ON THE POLYMERIZATION OF LIQUIDS AND A GENERAL METHOD FOR DETERMINING ITS RELATIVE VALUE' RY M. M. CARVER
In a recent paper' presented before the Royal Society of Canada at its May meeting, 1912, I applied a new molecular thermodynamic theorem t o test, theoretically, the degree of polymerization of liquid substances. The dynamic basis of the theorem independent of its thermodynamic implications may be concisely stated as follows : The attractive forces which maintain a substance in the free liquid phase are just numerically equal to the pressure the same substance would exert, were i t a perfect gas, at the same temperature and deasity. The thermodynamic significance of this theorem lies in its furnishing the force function which may be applied in all isothermal changes of phase between liquid and vapor. From a
the fundamental equation for work, W
=
(- Fds, we may alI
ways, theoretically, find the heat equivalent of the work when F can be expressed as a function of s the displacement. When this function is expressed in terms of pressure and volume the above theorem leads directly t o the conclbsion that
1
I
which follows because it implies that the force, whether attractive, or exerting a pressure, is always proportional t o the number of molecules acting on, or through, unit area. If we ignore the difference in sign which alone distinguishes an A paper read before the Physical Chemistry Section of the Eighth International Congress of Applied Chemistry. Jour. Phys. Chem., 16, 454 (1912).
M . M . Gaiver
670
attraction from a pressure, the force function is the same as that of the pressure exerted by a perfect gas. A number of experimental facts and theoretical reasons were advanced t o establish the validity of the theoretical applications made. For details the former papers1 may be consulted. However, there is another aspect of the theorem t o which attention never has been directed, so far as the writer is aware; but as it has special significance in connection with polymerization as the fundamental source of liquid stability it should be considered in connection with estimates of the degree of polymerization.
Polymerization and Liquid Stability Probably the first impression made on most minds by the bare statement of the equality of the attractive and pressure relation of molecules contained in the above dynamical theorem is one of instability. If the forces are so delicately balanced whence comes the liquid stability with which we are all familiar? Why should the forces be just equal arid not more or less? Ifi a vague sort of way the molecules are generally regarded as held by some powerful attraction represented by thousands of atmospheres. It is perfectly evident on reflection that the whole attraction cannot exceed the sum of its parts or that the attraction per unit area cannot exceed the sum of the pressures of the separate molecules acting in one direction through any element of area. The tension in a spring is measured by the pull on one end and not by the sum of the pulls on the two ends. Likewise the force should be measured by the rate of change of momentum noirnal to’one side of an element of area. This is what we call pressure and is what we should regard as the attraction also, since it is just sufficient t o neutralize any pressure in the opposite direction. This view leads t o the preceding theorem. The mathematically inclined may also doubtless reach the same conclusion through the application of d’Alembert’s principle to a system of free molecules in motion. LOC.
c t.
.
