graphical representation as negative pressure is, I believe, great. One would espcct further studp of these relations to throw additional light L1i7011 the equation of state, thermodynamical formulae, and entropy relations.
Sti mm a r y It is pointed out that negatil-e internal pressure can be diagramed and studied, and that this method of representation leads t o relations not hitherto recognized. l~i11;evsLf?'a t .solii:l C'ill OIlilCL S o i o i i b e i 23, r y r j
