REINOW. HAKALA
1880
The Law of Corresponding States in Its Most General Form
by Reino W. Hakala Department of Chemistry and Chemical Engineering and Department of Mathematics, Michigan Technological University, Houghton, Michigan 49931 (Received November 16, 1966)
It is shown that any two-body intermolecular potential function can be put into a dimensionless form involving only one energy parameter, eo, and only one length parameter, rot in addition to any dimensionless intermolecular potential function-shape parameters C1,Cz, . . . and any dimensionless electrostatic parameters p2/e0~O3,Q2/eor$, . . ., where p , &, . . . are dipole, quadrupole, . . . moments. Use is made of this result to derive the most general form of the law of corresponding states for PVT behavior at moderate densities by means of a completely general and systematic dimensional analysis procedure. The result, readily transferable to any equilibrium or transport bulk property in reduced form, is Pv/nRT = f(v/ro3, kT/eo, C1,C2, . . ., a/ro3,p2/eoroa,Q2/eoro6,. . ., M , A*) where A* is a new quantum-deviation parameter, combining into a single variable the quantum deviations from the classical law of corresponding states due to translation, rotation, vibration, and electronic transitions, the Ci’s are intermolecular potential function-shape factors, and the , w1,102, . . . , other symbols have their usual meanings. Alternatively, PV/RT = f [ V / V c T/T,, a N / V C p2N/VckTc, , Qz(N/V,)”/’/kTc,. . . , M , A(’)] where A@) is another general quantumdeviation parameter related to A*, wl,w2, . . . are molecular geometry parameters, and the other symbols are standard. Inclusion of the molecular weight, M ,is considered here for the first time.
Introduction The law of corresponding states has been extended considerably since it was introduced by van der Waals. Parameters are now included to account for molecular geometry and polarity and for a translational quantum effect. Because the law of corresponding states is an essentially universal relationship among dimensionless variables, it is natural to examine it from the viewpoint of dimensionless analysis. This has been done in the past, but not completely systematically and not including all of the important variables. We shall present a fully systematic dimensional analysis of PVT behavior accounting for all of the known variables. The results can readily be transferred to any bulk property.
potential function point of view have been restricted to two-parameter, two-body intermolecular potential functions. This restriction to two-parameter functions is not necessary because any two-body intermolecular potential function, no matter how many parameters it contains, is reducible to a dimensionless form involving only one energy parameter, eo, and only one length parameter, r0, the remaining parameters all being dimensionless. This important fact, apparently not taken note of previously, will now be proved. (Multibody intermolecular potential functions can be treated the same way; more energy and length parameters will then be involved.) According to the fundamental theorem of dimensional analysis,’ a necessary and sufficient condition
General Form of the Intermolecular Potential Function In order that our treatment be fundamental, we shall carry out the dimensional analysis in the light of the two-body intermolecular potential function. Other discussions of the law of corresponding states from the
(1) This is traditionally referred t o as “Buckingham’s theorem,” though his statement of it (E. Buckingham. Phus. Rev., 4, 354 (1914)) lacks generality. For completely general proofs, see H. L. Langhaar, J. Franklin Inst., 242, 459 (1946); H. L. Langhaar, “Dimensional Analysis and the Theory of Models,” John Wiley and Sons, Inc., New York, N. Y., 1961; E. R. Van Driest, J . A p p l . Mech., 13, A-34 (1946) ; G. Birkhoff, “Hydrodynamics,” Princeton University Press, Princeton, N. J., 1950; R. W. Hakala, J . Chem. Educ., 41, 380 (1964) (elementary proof).
