A new hypothesis for molecular behavior

J. theor. Biol. (1981) 93, 829-853

A New Hypothesis for Molecular

Behavior

PETER A. H. MATTSON Department of Botany, University of Minnesota, 220 Biological Sciences Center, 1445 Gortner Avenue, St Paul, Minnesota 55108, U.S.A. (Received

14 February

1979, and in revised form 17 April

1981)

In order to understand the living cell, biologists need workable explanations of molecular phenomena. From data in the literature, it is possible to construct a mathematical hypothesis: molecular behavior in the continuum can be represented by a set of simple, logarithmic functions. These functions apply not only to biology, but also to chemistry and to physics. Although empirical in its origins, this hypothesis satisfies criteria which are often associated with good theories. It is comprehensive in its application, and it is mathematically simple. It is easily tested; and it is accurate, when tested. Because it can be correlated with classical atomic and molecular theory, the hypothesis can be used to make strong predictions. The proposal also appears to be consistent with quantum mechanics.

1. Empiricism Consider a single, isolated, molecular phenomenon in a homogeneous, isotropic medium. Examine a quasi-steady state in this medium with appropriate, intensive variables of state 2. Examples of appropriate variables are absolute temperature T, absolute pressure P, concentration p and rate of reaction dp/dt. Because a choice may not always be as obvious as

absolute temperature, “appropriate” variables must be selected carefully. Assign these variables, as needed, to an independent, intensive variable of state X and a dependent, intensive variable of state Y. A plot of experimental data for X and Y which meets the stated conditions will describe a continuous, single-valued curve in X-Y space. Moreover, a complete description with a plotted curve will show either a maximum, a minimum, or a point of inflection in this X-Y space. Find initial values X0 and Y0 at the maximum, minimum or point of inflection; and take reduced values X, =X/X0 and Y, = Y/Y0 for the curve. These reduced values are dimensionless, and they simplify the problem of quantification. Dimensional values X and Y can be recovered easily, if needed. 829

0022-5193/81/240829+25

$02.00/O

@ 1981 Academic

Press Inc. (London)

Ltd.

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Transformations of X, and Y, are sought next, such that curves of this type exhibit properties which are invariant. There is no particular reason to treat X differently from Y. Therefore, both the transformations and their associated parameters are assumed to be mathematically symmetrical for X and Y. This symmetry introduces an ambiguity into the search for a simple description of a molecular phenomenon. Any specific description under this ambiguity must necessarily be a model. The needed transformations of X, and Y, are simply their natural logarithms. For each phenomenon, the resulting curve exhibits a number of invariances. Specifically, the curvature is constant over the entire domain; and the coefficients are constant within the subdomains to either side of X0. Examples of these invariances will be shown later in this paper. This means that such a curve in each subdomain of In-ln space is describable by a symmetric, algebraic expression of the simplest possible form. This empirical generalization can be expressed mathematically as (*C In Yr)” = (-A In X,)“,

O
(*D In Y,)” = (B In X,)“,

l
(1)

where positive m and n are constant, as are positive A, B, C and D. The signs correspond to those of the Cartesian quadrants in a graph of In X, vs. In Y,. Each of the equations of function (1) is part of a generalized parabola. One equation is specified for each subdomain of this function. This is desirable, because the coefficients of the function are not always constant over the entire domain. Occasionally, a shift in the values of a coefficient occurs at (X0, YO). In other words, function (1) is non-analytic. It is conceivable that the X0 or Y0 values might display a shift between subdomains. However, in the examples studied, no convincing evidence has been found for these possibilities; and the initial values X0 and Y0 are assumed to be the same for both subdomains. (For computational purposes, one of the boundaries must remain open. Here, X >XO for X, > 1.) For the independent variable X, both a given variable and its inverse are correct representations, because In (l/X) is simply -In X. With multiple phenomena, this is not true for the dependent variable Y. Note that each term of function (1) can be generalized as a function of 2,: F(Z,) = (*K In Z,)“,

o
(2)

Because equations (1) meet the general requirement for mathematical symmetry, there are too many parameters for the function to be defined

MOLECULAR

unambiguously. by a model.

Additional

information

2. Constructing

BEHAVIOR

831

is needed, and it must be provided

a Hypothesis

For stoichiometric, chemical phenomena, one invariance is highly specific. Ratios m/n of the exponents are found to be integer: usually integer two, sometimes integer three. That is, for reasonably good data, the continuum of values yields the m/n ratios of ~.OXX, 2.00~~ and so on. This fact implies that a macroscopic pair of measures, which are intensive variables of state, are monitoring some countable entity or entities. In molecular science, the only available entities which have countable values are electrons, molecules, molecular events, and the like. A molecular event is defined here as a transient set of interacting molecules. Quantum mechanics is set aside, for the time being, in favor of the simpler viewpoint of classical atomic and molecular theory. The atoms, chemical bonds and other aspects of a molecular event are supposed to be countable as 1,2,3,. . . in the conventional chemical formulations. Assume that the m/n ratios of 2*00,3*00, etc. correspond to the number of molecules which are involved in the molecular event. This implies that an intensive variable of state is referenced to a molecular event. In practice, of course, a number of similar events must be observed by such a measurement; the parameters associated with such data then take on average values. Therefore, as a working hypothesis, suppose that an intensive variable of state monitors an average, molecular event. Depending upon the species of event, chemical change may or may not occur. A monitoring of multiple species of events is also possible. This concept is consistent both with classical thermodynamics and with molecular science. Classically, an intensive variable of state is measured at a point in a structureless substance (Daniels & Alberty, 1975). However, if the existence of molecules is assumed and if the consistency of definition is maintained, then, in the limit, each such measurement must be the least description of a total phenomenon which retains the properties of that phenomenon. For a single, molecular phenomenon, this description is assumed to be, in the limit, an average, molecular event. Yet this simple concept differs from prevalent opinion in molecular science. In statistical mechanics, emphasis is on ensembles of particles, not individual molecular events (Furth, 1970). Even in the quantum mechanical realm, in which there is an explicit recognition of individual events, some idealized model or limiting state is often assumed. This can be a photon gas, an electron gas, a scattering cross section, and so on (Heer, 1972~).

