Volume 23, Issue 6, 01 June 1955

Statistical Mechanics of Polymer Solutions. Parts I and II
View Description Hide DescriptionSome contributions are made to the refining of the lattice modeltheory of Flory of chain polymer solutions by introducing the idea used in the recent theory of regular assembly that combinatory factor constructed in terms of multiple site clusters can be properly used to take long‐range correlations between a given site and its far distant neighbors into account.
In Part I, the combinatory factors of Flory and others in current use are reconstructed to exemplify the pseudo‐assembly method which has been presented by Kikuchi and the present authors for an easy construction of combinatory factors of higher‐order approximation for regular assembly, and to give a thorough understanding of the succeeding uses in Part II.
In Part II, a refined theory of chain polymer solutions is developed. The final expressions for the entropy of dilution and the osmotic pressure depend on the flexibility and the degree of branching of the polymer chain, and give a better agreement with observed data even at high dilution.

Spectra of Transition‐Metal Complexes
View Description Hide DescriptionThe electronic transitions observed in complexes of the transition‐metal ions are interpreted in terms of a slightly modified crystal‐field theory. Parameters of chemical interest are derived.

Studies of the Interaction between Stable Molecules and Atoms. I. A Molecular Orbital Theory of the Activation Energy between Molecules and Atoms
View Description Hide DescriptionThe molecular orbital method of calculation is applied to the interaction between closed‐shell species; the results discussed in this paper and presented in the following papers of this series indicate that this method will yield quantitative results for energies of a system as function of internuclear parameters if one is interested in the differences in energy for two different geometric ground state configurations of the system. The molecular orbitals are used as a basis of constructing correlation diagrams for predicting mechanisms and calculating activation energies for chemical reactions. The orbitals of the reactants and products are correlated with the orbitals of a state defined as the ambivalent complex. The low‐lying excited states of the interacting molecules seem to be of importance in determining the activation energy of a chemical reaction. Considerable qualitative insight into the activation process is gained by this method; the relative values of the activation energies for the interaction between two closed‐shell molecules, a molecule and a radical, and two radicals are predicted in agreement with experiment.

Studies of the Interaction between Stable Molecules and Atoms. II. Interaction between Two He Atoms
View Description Hide DescriptionThe repulsive energy of two helium atoms has been calculated using molecular orbital theory. The equations of Roothaan were used to break down the energy into integrals involving atomic orbitals. The necessary integrals are tabulated in the literature for a wide range of parameter which enabled the potential energy curve to be calculated from R=0.6 to 7.0 a _{0}. The Slater (1s) orbital with effective nuclear chargeZ=27/16 was used. The potential curve is lower at all distances from those of other investigators in the range where Van der Waals forces give attraction although the curves agree at closer distances. Ionization energies have been calculated over these distances and extrapolated to infinity to give a value which is much better than that calculated with the same orbital by ordinary means. The addition of 2pσ character to the molecular orbitals by constructing hybrid orbitals of the type N {(1s)+λ(2pσ)} and the determination of λ by a SCF calculation according to the scheme of Roothaan, gives an attractive energy of 2.14×10^{—5} a.u. with λ=1.4×10^{—3} at R=3.11 A. Experimental values are approximately 3×10^{—5} a.u. The calculated results are compared with experimental data in the exchange and Van der Waals region of the values of the internuclear parameter.

Studies of the Interaction between Stable Molecules and Atoms. III. A Molecular Orbital Treatment of the Quadratic Form of H_{4}
View Description Hide DescriptionMolecular orbital calculations of the quadratic arrangement of four hydrogen atoms have been carried out as a function of the internuclear distance. It is shown that the complex has orbital degeneracy in the ground state. An approximation, which includes the lowest orbital and the degenerate orbitals only, yields a triplet and three singlet states, the triplet being the lowest one in accordance with Hund's rule. The actual calculations show that the quadratic H_{4} is unstable with respect to a totally symmetrical displacement at all points of the potential energy surface.