The Journal of Physical Chemistry
THELAWOF CORRESPONDING STATESIN ITSMOSTGENERAL FORM
that an equation be dimensionally homogeneous is that it be reducibje to a relationship among dimensionless power products. Any intermolecular potential function is dimensionally homogeneous, since each term has the dimensions of energy. Dividing through by eo evidently renders each term dimensionless. According to the fundamental theorem, the resulting equation is such that, or it can be reduced further so that, the intermolecular separation, T , always occurs in the form of a dimensionless power product arb, where the power b is a positive or negative pure number and the factor a necessarily possesses the same dimensions as T - ~ ,in whatever types of terms (linear, exponential, etc.) r occurs. Let us now multiply a by rob and divide rb by rob. The general result is (mob)(r/rOb). The power product(s) arObis (are) evidently dimensionless. This completes the proof that only one energy parameter and only one length parameter are required in any intermolecular potential function. We shall now inquire into the general form of the intermolecular potential function. In the nonelectrostatic part of the intermolecular potential function, the dimensionless power products arobare obviously related to the shape of the potential function. We shall therefore call the factors arob the "intermolecular potential function-shape parameters" and shall denote them by C1, CP, . . . . Each of the electrostatic terms, if any, of the original nonreduced potential function is of either of the two general forms
{ c [p(2")]2/~2~ +l] F { d p ( z w p c z n ) / r m +* +1 I G where c, d, ni, and n, are pure numbers, p@") and p(2n) are 2"- and 2"-pole moments, and F and G are the dimensionless angular dependences, including the signs, of the corresponding electrostatic interactions between two molecules. Reduction of these terms to dimensionless form is a simple matter. Proceeding as before, the results are
{ c [ p ( 2 n )]2/eor02n
*l
1(ro/r) +lF 2n
and
{ d p ( z m ) p ( 2 n ) / e o r o m +*+I 1 ( ~ ~ / r ) ~ + " + l G Thus, for the electrostatic part of the intermolecular potential function, the dimensionless factors arObare identified with
{ c~p(2n)]2/eOr02n+1} F and
{ dp(2m)p(2n)/e,,Tom+n
+I
JG
1881
Therefore, completely generally, the mutual potential energy of two molecules is given by the dimensionless expression @(r; OA,
OB, 4 A , $B,
.
9
.)/e0
=
. . . ; [p(zrn)]2/eorom+1, [p(2n)]z/eO~02n+1, . . . ; OA, OB, @A,
j { r/ro;
~ 1 cZ,,
$B,
.. ]
OB, +A, $B, . . . are the angles involved in F , G, . . . . It is not necessary to include any multipole-
where OA,
moment-product parameters p ( 2 m ) p ( 2 n ) / e o r o m + n + 1 among the variables in the general form of the intermolecular potential function because they are the geometric means of the multipole-moment-square parameters [ p ( 2 m ) ] 2 / e O ~ 0 2 m + n + 1 and [p(2n)]2/eOr02n+1 and are therefore not independent. It should be noted that the physical meanings of eo and ro (the depth of the potential well, the value of r for which @ = 0, etc.) will generally differ from one type of potential function to another. By way of example, we shall superimpose point dipole-dipole, dipole-quadrupole, and quadrupolequadrupole interaction terms on a BuckinghamCorner potential function. The resulting potential function can be written in the dimensionless form
@(r;OAj
b
4-4, 4B)/e0 exp( - CIr/rO) - [ C & O / ~ ) ~ C~(~O/T X) exp[--.l(ro/r ~I - l ) a ]j
+
+ (3cc&/4eor0~)(~o/r)~G(~~, OB, 4 ~4, ~ + ) ( c c 2 / e o r o 3 ) > ~ O / ~ ) a F ( ~ AOB, ~ ~ A @B) J
( 3 & 2 / 1 6 e O ~ 0 6 ) ( ~ O / ~ ) 6 ~OB, (~A $A, ,
4B)
Dimensional Analysis of PVT Behavior Besides the parameters eo, TO, CI, Cz, . . ., and the various multipole moments (if not zero), other important molecular parameters that we shall take into account are the polarizability, a ; the molecular mass, m; the moment of inertia, I , in the case of linear molecules, or, for nonlinear molecules, the product, ABC, of the principal moments of inertia; the various vibrational frequencies, VI, v2, . . . ; and the various electronic energy levels, €1, €2, . . . . Since translational, rotational, vibrational, and electronic energies are all quantized, Planck's constant, h, must also be included as a variable. Although the law of corresponding states applies to any bulk property of the system, for both equilibrium and transport phenomena, we shall find it convenient to center our attention on PVT relationships. Thus, we shall also include among our variables the pressure, volume, and absolute temperature of the Volume 71, Number 6 May lBB7
REINOW. HAKALA
1882