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For example, in scattering theory, asymptotic behavior is studied, because “ . . . the region of interaction is certainly no larger than a few atomic diameters and so is, in practice, completely unobservable” (Taylor, 1972). In such a climate of opinion, it is understandable that the molecular event itself has had no counterpart in a specific, mathematical structure. Despite this, the m/n ratios are awkward to explain in any other way, except by referencing them to the molecular event itself. The countability 1,2,3, . . . of the exponents is assumed to extend to the coefficients as well. The coefficients of equations (1) pose problems in the construction of such a model; but these problems can be resolved by replacing the original coefficients A, B, C and D with constants a, b, c and d, where one of two possibilities is chosen for each coefficient: A = a,

arl;

A = l/a,

a21;

(34

B = b,

brl;

B = l/b,

brl;

(3b)

c = c,

czl;

c = l/c,

crl;

(3c)

D = d,

d>l;

D=l/d,

dsl;

(34

kzl.

(3e)

for equations (1) or, generally, for equation (2), K = k,

krl;

K = l/k,

Also, the minimum values of m and n are unity. For all possible combinations of equations (3) with equations (l), a large set of functions results: this set will not be listed here. However, for ordinary, chemical reactions of the type which are reviewed in this paper, it appears that the proper values of c, d and IZ are all the minimum value of unity. This specification immediately removes the indeterminacy of equations (l), and it is possible to write two restricted functions which are useful. For X = 7’, it is hypothesized that *ln Yr=(-a

InX,)“,

O
*ln Y, = (b In X,)“,

1
For X = P or X = p, it is hypothesized that *ln Y, = (-l/a

In X,)“,

O
*ln Y, = (l/b In X,)“, l
(5)

For some phenomena, there is an empirically observable linkage between the exponents and the coefficients. A consistent interpretation for all phenomena requires constants a, 6, c, and d of equations (3) to be expressed

MOLECULAR

more completely

833

BEHAVIOR

as

a=a+m-2, c=y+n-2,

CUZ-1; yzl;

/3?1;

b=P+m-2,

6~1;

d=6+n-2,

(6a, b) (6~

4

or, generally, for constant k of equation (3e): k=tc+p-2,

K21.

tee)

Equations (6) are mathematical analogues of Gibbs’ phase rule. The conditions under which they are found are the same as those for the phase rule: homogeneous, isotropic space, and so forth (Ferguson & Jones, 1966; Findlay, 195 1). Substituting equations (6a) and (6b) in function (4), we obtain, for X = T, *In Y,=(-(a+m-2)lnX,)“, *ln Yr=((P+m-2)lnX,)“, Substituting or X = p,

O
(7)

equations (6a) and (6b) in function (5), we obtain, for X= P *ln Yr=(--(l/((~+m-2))lnX,)“, *ln Y,=((l/(P+m-2))lnX,)“,

O
(8)

Equations (7) and (8) represent restricted forms of a more general hypothesis. By substituting equation (6e) in equations (3e) and by substituting the results in function (2), one can construct a complete set of functions for X and Y which display all possible combinations of constants a, 6, c and d. This set is hypothesized to describe all simple molecular behavior in the continuum under the specified conditions. Descriptions for multiple phenomena of number j can be obtained by solving the jth phenomenon for Yi and then adding the Yi algebraically. The mathematics is obvious, but lengthy; and it will not be listed here. 3. Examples

Sets of physiological data which have extensive range and low scatter are hard to find. For a complete review, resort must be made to chemical studies. A search of the chemical and physiological literature disclosed a variety of simple, clearly defined examples as candidates for fitting by means of a computer program. Unless mentioned otherwise, tabular values are used. Co-operation from the experimenters made an adequate survey possible. A computer program in FORTRAN IV uses some borrowed subroutines (Bevington, 1969) for the fitting. The program is designed to

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sum the Y-values of one to three phenomena, although the simple illustrations used here are best described by functions (7) and (8). The chi-square method is used for fitting in logarithmic space, but the statistical parameters are reported with reference to Y-values. An Information International FR-80 microfilm plotter generated negatives for the graphs. For single species of molecular phenomena, these graphs display logarithmic, reduced values. For compactness with clarity, the fitted curves are sometimes “stacked” by adding arbitrary constants to the logarithmic measure in Y. Values of the data which have been rejected for fitting are shown graphically as points enlarged in diameter l$ times. (A)

PRESSURE

A few examples of unimolecular reactions in a pure gas are shown in Figs l-4. Figure 1 shows the fit of function (8), with m = 2 and (Y= 6, to data for the isomerization of 3-methylcyclobutene, as affected by P (Frey & Marshall, 1965). Because of inadequate data at higher pressures, the curves beyond X0 are arbitrarily zeroed. The molecule has six carbon-tocarbon bonds (Frey, 1964). I.40

0.80

-1.60

I

I

FIG. 1. Isomerization of 3-methylcyclobutene. x, Values at 396.7 K from Frey & Marshall Marshall (1965).