Studies of the Interaction between Stable Molecules and Atoms. IV. The Energy of the Linear H_{4} Complex
View Description Hide DescriptionRoothaan's LCAO—SCF treatment has been applied to the linear symmetrical H_{4} complex in an attempt to test this method for calculating the energies of activated complexes. Calculations have been made for internuclear distances of 1.4, 1.6, 2.0, and 2.2 a.u. The SCF method gives a lower value for the energy of 1.4 a.u. than was obtained by a simple molecular orbital method without configuration interaction, but gives a higher value than the best configuration interaction treatment. The values of the energies that were obtained are —55.48, —55.99, —56.50, and —56.54 ev at distances of 1.4, 1.6, 2.0, 2.2 a.u., respectively. The changes in the wave functions as one adiabatically changes the internuclear distance are of particular interest. Although the results cannot be directly compared with experiment, reasonable values for the energy are obtained indicating that the molecular orbital SCF method with a limited amount of configuration interaction is a promising method for treating closed‐shell molecules and interactions between such molecules for small values of the internuclear parameters.

Studies of the Interaction between Stable Molecules and Atoms. V. Molecular Orbital Approach to the H+H_{2} Reaction
View Description Hide DescriptionIn an attempt to find a relatively simple way of calculating activation energies for chemical reactions, Roothaan's LCAO‐SCF method has been applied to the linear H_{3} complex. Since Roothaan's method has been developed for closed shells, a self‐consistent field treatment has been done on the linear H_{3} ^{—} complex. According to Roothaan, the energy of the H_{3} should be equal to a first approximation to the energy of the H_{3} ^{—} minus the energy required to remove one electron. Calculations have been made at distances ranging from 1.4 to 2.5 atomic units and for various values of the screening constant. Correlation diagrams for the reaction H+H_{2} have been constructed and the activation energy calculated. The method gives considerable insight into charge distribution but is unsatisfactory for calculating good energy values. The results are compared with experiment and with values calculated by the MO‐configuration interaction method which yields a value of 8.76 kcal/mole.

Long Time Phosphorescence of Organic Crystals
View Description Hide DescriptionLong time phosphorescence (of the order of minutes) of anthracene, stilbene, durene, and naphthalene crystals at room temperature is reported. The decays are approximately hyperbolic and the observed phosphorescence is shifted to longer wavelengths than the usual fluorescent emission. This phosphorescence is not induced by the light strongly absorbed in the substance but mainly by the weak absorption of light of wavelengths corresponding to their own fluorescent light. There are indications that this phosphorescence stems from an approximately uniform, continuous trap distribution and that such a mechanism permits a description of some of the observed behavior. Wavelength measurements seem to indicate that this phosphorescent light emission does not originate from triplet to ground transitions.

Infrared Absorption Spectra of Monodeuterated Pyridines
View Description Hide DescriptionFor this work 2D‐, 3D‐, and 4D‐pyridine have been prepared from corresponding halogen compounds. The infrared spectra of the compounds in the liquid phase are reported over the spectral range 400—3200 cm^{—1}. Interpretation of the spectra is carried through by analogy with the spectra of isotopic benzenes, pyridine, and pyridine‐d _{5}.

Vibrational Spectra of Molecules and Complex Ions in Crystals. VIII. The Infrared Spectrum of a Mixed Crystal of 1‐Deutero‐Ammonia and 2‐Deutero‐Ammonia
View Description Hide DescriptionThe infrared spectrum of a mixed crystal containing NH_{3}, NH_{2}D, ND_{2}H, and ND_{3} was observed at —190°C. All of the fundamental frequencies of NH_{2}D and ND_{2}H were assigned. A calculation of the 12 frequencies of these two molecules was made, using the four nonzero force constants previously obtained from NH_{3} and ND_{3}. The average deviation from the observed frequencies was 0.6 percent and the maximum deviation 1.6 percent. It is shown that the coupling between the motions of the various molecules can be neglected.