ables of the above matrix, M1/‘L’/aT-l. Therefore, [ p ] = M1’aL”/’T-l,[&I = M’/’L7’2T-11 and similarly for higher multipole moments. The dimensional formulas of the remaining variables are all obvious. A corollary of the fundamental theorem of dimensional analysis is that the number of independent dimensionless power products in a complete set is equal to the total number of variables (listed at the top of the dimensional matrix) minus the rank of the dimensional matrix.2 The rank of a matrix is defined as the order of the greatest order nonzero determinant of a square submatrix obtained by deleting columns and,
system. As any one of these depends on the other two and on the quantity of material present in the system, the number of moles, n, likewise must be included. If we were to carry out a theoretical treatment of the problem, the temperature would enter solely as the product kT with Boltzmann’s constant, and Avogadro’s number, N , would be involved; these will therefore be included among our variables. Denoting the dimensional variables mass, length, and time, as is customary, by the respective symbols, M , L , and T , the dimensional matrix of the variables we shall consider is
P M L
T
v
kT
n
N
eo
ro
C c u
p2
1 2 -2
1 0 0
-1 0
1 2 -2
0 1 0
0
1 5 -2
1 0 - 1 3 -2 0
0
For simplicity, instead of listing CI, CZ, . . . , VI, v2, . . . , and el, ezl . . . , only C, v, and e are given and I is used to represent either I or (ABC)’/’, whichever applies. Each column of this matrix represents a dimensional equation, such as [PI = ML-’TF2 for the first column on the left. It may be helpful to indicate how the dimensional formulas of certain ones of the variables were determined. One might think, offhand, that the number of moles of a substance is a dimensionless quantity, but further consideration shows that such is not the case. The number of moles is calculated by dividing the number of grams of a substance by its molecular weight, whence the dimensions of the number of moles are mass divided by the dimensions of molecular weight. The latter is a relative mass (currently, 12 times the mass of a given molecule divided by the mass of one carbon 12 atom) and is therefore dimensionless. As a consequence, we have the dimensional relationship [n] = M . Avogadro’s number, the number of molecules per mole, then has the dimensions M-l. (We might remark that molecular weight has units, grams per mole, although it has no dimensions. This seeming paradox, stemming from a confusion between units and dimensions, has caused much needless trouble in dimensional analysis.) The dimensions of the dipole and quadrupole moments depend on the dimensions of electric charge. The latter is usually taken to be a fundamental dimensional quantity but shall be regarded here as a derived physical quantity. According to Coulomb’s law, assuming the proportionality constant to be dimensionless, the dimensions of electric charge are the dimensions of distance times the square root of the force or, in terms of the dimensional variThe Journal of Phyaical Chemistry
0
0
0 3 0
Q 2 m I 1 7
-2
1 0 0
1 2 0
v
0 0 -1
h
e
1 2 -2
1 2 -1
if necessary, rows from the matrix. The rank of the above dimensional matrix is 3 because, for example, the determinant of the second, third, and fourth columns does not vanish (even though those of the first and last three columns and of various other columns vanish). Therefore there are 12 independent dimensionless power products of the variables in a complete set, not counting C which is dimensionless to begin with. There are, of course, any number of complete sets from which we are to choose one particular complete set. This choice is best dictated by physical considerations. With as many variables as we have decided to relate to one another, it would be difficult to find a complete set of independent dimensionless power products without some kind of systematic procedure. The only fully systematic procedure in the previous literature appears to be that of LanghaarJ1q3but it is not suitable in the present application. The simplest and most productive way to proceed is first to consider the smallest possible physically significant submatrix and then to increase the number of columns of the sub-
(2) H. L. Langhaar (see ref 1 ) . We shall refer to this corollary as Langhaar’s theorem. It is completely general whereas the corresponding much-quoted principle of Buckingham, which substitutes the number of dimensional variables involved for the rank of the dimensional matrix, lacks complete generality, as was first pointed out by P. W. Bridgman, “Dimensional Analysis,” Yale University Press, New Haven, Conn., 1922. (3) If the total number of variables is u and the rank of the dimensional matrix is r , then each of (v - r ) of the variables is combined into a dimensionless power product with the same remaining r variables. These ( v - r ) power products are independent because the rank of the matrix of the solutions is necessarily equal to the number of rows in the matrix, which is ( v - r ) . Langhaar’s procedure, for which he has developed an automatic numerical scheme, therefore always generates a complete set.