I

I

Pin mm Hg. Y = rate of reaction k in sec.-‘. (1965); t, values ai 421.7 K from Frey &

MOLECULAR

835

BEHAVIOR

I.50

-0.50 P 5 -2.50

-4.50

-6.50 -12.5

-9-5

-6.5

-3.5 In

-0.5

2.5

P,

FIG. 2. Isomerization of methylisocyanide. P in mm Hg. Y = rate of reaction k in set-‘. X, Values at 472.6 K from Schneider & Rabinovitch (1962); A, Values at 503.6 K from Schneider & Rabinovitch (1962).

The treatment for the isomerization of methylisocyanide (Schneider & Rabinovitch, 1962) is similar. In Fig. 2, curves A and C represent applications of function (8), with m = 2 and LY= 4.5. If the side hydrogens are not counted, but if the carbanion is assigned a statistical weight of O-5, there are then 4.5 bonds in the main structure of this molecule (Schneider & Rabinovich, 1962). Curves B and D in Fig. 2 represent the theoretical curves of the experimenters. Of all the examples which are shown in this review, these latter two curves are the only theoretical explanations by the experimenters which result in accurate fits. The equations for curves B and D contain over 40 parameters each. Insufficient information was given for a quantitative evaluation of these equations (Schneider & Rabinovitch, 1962; Schneider, 1962). A similar application of function (8) to the isomerization of cyclopropane (Pritchard, Sowden & Trotman-Dickenson, 1953~; Pritchard, pers. commun.; Chambers & Kistiakowsky, 1934) in Fig. 3 uses values of m = 2 and

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0.:

-0.2

i-

-0

E)-

* 5 -1.5

-2 I +

-2 7

I

-9.5

-7.5

I

I

-5.5

-3.5

I

-1.5

C5

Ln P, FIG. 3. Isomerization of cyclopropane. x, Values from Chambers & Kistiakowsky Pritchard, pers. comm

P in cm Hg. Y = normalized rate of reaction k/k,. (1934); f, values from Pritchard et al. (1953a);

0.4

0

-0.4 b-5 5 -0.8

- I .2

-1.6

I

6

-4

I

I

I

-2

0

2

In P, FIG. 4. Decomposition of cyclobutane. Values from Genaux & Walters (1951); pers. comm.

P in cm Hg. +, values from

Y = rate of reaction k in set-I. X, Pritchard et al. (19536); Pritchard,

MOLECULAR

BEHAVIOR

837

(Y= 6. There are three carbon-to-carbon bonds in this molecule, which breaks one of these bonds to form propylene (Pritchard et al., 1953~). The decomposition of cyclobutane (Pritchard, Sowden & TrotmanDickenson, 19536; Pritchard, pers. commun., Genaux & Walters, 1951) in Fig. 4 is slightly more complicated. This decomposition appears to consist of two linked steps: a rupture of the cyclobutane molecules and a reorganization of the intermediate to form two molecules of ethylene. Fitting function (8) to the data in a similar manner to the previous examples gives m = 2 and (Y= 5. (B) TEMPERATURE

The effect of T on the ionization of some aliphatic acids (Harned & Ehlers, 1933~1, 6; Harned & Embree, 1934; Harned & Sutherland, 1934) is shown in Fig. 5. Here, Y is not a rate of reaction, but a dissociation constant, Also, both subdomains are fitted, with cr = p, but the function used is function (7), with m = 2 and a = p = 3 for all four curves. The data for formic acid do not agree closely with the other three sets, and this example, curve A, is assumed to have some systematic error. The other

FIG. 5. x, Values Harned & values for

Ionization of aliphatic acids. T in degrees Kelvin. Dissociation constant K in mol/l. for formic acid from Harned & Embree (1934); f, values for acetic acid from Ehlers (1933~1); A, values for propionic acid from Harned & Ehlers (19336); q, butyric acid from Harned & Sutherland (1934).

838

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MATTSON

three sets yield integer values for the coefficients, to a high degree of accuracy. In Fig. 6, the initial denaturation of ovalbumin by urea (Simpson & Kauzmann, 1953; Simpson, pers. commun.) shows a thermal minimum; but these graphical data, too, are fitted by function (7), with m = 2 and (Y= /3 = 21 f 1. The truncate nature of values of an intensive variable of state is intuitively obvious in this example. The logarithm of the maximum value of a rate of reaction cannot be infinite. 3.6

2.8

-

0.4-

-0.4 -10

1 -6

I -2

I 2 tn

I 6

7y (x10-2)

FIG. 6. Denaturation of ovalbumin by urea. T in degrees Kelvin. Y = reciprocal half-time in min-‘. X, Values from Simpson & Kauzmann (1953); Simpson, pers. comm.