O n the Polymerization of Liquids
671
The Source of Liquid Stability Since the molecular forces in a free liquid must be in a state of equilibrium and all the molecules are relatively in motion with the same average speed of translation that they would have a t that temperature were they perfectly gaseous, it is a t once obvious that any greater force than that specified in the theorem would drive a t least some of the molecules into contact, or union, and thus destroy their relatively free motion. It is also obvious that any less attractive force than that just sufficient t o hold the molecules at the proper average distance would allow a t least some molecules t o escape and thus cool the remaining liquid. Let us call the union of two or more exactly similar molecules to form one, polymcrization and the heat given out by the union, the heat of polymerization. (Since by Maxwell’s law the average kinetic energy of translation is independent of the mass, such union must either increase the average energy per molecule of the system, or the system must give out heat.) With this in mind the part played by the polymerization in producing stability in the phase equilibrium is manifest. The union, or coalescence, of molecules must raise the average temperature of the remaining molecules and the escape of free vapor molecules must lower the average temperature by permitting additional molecules t o de-polymerize. I n consequence of this relation of polymerization to heat, the stability of the system of balanced forces which would otherwise be unstable, is maintained, so t h a t the actual number of molecules can be disturbed, or changed, only by a transfer of energy in the form of heat or work to, or from, the liquid. The fiolymerization thus $erjorms the function of a ( ‘ source and sink” of heat energy and renders possible a stable equilibrium of freely moving i n d e pendent molecules acted o n by attractive forces, without the intervention of any hypothetical repulsion. Resolution of the Heat of Vaporization into Components During an isothermal change of phase from the vapor to the liquid phase, heat must be removed from the system
M . 111. Garver
672
of molecules in order t o permit of the change. This heat, as we have seen, may consist of two parts, first, the heat of polymerization due t o the decrease in the number of molecules, and secondZv, the heat equivalent of the work due to all the forces acting during the process. It is easily seen that these two separate sources of the heat of vaporization, I,, may be regarded separately and treated as independent of each other provided we are able to estimate the work du t o the forces concerned. This we are able to do in the case of isothermal processes, by applying the theorem previously given, since the force at any instant depends upon the number of free molecules independent of polymerization. (The ‘‘ force ” is to be understood as independent of whether due t o an external pressure or internal attraction.) Suppose we have N molecules occupying a volume V at a temperature T. Now suppose the volume t o be decreased isothermally during which the heat I, is given out and the combined external and internal forces do the amount of work W and the number of molecules is diminished by an amount rz, the n molecules uniting with others, so that the number becomes N-n. From Maxwell’s law and the principle of conservation, using the same units for heat and work W
+
Nm,zc,*
=
L
+ 1/2(N-n)m,u,2
where I,is the heat given out during the isothermal process. By Maxwell’s law l/2m,u,2 = 1/2mp22 = ‘/,mu2,
therefore L-W
=
1/2nwm2 = H
. . , I, = W
+H
where H is the heat of polymerization due to the decrease in the number of molecules occurring during the isothermal change from a vapor t o a liquid. A General Method of Finding the Relative Polymerization of Two Liquids A t the time the previous paper was written it was not possible for me to do more than t o show that all the liquids
O n the Polymerization of Liquids
673
examined (many of which are regarded as non-associated,' or as consisting of simple vapor molecules) are, according t o the present theory, considerably polymerized. For the details of the theory and method of computation the former paper may be consulted. The quantities H and W and the ratio r = I,/W were found for thirteen different liquids of various different chemical types, but except in the case of water no estimate of the ratio of the weight of the liquid t o the vapor molecule could be made, for no method of general application for such purpose had been found. Since then, however, further study of the subject has shown how the theory may be applied and used t o determine the relative molecular weights of the liquid and vapor phases of any other liquid substance when these values are known for any one substance which may be used as a standard of comparison. In the case of water, as previously mentioned, it was possible from certain available experimental data, t o make a fair estimate of the relative molecular weights of the liquid and the vapor as lying between 2 . 5 and I . 9 a t o o C. The minimum value, I . 9 a t oo C, agrees with the value found by van der Waals2 from entirely different considerations (i.e., Ramsay's values). In an early paper, W. Ramsay and J. Shields,3 from surface tension experiments, found a value lying between 3 and 4; but in a later paper, Dr. Ramsay4reduced this estimate t o I . 707 for water at o o C. For the present paper, the polymerization of water has been computed and tabulated a t intervals of 20' from o o t o 200' C and will be used later as a standard of comparison in computing the relative polymerization of the remaining dozen liquids discussed in the previous paper. The method will be found to be a perfectly general one and is not necessarily confined to the use of water as a standard. When any Nernst: Theoretische Chemie, 6th Zeit. phys. Chem., 13, 7 1 5 (1894). Ibid., 12, 433 (1893). Ibid., 15, 1 1 5 (1894).
Ed.,282
(1909).