THELAWOF CORRESPONDING STATES IN ITSMOSTGENERAL FORM
matrix, by one column each time, until all of the variables are accounted for. The smallest physically significant submatrix consists of the first five columns of the dimensional matrix, which contain only general physical variables as opposed to molecular variables (except for Planck’s constant which, however, we do not need to consider until we take into account the variables m,I, Y , and E ) . Langhaar’s theorem indicates that two independent dimensionless power products are related to one another. One of these is evidently nN, the total number of molecules in the system. Both kT and Pv have the diinensions of energy, whence Pv/kT is the other dimensionless power product that is sought. The combination NkT occurs in theoretical studies. Hence we shall form the combination PvlnNkT. This is the same as P V / R T , the compressibility factor. Had we a t the outset combined N and k to give R , there would have been only four variables and therefore only one possible dimensionless power product, P V / R T (or any power thereof), which must be equal to a universal constant if no other variables are important. We have thus derived the ideal gas law, except that the actual value of the universal constant (= 1) is not given by dimensional analysis. According to statistical mechanics, the pressure of a system is given by
where QN is the total molar partition function. Assuming that
E
=
Etr
-I- E r o t
Evib
-I- E e l
+
Epotential
which is adequate a t moderate densities, whence QN
= (QN) tr(&N)
rot(Qhr)vib (QN)el (&N)potential
we find that
since only the translational and potential partition functions depend on the volume. Consequently, deviations from the ideal gas law depend on the parrtmeters of the intermolecular potential function. Hence we shall next consider combinations of P , v, and kT with these parameters. At first, we shall disregard any electrostatic interactions. The only possible dimensionless combinations of eo and ro with P , v, and kT taken separately are Proa/ eo, v/rO3,and kT/eo. Only two of these are independent, however, because the product of the first two divided by the third equals Pv/kT. A convenient pair to take
1883
for later discussion is v/rO3and kT/eo,although any two can be selected. The value of P V / R T will also depend on the values of any intermolecular potential function-shape factors, CI, CZ,. . . . Thus far, we have derived the relationship4 P V / R T = f(v/roa, kT/eo, CI, Cz,. . .) which must be a universal function, according to dimensional analysis, if no other variables are important. If we omit C1, Cz,. . . , as is customary, then the result can apply only to substances whose potential functions are of the same type6and the same shape. Since roa and eo/k are proportional, respectively, to the critical molar volume, Vc, and the critical temperature, T,, it is permissible to make these substitutions in the above relationship. As the proportionality constants depend on the molecular geometry, the latter is also a ~ a r i a b l e . ~We can take it into account by replacing C1, CZ,. . . by the molecular shape factors w1,wz,. . . Thus, we have
.
P V / R T = f ( V / V c ,T / T c ,W,
WZ,
. . .)
In general, there will be more fhan me molecular shape factor. If the dependence of P V / R T on molecular geometry is neglected, this result retrogresses to the primitive form of the law of corresponding states. Commonly, for experimental convenience, V / V , is also replaced by P / P c , in analogy to the permissible substitution of v/vo3 by Pro3/eo,but the analogy is not exact. The difficulty is that Pro3/eois directly proportional to PV,/NkT,, which is not directly proportional to P / P c since PcVc/NkT,is not a function of the molecular geometry alone. Thus, the fundamental critical constants in the law of corresponding states are T , and V,, not T , and Po, and the proper form for the reduced pressure is PV,/RT,, not P / P c , an important feature of the law of corresponding states which appears to have been overlooked previously. Since our study has thus far not taken into consideration any electrostatic interactions, it is necessarily restricted to substances of which the molecules possess identical values of the polarizability and their multipole moments in reduced form. There is no reason, however, to restrict the primitive form of the law of corresponding states to only nonpolar substances, as is often done. All that is required is that they be of equal, or nearly equal, reduced polarity, as well as of equal reduced polarizability. (The latter condition is more readily met.) We shall now find out how to compare substances of different polarizabilities and polarities. (4) Omitting CI, Co, . . . , this was first derived by J. de Boer, Doctoral Dissertation, University of Amsterdam, The Netherlands, 1940; PhYShl, 14, 139 (1948). (5) K. 8. Pitser, J . Am. Chem. SOC.,77, 3427 (1955),reached the same conclusion but in a different way.