(C)

CONCENTRATION

Function (8) is applicable to single, molecular phenomena which are affected by concentration p (or its inverse, specific volume v’>. The effects of hydrogen ion concentrate provide some good examples. For instance, the hydrolysis of salicyl phosphate (Hofstetter et al., 1962) is shown in Fig. 7. The fitted curve of function (8), with m = 2 and (Y= p =4-O for the initial fit, is simple and symmetrical, like those for Figs 5 and 6. However, the mathematically non-analytic nature of simple, molecular phenomena is clearly shown in Fig. 8, which displays the changes in the activity of p-amylase in the presence of progressively different buffersystems (Ballou, 1940; Ballou & Luck, 1941). Each set of data requires a

MOLECULAR

839

BEHAVIOR

0.4

C

-0.4 * 5 -0.0

-I

2

I

\ I

- I.6 -5

I

-3

-1

I

I

1

3

5

tn [H+l,

FIG. Values

7. Hydrolysis of salicyl phosphate. from Hofstetter er al. (1962).

k in see-‘.

X,

FIG. 8. Activity of P-amylase. [H+] in mol/l. Y = rate of reaction = activity in min-‘. data are from Ballou (1940). X, Values for formate buffer; f, values for acetate buffer; values for propionate buffer; 0, values for butyrate buffer; 0, values for valerate buffer; values for phenylacetate buffer.

All A, 0,

-4.0

-2.4

[H’]

-0.00

in mol/l.

0.0

Y = rate of reaction

24

L

Ln CH+lr

840

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MATTSON

progressive shift in the values of LY and p of function (8). The exponent m is 2 for all sets. The titration of various weak acids (Albert & Serjeant, 1962) is shown in Fig. 9. Here, Y is a molar concentration whose relative values are the amount of titrant per unit volume. In function (8), m = 3 and (Y= p = 4 for these curves.

-1.0

0.6

2.2

3.8

5.4

7-O

tn [H’l,

FIG. 9. Titration of 0.01~ weak acids. [H+] in mol/l. Y = [KOH] in mol/l, calculated as the amount of titrant added per unit volume of solution. All data are from Albert & Serjeant (1962). X, Values for boric acid; +, values for benzoic acid; A, values for p-cresol.

Curve A and its data in Fig. 10 (Albert & Serjeant, 1962) are similar to those of Fig. 9. However, curve B in Fig. 10 describes the titration of a moderately weak acid and curve C describes the titration of a strong acid (Klingenberg & Reed, 1965). These progressively elongated curves are all described by function (8). The value of LY= /3 = 4 is common to all examples, but curves B and C show progressively higher values of m. (D)

SUMMARY

OF EXAMPLES

Table 1 gathers together information on the accuracy of the preceding fits. These fits are reported as percentages with respect to the Y-values of the sets of data. None of these sets have replicated values for their data; and a thorough, statistical analysis of them is not feasible.

MOLECULAR

841

BEHAVIOR

tn [H+l, FIG. 10. Titration of various acids. [Hf] in mot/l. Y = [KOH] in mot/l, calculated as the amount of titrant added per unit volume of the titrated solution. X, Values for 0.01 M glycine from Albert & Serjeant (1962); +, values for O*lOOO N acetic acid from Klingenberg & Reed (1965); 0, values for 0.0500 N HCI from Klingenberg & Reed (1965). TABLE

1

Percentage error with respect to the Y values of sets of data fitted by functions (7) and (8) P = X (equations (8)) and T = X (equations (7)) Figure

Curve

1

A B

2 3 4 5

6

A C

A B C D

p =X (equations (8)) Error W) 1.3 1.3 5.4 6.1 3.7 2.7 0.37 0.10 0.15 0.26 4.0

Source: Generated with the computer.

Figure 7 x

9

Curve A B C D E F A

B C 1I 0

A B C

Error (% 1 2.4 1.1 1.1 3.2 4.6 3.7 2.0 1.5 2.9 1.9 0.7 3.8 1.6

842

P. A. H. MATTSON

2

TABLE

Values of parameters [Y, /? and m for functions (7) and (8) P = X (equations (8)) and T = X (equations (7)) Fig.

Curve

1

A B A C

2 3 4 5

6

A B C D

(Y 6 6 4.5 4.5 6 :.21+ 3.02t 3,oot 2.97t 21

p = X (equations (8)) P 3.211 3.021 3.00t 2.91-I 21

m

Fig.

2 2 2 2 2 2 2 2 2 2 2

7 8

9 10

Curve A B C D E F A B C A B C

a

P

4.0t 5 5.5 6$ 6.5$ 7 8 4 4 4 4 4 4

4.ot 5 4 3 2 1 1 4 4 4 4 4 4

111 2 2 2 2 2 2 2 3 3 3 3 8.01 11.18

Source: Original. Unless noted, values have been selected by the programmer. t Values of equated constants for the initial fits, as found with the computer. These are set equal to the nearest integer or half-integer for the final fits. $ Value for best fit is 5!. 8 Values found with the computer.