111. ill. Garver
674
a
other substance becomes better known than water it may be used instead of water. Let us consider any two substances S , and S , of which the molecular weights as vapor and also the values I, and W previously explained, are known. Let m represent the molecular weight of the vapor and m’ the molecular weight of the liquid, H, and H, the heats of polymerization where H = I,-W. The values I, and H refer t o one gram of the substance. Hm will then represent the heat of polymerization of a gram mol and must be proportional to the number of molecules that unite with others to form complex ones. Hence we have generally H,w, --
Ir., = _
n,
H,ni,
The quantity we wish to find is
111
.................... which for brevity will be
written x. Since the total mass of n molecules is unchanged by polymerization we must have in the two cases nni,
where
92’
= n f l n i f l and
nnz,
=
n’2n7‘2
represents the number of molecules of liquid. n-n1
=
n’, and n-n,
=
But
n’,
hence r~ = ( n - n J
n1
-’
= (n-np)
712 1
111 I , 7972
By substitution and elimination we get finally 31, - I
X2
where all the quantities are supposed t o be known except x,, that is, the ratio of the weight of the liquid t o the vapor
molecule perature, xl, of the The
of the substance investigated a t the specified temthe value of x z , however, depending upon the value liquid supposed known. method of finding the value of x for water was
Om the Polymerization of I i q u i d s
675
given in detail in the former paper' and need not be repeated here. The above method t o be strictly applicable must be used for the same temperatures. Hence the value of x
112
= TIL
for water was first found for each zoo from o o t o
zooo C and tabulated in the accompanying Table I. The value of I, at the different temperatures was found by applying Clausius' formula L = 607-0.7t as being sufficiently accurate although i t probably is not very reliable for high temperatures. The specific volumes for liquid and vapor water (saturated vapor) were taken from Winkelmann,' Matthiessen's values for the liquid being used from o o to 100' and the table of Waterston for values from 100' to zoo'. Battelli's values of the specific volumes of the vapor were used except for the temperature 1 6 0 O which was ,omitted. The value 307.3 was taken from the table of Knoblauch, Linde and Klebe (p. 997). The remaining values are taken from the author's previous paper.
TABLEI Polymerization of water,
m' -
?n
(with data)
____ ~ _ _ _ _ _ _ _____________- _ _ _ - _
, 1
Temp. '
o 20
40 60 80 100
I20
I 60
I80 2 00
1 I
1
I
I
I 1
1
I
I
Specific volumes Liquid --___
...
.
I ,0018 I . 0077 I ,0170 I . 0290 I . 0432 I .0600
1.0795 I . 1015 I . 1268 I . I578
I
I F
Vapor
I1-
I--
1 204000 57730 19484 7650 3401 1667 893 * 511.4 307.3 197.1 130.6 '
__
.
_= -
Calories
Ur
--
607 593 5 79 565 551 537 523 509 495 48 I 46 7
365.6 351.7 338,2 325 5 313.3 301.5 290.0 278.6 266.0 256.2 244.8 '
229.0 224.8
' LOC. cit. Winkelmann: Handbuch d. Physik., 111, 92, 94, 996, 997.
1.61 I
57
M . M . Garver
676
The values of I, for most substances are most accurately known a t their normal boiling points; hence it is desirable t o have a standard liquid covering a wide range of temperatures. Hence the utility of the water table. From the degree of accuracy at present possible it was deemed sufficient t o use 20' intervals in the construction of the table. The entire data are given in the table in order that the theory and results may be conveniently checked by those interested. Two illustrativc examples of the method of computation will be given in some detail and the results of the computation of the remaining twelve substances will be found in the column
m'
--. m
The data for the computations are included in
the table under appropriate column headings. Attention was called in the previous paper to the necessity of taking into account possible polymerization of the vapor when determining the value of H = I, - W as the result measures only the change in the polymerization; but in case of acetic acid I failed t o profit by my own warning and used the theoretical molecular weight 60 instead of the actual which is about 97 according t o its vapor density. The result was a value much too low and is corrected in the piesent table. Let us take ethyl alcohol as the first example. I t s boiling point 78' is near enough t o 80' to allow us t o use the data of water a t 80'. Taking the necessary data from the tables we have for alcohol a t 78' XI __I
XZ
x2
For acetic acid a t 118' we may use water at 233 49.8
x x
18 97
-
0.868.=
0.412 ~
x,
-I
. , . x,
120' =
1.90.