Volume 71, Number 6 M a y 1967
REINOW. HAKALA
1884
We extend our submatrix further to include the a, and Q 2 columns. Langhaar's theorem indicates that we should thereby obtain three new dimensionless power products, one for each new variable introduced. The most convenient combinations are with the intermolecular potential function parameters ro and eo: a/ro3,p2/eoro3, and Q2/eOr05. As these are mathematically independent, our subset of dimensionless power products is complete. An infinite number of complete subsets is possible; we have selected the most convenient one, e.g., we could have included the dipole-quadrupole interaction parameter pQ/eoro4 in place of either the dipole-dipole or quadrupole-quadrupole interaction parameter, but the last two are more convenient and inclusion of both in t h e subset also accounts for dipole-quadrupole interactions. The dimensionless power product p2/eoro3 is well known as a correlation parameter for polar substances, whereas the quadrupole-quadrupole interaction parameter is not. Higher multipole interactions can be included by inserting the corresponding higher multiple moment columns into the dimensional matrix. The corresponding dimensionless power products with ro and eo are very easy to obtain. According to the statistical mechanical formula given earlier for the compressibility factor, we have completed our task. But quantum mechanical effects have not been accounted for because we have not yet considered Planck's constant as a variable. The variables m, I , v , and E are all related to h, as has already been indicated. We must therefore find dimensionless combinations of each of these variables with h wherever possible and eo and ro where required. These dimensionless power products are readily found to be6 h2/meoro2, hL/Ieo (or h L / ( A B C ) 1 / 3 e ohv/eo, ), and €/eo. It turns out that, even though electronic energy is quantized, h does not appear in the simplest possible dimensionless power product for E . There will, of course, be a separate combination hv/eo and € / e o for each vibrational and electronic energy level. If we replace ro by (Vc/N)''alwhich has the same dimensions as rot and eo by kTc in the quantum-deviation parameters for the molecular mass and the moment(s) of inertia, we obtain parameters which are evidently directly related to the translational and rotational molecular partition functions at the critical point' p2,
Qt,(.)/nN = ( V , / N )(2smkT,)'/'/h3
(linear) (nonlinear)
=
=
8n21kTc/ah2
(8?r2kTC)"'(~ABC)'/'//ah3
The quantum-deviation parameter for vibration beThe Journal of Physical Chemistry
comes h v / k T c which is the exponent in the vibrational molecular partition function at the critical point Qvib(')(permode of vibration) =
[l - exp(-hv/lcT,)]-' Finally, €/eo becomes c / k T c which is the exponent in the electronic partition function at the critical point &el(@)
= Cgi exp(
- Ei/kTc)
Since all bulk properties of a given system are functions of its total partition function, which is the product of the partition functions corresponding to the various kinds of energy (assuming the energies to be additive), it follows that the various quantum-deviation parameters can be combined into a single quantumdeviation parameter A") = nN/(&trQr,tQ,ln&vib)'
where the product l l Q v i b is over all modes of vibration. The reciprocal of the partition function product at the critical point is used, rather than the partition function product itself, because h then occurs in the numerator whence the smaller the quantum-deviation parameter, the smaller is the quantum effect. If the intermolecular potential function parameters ro and eo are used in and k T c in A(@),then, by analogy to place of (VC/N)'Ia de Boer's translational quantum-deviation parameter which is denoted by A*, we shall employ the notation A*. We have found a total of eleven independent dimensionless power products (nN, P v / k T , v / r O 3k, T / e o , a / r O 3 , p2/eoro3, Q2/eoro6, h2/meorO2,h2/Ieo, hv/eo, and € / e o ) whereas Langhaar's theorem indicates that there must be altogether 12 (15 variables, not including C which is dimensionless to begin with, minus the rank, 3, of the dimensional matrix). In our systematic procedure for generating physically significant independent dimensionless power products, we divided the variables into four classes: P , v, k T , n, and N ; eo and ro; ~