Table 2 is a compilation of the values of (Y,p and m for Figs l-10. Regularities in the values of these parameters are apparent. The values of X0 and Y0 are not listed. These values are of less theoretical interest at the moment; but they can be generated from the information given, if necessary. 4. Discussion The proposed hypothesis, in its limited form of functions (7) and (8), has been illustrated by the examples of Figs l-10. These examples include common types of chemical reactions. Other selections under equations (1) await study of other types of molecular behavior. This limited form of the hypothesis shows traits which have long been associated with good theories in other fields. However, discussion of a few of these traits will show that the present paradigm (Kuhn, 1970; Masterman, 1970) of the field emphasizes different qualities. The discussion will follow Holton (1979), will consider first the promise of internal perfection and then take up the more complicated question of external validation.

MOLECULAR (A)

843

BEHAVIOR

COMPREHENSIVENESS

AND

SIMPLICITY

Many qualities, or themata (Holton, 1979) of internal perfection can be considered. Just two will be mentioned here. The proposed hypothesis is as broadly applicable as it is mathematically simple. Usually, these themata have been considered to be desirable attributes of hypotheses. However, even a slight aquaintance with present knowledge in physical chemistry and molecular physics is enough to persuade one that molecular behavior in the continuum is a highly specialized and complex subject. In this intellectual environment, comprehensiveness and simplicity become highly suspect, and these qualities argue against a hypothesis, rather than for it. It is evident that any hypothesis is undecidable in such an ambiguous environment, and we pass on towards the clearer light of scientific fact. (B) TESTABILITY

AND

ACCURACY

Any scientific candidate for the status of a theory must be capable of external validation (Holton, 1979). “We must trust nothing but facts. . .“, wrote Lavoisier (1789). But what facts can be trusted? The challenging feature of molecular behavior in the continuum has been the extreme difficulty in obtaining facts of good quality or of excellent quality. A fact is defined here as the minimum correspondence of an idea with reality. A fact, thus defined, is the minimum quantity of information which, by itself, can support a hypothesis or aid in the validation of a theory. Irreducible facts are data; but, by themselves, data are incapable of supporting anything but the most trivial of hypotheses. The enormous accumulation of data in molecular science does not, by itself, guarantee that adequate theories of molecular behavior do indeed exist. The data must be incorporated into meaningful or significant facts which validate molecular theories. In most fields, it is a relatively simple matter to construct a significant fact. For example, a paleontological fact is not markings on a rock. It consists of markings on a rock, together with the idea that these markings represent a describable fossil. Similarly, a spectral line is an observational datum, nothing more; but a quantum mechanical fact is that spectral line, together with a predicted value for the line. Molecular behavior in the continuum presents much more difficulty in constructing meaningful facts, because the minimum requirement for an excellent fact, or even a good fact, in this science is an accurate fit of a function to data taken over a very broad range of values. If the data are taken over a narrow range, any number of theoretical alternatives can be supported. Having excluded little, these continuum facts prove little and,

844

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hence, they are of poor quality. Any hypothesis for molecular behavior in the continuum is decidable only with continuum facts of good quality or of excellent quality. With the generous help and co-operation of a number of experimenters, it was possible to conduct an adequate survey of ordinary chemical reacions. Even with their help, the best data which could be found in the literature had some statistical deficiences (the observed points were not statistical averages), but these data could and did provide some good facts, as shown in Figs l-10. These figures display good fits over broad ranges of the variables. According to Table 1, the calculated errors appear to approximate experimental errors which can be expected in investigations of this type. Therefore, the present hypothesis is testable; and it is accurate, when tested. These examples feature well-defined chemical reactions, and testing of other molecular phenomena is desirable. Hundreds of sets of experimental data were examined by the author, but he was able to find only one concomitant set of good, alternative facts. These are curves B and D in Fig. 2. The equations which generated these curves (Schneider, 1962) had over 40 parameters, some of which could not be determined by the author. The large number of parameters create the risk that some of them might be theoretically inexact. The search for facts was not exhaustive and, among the thousands of sets of experimental data in the literature, other good facts probably exist. However, the dearth of significant facts in this subject ought to be a matter of serious concern to molecular scientists. Compare this situation with, say, classical mechanics, where prediction and observation go hand-in-hand. (C)

CORRELATIONS:

STATISTICAL

MECHANICS

External validation can also extend to other theoretical structures which have an independent, factual basis. Some possibilities are the classical atomic and molecular theory, modern quantum mechanics and statistical mechanics. The use of intensive variables of state and the appearance of the mathematical analogue of Gibbs’ phase rule first suggest that the proposed hypothesis might be identified with statistical mechanics and its theoretical subsidiary, thermodynamics (Tolman, 1938). Also, the hypothesis expresses a functional dichotomy between energy and work. The internal energy of a molecular system is a function of T, which is quantified in equations (7), whereas the work of a system is a function of P or of p = l/v, which are quantified in equations (8). There is a curious correspondence of this dichotomy to the iteration of work and energy in the First Law of Thermodynamics.