Calculation of Thermodynamic Properties of Polyelectrolytes
View Description Hide DescriptionExpressions are derived which introduce an appreciable simplification into the calculation of the thermodynamic properties of solutions of polyelectrolytes in certain cases. For example, for a certain class of theoretical models of these systems it is found that the square of the mean ion activity coefficient of a uni‐univalent salt in the presence of polymeric ions is , the integration of the potential ψ being over a region whose volume is the volume of solution per macro‐ion, V, and whose symmetry is that assumed for the polyelectrolyte. The osmotic pressure of a salt‐polyelectrolyte system is, ignoring the contribution of the macro‐ion, estimated to be Σ_{ i } c_{i} ^{s}kT, where Σ_{ i } c_{i} ^{s} is the sum of the concentrations of all ions at the surface bounding the previously defined volume V. Other relations and various applications are given. The activity coefficient of salt in the presence of polyelectrolytes, calculated by extending the ``parallel rod'' picture of polymeric ions, is found to be in reasonable agreement with the experimental data. The use of the Poisson‐Boltzmann equation to estimate ψ in these systems is shown not to render inconsistent several alternative expressions for the electrostatic contribution to the free energy.

Photochemical Studies of the Porphyrins. III. Photoreduction of a Porphyrin by Benzoin
View Description Hide DescriptionAn examination of the kinetics of the photoreduction of zinc tetraphenylporphin by benzoin is reported. The products are successively the dihydro‐ (chlorin) derivative; a tetrahydro‐derivative and possibly a hexahydro compound as well. It is demonstrated that the primary requirement is the photoactivation of the benzoin; a secondary reaction, involving photoactivated porphyrin as well, is also demonstrated. The quantum yield based on light absorbed by benzoin varies between 0.01 and 0.06.

On the Logarithmic Rate Law in Chemisorption and Oxidation
View Description Hide DescriptionIt is shown that the logarithmic rate law is found very widely in chemisorption and oxidation. A theory of this law is offered, and comparison with experiment leads to reasonable values for the parameters. Although the theory is very elementary, it is hoped that the physical interpretation it provides of experimentally determined parameters will encourage a more detailed study of these, for instance as a function of temperature and ambient pressure.

Method of Calculating Cross Sections for Molecular Collisions
View Description Hide DescriptionThe calculation of cross sections for slow inelastic collisions of molecules, such as occur in chemical reactions, shows an interesting combination of classical and quantum‐mechanical features. The de Broglie wavelength is small compared with the range of the intermolecular potential so that there exists a relatively well‐defined quasi‐classical orbit; but in some cases the coupling constant is small compared with unity, and this gives the problem certain quantum‐mechanical characteristics. On account of the condition relating to the de Broglie wavelength, a calculation of the total collision cross section requires the knowledge of the partial waves for a large number of values of the angular momentumlh/.
A method is devised to suit these different features. The WKB method lends itself well to this application, with a quasi‐classical amplitude and a quantum‐mechanical phase for the wave function. We also use the concept of an impact parameter b rather than the orbital quantum number l: this is well suited to describing a quasi‐classical orbit, and hence for calculating the total cross section in a case like the present one. As an example of the method, the cross section for excitation of the first vibrational state of a hydrogen molecule resulting from collision with another hydrogen molecule is calculated.

Resistance in a Liquid‐Liquid Interface. III. The Effect of Molecular Properties
View Description Hide DescriptionThe diffusion of molecular sulfur across a saturated liquid‐liquidinterface has been investigated for several molecular pairs, using a method previously described. The interfacial resistance does not directly depend on polarity of the liquids, nor can it be correlated with interfacial tension. High resistances are associated with liquids which show a high degree of hydrogen bonding. With suitable assumptions it is possible to calculate a free energy of activation for transfer across the interface. It appears that transfer in systems involving negligible resistance is analogous to van der Waals adsorption, while for systems involving high resistance the process is comparable to chemisorption.