~~
(6) C. S. Wang Chang, Doctoral Dissertation, University of Michigan, 1944, showed that for linear rigid rotors there are quantum deviations from the second virial coefficient which depend upon h*/m and h*/I, in agreement with the above results. The complete translational parameter h/(meo)'/% was later found and was applied successfully by deBoer and his co-workers: J. deBoer and B. S. Blaisse, Physka, 14, 149 (1948); J. deBoer and R. J. Lunbeck, ibid., 14, 520 (1948); R. J. Lunbeck, Doctoral Dissertation, University of Amsterdam, 1951. Pitzer6 also obtained the complete translational parameter. The remaining two parameters have not been considered previously. (7) The quantum-deviation parameter h ( N / V,)'!a/(rnkTc)'/n,which we see is equal to (2*)'/2[nN/&tr(0)]'/a, was introduced, with the aid of dimensional analysis, by A. Byk, Ann. Phys., 42, 1417 (1913); 66, 157 (1921); 69, 161 (1922);Physik.Z., 22,15 (1921); Z.Phy8ik. Chem. (Leipaig), 110, 291 (1924). Byk was the first author to consider quantum deviations from the classical law of corresponding states.
THELAWOF CORRESPONDING STATES IN ITSMOSTGENERAL FORM
. . . ; and m, I, v, e, and h. We related the variables in the first class to one another and to those in the second class and those in the second class to those in the third and fourth classes, but we did not relate any of the variables in the first class to those in the third and fourth classes, whence this remains to be done. One such simple conibination is Nm, the molecular weight. This has not been suggested previously as a parameter in the law of corresponding states. As it is independent of the other dimensionless power products, its inclusion completes the set. We shall now check whether the twelve dimensionless power products we have found are indeed independent. We shall employ the theorem from the theory of equations according to which the number of independent equations in a system of linear equations is equal to the rank of the matrix of the coefficients. (Langhaar’s theorem is also a corollary of this one.) In the present application, the rank of the dimensional matrix of the 12 dimensionless power products will be 12 if the power products are independent. The matrix of the set of dimensionless power products that we have chosen is
pz, Q z ,
P 0 1 0 0 0 0 0 0 0 0 0 0
v 0 1 1 0 0 0 0 0 0 0 0 0
I
c -
T 0 1 0 1 0 0 0 0 0 0 0 0
n 1 0 0 0 0 0 0 0 0 0 0 0
N 1 0 0 0 0 0 0 0 0 0 0 1
eo
0 0
0 -1
0 -1 -1 -1 -1 -1 -1 0
(It should be noted that the roles of the dependent and independent dimensional variables are reversed in the two matrices.) If a 12 X 12 submatrix with a nonvanishing determinant occurs in this matrix (which contains altogether 15!/12! 3!, or 455, different 12 X 12 submatrices), then the rank of the matrix is 12, whence each member of the chosen set of 12 dimensionless power products is independent and the set is complete. The sparsest 12 X 12 submatrix (obtained by deleting the eo, To, and h colunns) can be partitioned into upper pseudo-triangular form, and the two subdeterminants lying along the principal diagonal of this submatrix are readily reduced to lower order and diagonal form by Laplace’s development. In this way,
1885
the value of the determinant of the sparsest submatrix was very quickly found to equal unity, thereby proving that our set is complete.