MOLECULAR

845

BEHAVIOR

Even so, a fundamental, mathematical incompatibility exists between the mathematical structure of these fields and that of the proposed hypothesis. By assuming no energy of interaction among individual molecules, it is possible to derive the Boltzmann distribution (Denbigh, 1964). This distribution, a simple exponential form, cannot be made equivalent to the equations of the proposed hypothesis. Conversely, if the logarithmic X-terms of the equations of the proposed hypothesis are approximated by simple algebraic terms, the equations of the hypothesis can be identified with simple exponential forms, such as the Boltzmann distribution. However, this identification does not follow naturally from the mathematics. Evidently, the energy of interaction among individual molecules cannot be neglected in a formulation of the hypothesis. The reconciliation of these differing approaches presents an unresolved puzzle. (D)

CLASSICAL

ATOMIC

AND

MOLECULAR

THEORY

Another possibility is a correlation of parameters of the proposed hypothesis with those of classical atomic and molecular theory: atoms combine in ratios of small, whole numbers, etc. If the proposed hypothesis were a purely empirical formulation which had no relation to molecules, electrons, molecular structure or molecular events, then its exponents and coefficients would have computed values which encompass the whole set of positive numbers. No pattern of numbers would be evident, and integer and half-integer values would occur with no greater frequency than other numbers of the set. There would be no correlations between computed values and any molecular entities. A study of Table 2 shows that what is randomly improbable in the highest degree has indeed occurred. The numerical pattern is formally symmetrical, rigorously consistent and highly predictable. Integer and half-integer values occur with improbably high frequency, and these numbers can be correlated with molecular entities. Lest the virtues of Table 2 be overstated, its defects should be noted first. Data for Figs l-4 did not extend to high pressures; and the assignment of m = 2, consequently, is arbitrary. More complete curves would have higher values of m, such as the m of curves B and C of Fig. 10. Also, for Figs 1-9, Table 2 shows values of m set equal to 2 or to 3 for the final runs. The computed, integral values of 2.OOxx, etc., for some of these curves are not shown; but they ought to have been included. For stochiometric events, if an assigned m differs from integer by even a tenth of a unit, the fitted curves show gross errors, and it saves computer time to set m equal to integer.

846

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H.

MATTSON

Nevertheless, what remains is curious, indeed. For these ordinary chemical reactions, all independent T require function (7), and all independent P and p require function (8). Such a rigorous pattern is randomly improbable. Furthermore, computed values of (Y and p are integer (or rarely, halfinteger, as in Fig. 2). As examples, curve C of Fig. 5 has a computed value of (Y= /3 = 3.00; and Fig. 7 has a computed value of 1y= p = 4.0. Note that data for these curves are fairly complete and, hence, the fitted curves are strongly convergent to these integers. These two examples alone require a triple improbability: specification of either function (7) or function (8) by the independent variable, specification of an integral value of 2 for the exponent and computation of an integral value for the constant. The occurrence of these results by chance is slight, The distinction between function (7) and function (8) is completely rigorous. For example, in Fig. 8, which employs function (8), p = 1 is supposed to be the absolute, lower limit for this constant. That is, one can predict that caproate buffer will yield p = 1, the same value yielded by valerate buffer in curve E. Note that for curve F, B = 1 for phenylacetate buffer. Phenylacetic acid is a larger molecule than valeric acid. There is yet another improbability displayed in Table 2, although it is one which is more difficult to interpret. This improbability is the apparent correlation between molecular structure and observed values of (Y and p. For example, Fig. 8 and Table 2 show that increasing the number of carbons in the aliphatic buffer by 1 concomitantly decreases the value of @ by 1. This observed sequence of values 5-4-3-2-l is improbable. The unimolecular reactions of Figs 1-4, whose fitted parameters are listed in Table 2, show some correlation between structure and observed values of (Y. For reactions which involve simple, internal rearrangements, such as the examples of Figs 1 and 2, the observed u is equal to the number of major structural bonds in each case. However, for an event which involves the breaking of a bond, as in Fig. 3, the value of (Y is double that of the number of major, structural bonds. Figure 4 displays results from a two-step reaction, which might combine both a breaking of a bond and internal rearrangement of the resulting intermediate. Apparently, the side hydrogens are not involved in any of these examples. The dissociation of aliphatic acids in Fig. 5 appears to involve just the carboxyl part of each molecule, because the constants are all the same. These dissociations have a “countable” one and one-half bonds for the group: (Y=p = 2(1*5) = 3. However, the existence of resonating forms in the dissociated acid complicates this simple picture of “countable” bonds.

MOLECULAR

BEHAVIOR

847

Some biological phenomena display high values for (Y and p, and Fig. 6, as listed in Table 2, shows this. In Fig. 7, one of the acid groups in salicyl phosphate determines the rate-limiting step for the observed phenomenon. Figure 8 appears to display a complex type of reaction which involves at least two steps. Such complexity is not surprising in enzymatic reactions. For titrations of weak acids in Fig. 9, the value m = 3 in Table 2 suggests that one molecule of water is involved in the reaction of acid and base molecules. Curves B and C in Fig. 10 force the hypothesizing of equations (6), and their values of m suggest a correlation of an increasing involvement of ambient water molecules in the reaction with an increase in the strength of the acid. From the empirical base provided by these examples, it is possible to make strong predictions. For ordinary chemical reactions, at least, the pattern exhibited in Table 2 is predicted to continue. The probability that this pattern will continue by chance is virtually zero. However, for some phenomena in molecular physics, recourse might be needed under function (1) to choices other than equations (7) and (8). Conversely, these numerical relationships within continuum phenomena are a very simple means of demonstrating the molecular theory itself: simpler, even, than the historical accomplishment of Einstein and of Perrin (Nye, 1972). The existence of molecules was in dispute throughout much of the nineteenth century, and the demonstration of their existence is a century late, but the curiosity is that such an exposition was within the experimental and mathematical capabilities of the late-nineteenth and early-twentieth centuries. Today, the barriers to understanding are quite different. The precedence of the theoretical approach of quantum mechanics is commonly assumed (Fowler, 1966) without an explicit recognition that the modern form of that science has been inspired by a limited type of experimental data: line spectra. Yet the natural occurrence of continuum phenomena is the rule, rather than the exception, on the terrestrial surface. Cell biology is dominated by such phenomena, for example. There is always danger in generalizing from the part to the whole. The peculiar scarcity of continuum facts in molecular science does not support the idea that quantum mechanical generalizations to ordinary, continuum phenomena have been successful. (El