One‐Dimensional Model of the Hydrogen Bond
View Description Hide DescriptionA one‐dimensional model for hydrogen bonding is proposed based on the potential function V = D[1—exp(—nΔr ^{2}/2r)]. The energy associated with both the weak and strong bonds of the configuration RO – H – – – OR _{2} is obtained through application of this function. A repulsive Van der Waals potential and an attractive electrostatic potential are also assumed to exist between the two electronegative atoms of the hydrogen bond. Through application of the conditions describing a stable equilibrium, relations are obtained which permit a calculation of OH frequency shifts, OH bonded distances, hydrogen bondenergies and k _{o — — — o} force constants, all as a function of the O – – – O distance R. The calculated quantities agree well with those obtained from neutron diffraction, infrared, and other experimental studies. Because of the assumptions involved, this model is best used to describe the properties of hydrogen bonds in crystals. The model is also capable of extention to describe the properties of other types of hydrogen bonds such as HFH, NHN, NHO, OHN, etc.

Treatment of Isotopic Exchange Reactions Having Complex Mechanisms
View Description Hide DescriptionThe reaction rate vs time data of isotopic exchange reactions have been shown by previous work to obey a first‐order rate law, neglecting differences in isotopic reactivity. A simple method, of use in reactions having complex mechanisms, is developed for relating this first‐order rate constant to the rate constants of elementary steps in the process. The method involves examining the progress of specified atoms through the reaction sequence, and is applicable regardless of the number of exchange positions of the reactants, or of whether or not one isotope is present in tracer amounts. These results are used to reinterpret some data on the deuteration of diborane.

Color Centers in Alkali Halides Containing NO_{2} ^{—} Ions in Small Concentration
View Description Hide DescriptionAlkali halide single crystals (KBr, KCl, NaBr, and NaCl) containing NO_{2} ^{—} ions at various concentrations, from 0.5 to 0.007 mole percent, were grown by the Kyropoulos method. At high NO_{2} ^{—} concentrations the typical NO_{2} ^{—} band appears in the absorptionspectrum with peak wavelengths between 280 and 260 mμ depending on the nature of the base material. The production of F‐bands by exposure to x‐rays is completely suppressed in these crystals at room temperature and greatly inhibited at —190°C. Instead of the F‐band, a continuous band (C‐band) is produced with a wavelength nearly coincident with the NO_{2} ^{—} band, but not showing at low temperature the vibrational structure characteristic of the latter. Other less important new absorption bands appear in the red and the ultraviolet. At lower NO_{2} ^{—} concentrations F‐bands begin to appear in competition with the C‐bands. The bleaching of the various absorption bands by light absorbed in them is investigated and discussed. NaCl seems to differ from the other halides in so far as in this crystal the production of the F‐band is greatly enhanced by NO_{2} ^{—} in concentrations which in KBr greatly inhibit the formation of F‐centers.

Collision Excitation of Molecular Vibrations in Halogen‐Substituted Methanes
View Description Hide DescriptionDepartures from the equilibrium between vibrational and translational energies in gases are characterized by relaxation times which depend upon the ease with which vibrations may be excited or de‐excited by collisions. The de‐excitation process has been discussed by various authors on the basis of the relative velocity of approach of two colliding molecules and on the basis of the relative energy of approach.
The dispersion of ultrasound has been measured in fourteen halogen‐substituted methanes and compared with existing data. Binary collisions were found to be responsible for the excitation of vibrations, and all the gases were found to have a single relaxation time indicating a strong intermodal coupling. The probability, in a collision, of exciting or de‐exciting molecular vibrations was found to depend upon the relative energy of approach of the molecules. Molecules with the same number of halogen substitutions resemble each other in this respect. The data suggest that the colliding molecules might form a complex molecule with a short lifetime during which energy exchange takes place readily.

Theory of Normal Paraffin Liquids
View Description Hide DescriptionA hole theory of n‐paraffin liquids, which is the same in mathematical form as Flory's theory of polymer solutions, is developed to derive the relationships between the physical constants and the number of carbon atoms n. If the number of segments in a molecule is equal to n ^{⅔}, quantitative agreements between the theory and the observations can be shown for the critical temperature, the critical pressure, the boiling temperature and the heat of vaporization. It is also possible to give theoretical foundations for some empirical rules recently proposed by Grunberg and Varshni.