.
O n the Polymerization, of Liquids
67 7
I n this case, since the vapor was already I . 6 X 60 and underwent a further increase of I .9, the total increase or actual liquid molecule is I . 6 X I . 9 = 3 . I . The following table gives data and values of
e for
all
W L
the substances for which I could obtain the necessary experimental data: Calories
Temp.
C
Benzene Carbon tetrachloride Carbon tetrachloride Carbon bisulphide Ethyl ether Methyl formate Methyl propionate Ethyl acetate Propyl acetate Methyl alcohol Ethyl alcohol Propyl alcohol Acetic, acid Water Water
80 20
77 20
w
L
93.5 5 1 .I 44.3 88.0 90.0
35 3 1 . 8 110.0 89.0 79.7 92.7 77.0 83.2 140.0 6 6 . 0 262.0 78.4 216.4 9 7 . 4 166.3 118.0 9 7 . 0 0 606.5 100.0 5 3 7 . 0
H
51.3 42.2 77.4 22.5 '53.7 28.6 25. I 1 9 . 2 '53 7 52.9 35.1 75 5 45 7 73.6 44.3 6 0 , 4 49.6 59t 6 44.3 44.7 87.4 48.8 43.9 87.4 3 9 . 4 43 8 1 0 1 . 3 1 3 4 , o 128.0 3 1 . 8 91.5 124.9 4 6 . 0 71.8 60.0 94.5 47.2 363.0 300.0 ' '
'
'
1.51 1.61 I .42 I .40 1.59 I .52
1.66 I 75 I , 71 I . 78 2.48 2.35 1 ,905 '
2.11
1
'
75
Conclusion Particular attention is called t o the functional part played by polymerization in maintaining equilibrium stability. The current impression t h a t there exist immense internal pressures in liquids may conveniently be abandoned. They are dynamically impossible and otherwise serve no useful purpose. Unless we abandon entirely the Newtonian mechanics and the principle of d'Alembert the inwardly directed pressure cannot exceed the outwardly directed reaction, in a system of freely moving particles. A very obvious objection that will no doubt'be made a t once by some physical chemists is, t h a t the assumption of
M . M.. Garvey
678
the perfect gas law as the law of force is unverifiable by experiment and is “ too theoretical.” The first reply t o this objection should be that “ the perfect gas law” actually assumed is entirely different from assuming that a given substance acts like a perfect gas or that it may be treated as a perfect gas because the internal attractions are negligible. The fundamental assumption that i s made is one that is supported by the universally accepted principles of dynamics; and that is that the force of attraction which diminishes a pressure must be numerically equal, but opposite in sign, t o the diminution in pressure produced by it. The direction of the action can have no influence on the magnitude of the work performed by it. So far as work is concerned it is immaterial whether it be done by a push or a pull or by a combined push and pull; and if the only effect of a pull is to diminish the push required and we know how muc% push would be required if there were no pull, then we may assume all push or all pull at pleasure without affecting the computed amount of the work due t o the forces concerned. The heat equivalent of work done by forces i s entirely independent of the nature of the body upon which the work i s done. A second reply might be: Sippose we admit that the fundamental assumptions are erroneous. Then the difficulty would be t o explain the remarkabk consistency of the results obtained and the practical agreement in many ways with the best results obtained by other methods. The values obtained for water are of course only a first approximation, b u t the relative values obtained also depend upon the theoretically determined heat of polymerization. This is the fundamental basis of the whole theory; and the remarkable consistency of the results obtained would be even more wonderful considered as a system of accidentally balanced errors than as the “workings of a law of Nature.” State College, Pa.,
June,
1912