QUANTUM

MECHANICAL

COMPARISONS

A tentative identification of the proposed hypothesis with quantum mechanics is possible, but some differences in emphasis should be noted.

848

P.

A.

H.

MATTSON

If an intensive variable of state monitors an average molecular event in the continuum, then the variable measures just what the frequency of atomic line spectra measures: averages of single events. Line spectra arise from transitions of single electrons to stable orbitals, and they are the historical basis for the development of modern quantum mechanics (Jammer, 1966). Yet, by contrast, continuum molecular events must involve two or more electrons (from two or more molecules) in unstable orbitals. This latter type of event appears to require a different form for mathematical solutions of the fundamental, quantum-mechanical relationships. The characteristic, applied form for quantum mechanics (and for statistical mechanics, for that matter) is the simple, exponential function. Its prototype is the Normal distribution. The characteristic form for the present hypothesis is a generalized, logarithmic function. Its prototype is the lognormal distribution. This type of distribution arises when the rate of change of a quantity is in proportion to its absolute value. Emphasis on the older, Normal distribution is an historical accident (Aitchison & Brown, 1969). (F)

MOLECULAR

DIMENSIONALITY

The balanced form of equation (1) suggests some kind of principle of conservation for the initiating and terminating orbitals of a molecular event. Immediate selection of a principle for these unstable orbitals would be premature and we pass on to consideration of the general equation (2). Without an appreciation of the dimensionality of the problem, a clear interpretation is difficult. The form of equation (2) suggests a subset of Hilbert space (Blinder, 1974) which has a dimensionality equal to the degeneracy CL.Because the molecules of the event are statistically independent, the degeneracy is not too surprising. The key to the puzzle, then, is the representation of the one dimension which creates this degeneracy. Consider the line between the centers of mass of two interacting molecules. Regardless of what complicated figures the two molecules might describe in the three-dimensional space of the observer, it is true that, along this single dimension of molecular interaction, the two molecules merely approach or recede. In this space, the only component of a vector which appears to require analysis is a projection of that vector on the single axis of interaction. With respect to the interaction itself, the other vectorial directions appear to have spherical symmetry and a net vectorial representation of zero. The observable scalar 2, is assumed to map on to a field representation of this simple, one dimension of interaction. The initial value ZO corresponds

MOLECULAR

BEHAVIOR

849

to a unique value of the field. It is possibly associated with the maximum electron density along the one dimension of interaction. Conversely, the observable scalar (an intensive variable of state) is a report to the observer on this one dimension of interaction. The diversity of such molecular phenomena is great, but the relation of molecular phenomena to this one dimension is always the same. The multi-dimensionality for the orbital is in the degeneracy p exhibited by this one-dimensional space. It appears that this degeneracy is either unity (that is, there is no degeneracy) or it is equal to the number of molecules in the average event. (G)

QUANTUM

NUMBERS

Equations (2), (3e) and (6e) contain three parameters which need explanation: the 2, continuum, the K of the coefficient and the p of the coefficient and exponent. In the absence of strong, external magnetic fields, the four possible quantum numbers for an orbital are reduced to three. This number is the same as the number of parameters which need explanation. The unintegrated form of Z, is l/Z, which represents a Coulombic field of energy for one dimension in the Schriidinger equation. It appears that the other terms of the Schrbdinger equation can be neglected, because a simple electron-electron repulsion has become dominant (Heer, 1972b). In place of the eigenvalues of the principal quantum number, there is a continuum of values for Coulombic behavior. The other two quantum numbers which need analysis are the total orbital angular momentum and the total intrinsic angular momentum (the spin). If p represents the orbital degeneracy and K represents the spin degeneracy of a weakly coupled system (Saxon, 1968), then the curious appearance of the mathematical analogue of Gibbs’ phase rule is explained: k = K + p - 2 of equation (6e) is simply the total, dimensionless, angular momentum of the orbital. In effect, this total angular momentum statistically weights the Coulombic energy in either an intensifying manner, as in equations (7), or in a dispersive manner, as in equations (8). The parallel spins of a symmetric electron pair of an unstable orbital have a minimum assignable value of $ + $ = 1. (The total spin-wave function of the molecular event, of course, is anti-symmetric.) This minimum, symmetric value corresponds to the ground state for a stable molecule. However, a second, countable value appears to be possible for either electron of the pair. This second value corresponds to the excited state for a stable molecule. The countable numbers for an electron pair then show additional, possible values of $ and 2.