Conclusion We have found by means of dimensional analysis that the most general possible form of the law of corresponding states for P V T behavior a t all moderate densities is
P V / R T = f(v/ro3,kT/eo, kT/eo, CI, CZ,. . ., a/ro3,~ . c ~ / e oQ2/eor8, ~ o ~ , . . . , M , A*)
which includes several dimensionless power products not considered previously, including the molecular weight, M , and the new quantum-deviation parameter, A*. The latter is obtained from the reciprocal of the total molecular partition function a t the critical point by multiplying the reciprocal by n N , the product of the number of moles of substance present in the system and Avogadro’s number, and then replacing ( VC/N)”’and kT,, respectively, by the intermolecular potential function parameters and eo. The parameter A* accounts for translational, rotational, vibra70
(Y
0 0 -3 0 -3 -3 -5 -2 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0
p2
Qz
0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0
m
0 0 0 0 0 0 0 1 0 0 0 1
I
v
e
h
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 2 2 1 0 0
-
1 0 0 0
tional, and electronic quantum effects; previous quantum-deviation parameters are special cases of this one. In terms of the critical constants, this result can be written in the form
P V / R T = f [ V / V , , T/Tc, wi,W Z ,. . ., a N / V C ,pzN/VokTc,Q2(N/Vc)s’a/kTc, . . . , M , A(c)] where w1,wz, . . . are molecular geometry parameters, of which there may be more than one, A(c) is the quantum-deviation parameter A* with ro and eo replaced by their dimensional equivalents (V0/N)”’ and kT,, and the other symbols have their usual meanings. Volume 71, Number 6 M a y 1967
NORIO ISEAND TSUNEO OKUBO
1886
The parameter v/rO3 can be replaced by Proa/eo and correspondingly, V / V , by PV,/RT,, but not, without some loss of accuracy, by P/Polcontrary to current practice. These results may be carried over readily to any
bulk property which has been put into the appropriate reduced form. Acknowledgment. The author is grateful to the Faculty Research Committee of Michigan Technological University for financial assistance.
Mean Activity Coefficient of Polyelectrolytes.
V.
Measurements of
Polyvinyl Sulfates of Various Gegenions
by Norio Ise and Tsuneo Okubo Department of Polymer Chemistry, Kyoto University, Kyoto, Japan
(Received November 17, 1966)
The mean activity coefficients of various salts of polyvinyl sulfuric acid (PVAS) have been determined by the isopiestic vapor pressure measurements. It has been found that the logarithm of the mean activity coefficient decreases linearly with the cube root of polymer concentration up to 0.5 equiv/1000 g of water. The slopes of the cube-root plots are -0.60, -0.65, -0.97, - 1.31, and - 1.52 for lithium, sodium, potassium, calcium, and barium salts, respectively. This order of the slope is in accord with what is expected from Gurney’s rule. The magnitude of the slope for the sodium salt is found to be smaller than that of sodium polycarboxylate. This is accounted for in terms of the difference in the structural effects of sulfate ion and carboxylate ion.
Introduction In previous papers, the mean activity coefficients of polyelectrolytes have been determined by the emf measurements of a concentration cell with transfere n ~ e l - and ~ the isopiestic vapor pressure measurem e n t ~ . The ~ results showed that first, the logarithm of the mean activity coefficient decreased linearly with the cube root of polymer concentration (“cube-root” rule). The magnitude of the slope of the cube-root plot increased with increasing charge density and decreased with increasing degree of polymerization of macroion. This cube-root rule suggested the existence of a “linkage” between macroions through the intermediary of gegenions. Second, it was found that the mean activity coefficients of polyelectrolytes could not be equal to the single-ion activity coefficients of its gegenions and the discrepancy between the two The Journal of Phyeical Chemistry
coefficients could be small for the low molecular weight electrolytes. In the present paper, experimental data are reported on various salts of a polyvinyl sulfuric acid (PVAS). The isopiestic vapor pressure measurements were carried out in order to determine the osmotic and mean activity coefficients in a comparatively concentrated region of polymer concentration. One purpose of the present research is to see whether the cuberoot rule holds for the sulfates and to study the specific gegenion effects on the osmotic and mean activity coefficients. Gegenion effects are expected to arise (1) N.Ise and (2) N. Ise and (3) N. Ise and (4) N. Ise and
T. Okubo, J . Phys. Chem., 69, 4102 (1965).
T.Okubo, ibid., 70, 1930 (1966). T.Okubo, ibid., 70, 2400 (1966). T. Okubo, ibid., 71, 1287 (1967).