850

P.

A.

H.

MATTSON

This interpretation is consistent with the proposed hypothesis. For example, if n = 1 and y = S = 2 in equations (6~) and (6d), then c = d = 2 + 1 - 2 = 1, as implied in the construction of equations (4), (5), (7) and (8). For ordinary types of chemical reactions, this implies that a dependent orbital (usually the terminating orbital of an event) involves just the minimum of two electrons which are doubly countable. The independent orbital, generally, is affected by larger numbers of molecules and electrons in an average event. Because the countable electrons of each orbital are statistically indistinguishable from one another, they must all be weighted equally.

(H)

PRINCIPLE

OF CONSERVATION

Each observed pair of values (X, Y) of equations (1) represents a conservation principle between the initiating orbital and the terminating orbital of a molecular event. This principle requires both energy and angular momentum in its formulation. The space of the event must be homogeneous and isotropic, with no strong, external, magnetic fields. Also, the event must be measured in a quasi-steady state. The specificity of the single-valued continuum (X, Y) in the total space of X, and Y, is determined entirely by the invariance of the angular momenta of that species of molecular event. A discontinuous change in the average number of countable electrons or countable molecules indicates a different species of event. Thus, the form of function (l), as novel as it is to molecular science, behavior for independent merely measures conserved, Coulombic molecules. This class of function is in widespread use elsewherespecifically, as the lognormal distribution (Aitchison & Brown, 1969). It is surprising that the general form has not been applied to the problem of molecular behavior heretofore, especially when the results are so unequivocally accurate. Yet this essentially empirical approach has its limitations, and the entire subject needs to be treated from a fundamental, quantum mechanical point of view. Vector analysis, convergence, and boundary conditions are just a few of the many subjects which need attention. Questions of symmetry are also important. The most widely used class of functions in molecular science-the simple, exponential class-is intrinsically asymmetric with respect to its variables. The present argument is that any mathematical treatment of linked orbitals ought to be symmetric and ought to yield a symmetric class of functions. Particular models for application are selected from this symmetric set.

MOLECULAR

BEHAVIOR

851

4. Conclusion Molecular behavior in the continuum can be described by a set of generalized, logarithmic functions. These functions are comprehensive and simple. They can be employed to create and organize continuum facts effectively. That is, they are accurate, when tested; and their parameters show regular patterns of invariance. Their parametric values are strongly correlated with classical atomic and molecular theory. Their connections with statistical mechanics and thermodynamics are not obvious, but they appear to correlate with some of the simpler aspects of quantum mechanics. These generalized, logarithmic functions are the basis for a reasonable hypothesis of molecular behavior in the continuum. The hypothesis is quite incomplete; but it is recommended to molecular scientists as an effective, analytical tool in their work.

Nomenclature A

positive coefficient for X term, X, 5 1 positive constantfor X term, X, 51-9 angularmomentum il positive coefficient for X term, X, > 1 b positive constantfor X term, X, > 1+ angularmomentum C positive coefficient for Y term, X, 5 1 positive constantfor Y term, X, 5 1+ angularmomentum f, positive coefficient for Y term, X, > 1 d positive constantfor Y term, X, > 1+ angularmomentum K positive coefficient for 2 term k positive constantfor Z term + angularmomentum m exponent of X terms+ degeneracyof orbital angularmomentum exponent of Y terms+ degeneracyof orbital angularmomentum ;b absolutepressure T absolutetemperature t time v specificvolume X independent,intensive variable of state Y dependent,intensivevariable of state Z intensive variable of state a parameterfor X term, X, I 1+ degeneracyof spin P parameterfor X term, X, > 1+ degeneracyof spin parameterfor Y term, X, 5 1+ degeneracyof spin ; parameterfor Y term, X, > 1+ degeneracyof spin K parameterfor 2 term -+degeneracyof spin CL exponent of Z term + orbital degeneracy P concentration

852

P.A.H.MATTSON

Subscripts i r

0

number of species of phenomena reduced value initial value + maximum electron

in homogeneous,

isotropic

medium

density for interaction

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MOLECULAR PRITCHARD,

H. O., SOWDEN,

853

BEHAVIOR

R. G. & TROTMAN-DICKENSON,

F. F. (1953~1).

Pm.

Roy.

Sot. A 217,563. PRITCHARD, H. O., SOWDEN, R. G. & TROTMAN-DICKENSON, F. F. (19536). Proc. Roy. Sot. A 218,416. SAXON, D. S. (1968). EIementury Quantum Mechanics, p. 327. San Francisco: Holden-Day. SCHNEIDER, F. W. (1962). Ph.D. thesis, Seattle: University of Washington. SCHNEIDER, F. W. & RABINOVITCH, B. S. (1962). J. Am. them. Sot. 84,4215. SIMPSON, R. B. & KAUZMANN, W. (1953). J. Am. them. Sot. 755139. TAYLOR, J. R. (1972). Scattering Theory: The Quantum Theory on Nonrelativistic Collisions, p. 22. New York: Wiley. TOLMAN, R. C. (1938). The Principles ofStatistical Mechanics, p. 9. Oxford: Clarendon Press.