Showing papers in "Journal of Chemical Physics in 1947"

On the Quantum Correction for Thermodynamic Equilibrium

Abstract: The behavior of any system at high enough temperatures approaches that of its classical counterpart. The probability of any configurational position is then proportional to exp—U/kT, with U the potential energy. Wigner has shown that quantum‐mechanical deviations, as the temperature is lowered, may be approximated by multiplication of this with 1–f, where f is a function proportional to h2 and having terms in T−2 and terms in T−3. This type of approximation is unsatisfactory for a system of many degrees of freedom, that is, one of many dependent molecules. For such a system it is shown that instead of Wigner's approximation we may replace the classical potential U in the exponent by U—kTf, where f is the same as Wigner's function.

Calculation of Equilibrium Constants for Isotopic Exchange Reactions

Abstract: It is pointed out that the possibility of chemical separation of isotopes is a quantum effect. This permits a direct calculation of the difference in the free energies of two isotopic molecules. Tables and approximation methods are given which permit a rapid calculation of equilibrium constants if the frequency shifts on isotopic substitution are known. Several applications are discussed.

Physical Properties and Cation Arrangement of Oxides with Spinel Structures I. Cation Arrangement in Spinels

Abstract: By means of a detailed x‐ray study of a large number of oxides having a spinel structure, it has been established whether they crystallize in the ``normal'' spinel‐type structure, XY2O4, or in the ``inversed'' type of structure, Y(XY)O4, discovered by Barth and Posnjak. Certain rules can be derived which are also applicable to more complicated spinels or to solid solutions of different spinel oxides. Madelung constants have been calculated for different spinel types in order to investigate their stability on the basis of an ionic concept of the chemical bond (Born's lattice theory). The experimental results are only partially explainable on this basis. All exceptions to the electrostatic principles can be understood by assuming that Fe3+, Ga3+, In3+, Zn2+, and Cd2+ favor a configuration having the coordination number 4, because of a considerable covalent contribution to the chemical bond for that configuration.

Physical Properties and Cation Arrangement of Oxides with Spinel Structures II. Electronic Conductivity

Abstract: The relations between the electronic conductivity of certain spinels and the arrangement of the cations in the crystal structure (see preceding paper) are studied. Several arguments favor the assumption that Fe3O4 contains both divalent and trivalent iron in the 16‐fold position. The transition point in the neighborhood of liquid‐air temperature is probably associated with an increased degree of order at low temperature in the distribution of the 8 electrons between the 16 Fe‐lattice points per unit cell. The considerably increased conductivity below the transition points shows tetragonal anisotropy when the crystal is cooled in a magnetic field. The possible distributions of the electrons in the crystal at low temperature are discussed. In more complicated spinels, containing other metal atoms as well as iron in both the divalent and the trivalent state, the electronic interchange is more or less inhibited by the foreign metal atoms. The higher values of their resistance in comparison to that of Fe3O4 can be roughly described by an increased activation energy. The investigation of a number of substances with different arrangements of the cations shows that the activation energy (and therefore the electrical resistance) is lowest for those cases in which the electrons can travel, as in Fe3O4, along the Fe of the 16‐fold position.

Thermodynamics of Crystallization in High Polymers. I. Crystallization Induced by Stretching

Abstract: A theory of oriented crystallization in elongated polymers having network structures (e.g., in vulcanized rubber) is developed through the application of statistical mechanical procedures similar to those employed in rubber elasticity theory. Expressions are derived which, within the limitation imposed by the simplifying assumptions, relate the incipient crystallization temperature with the elongation, the degree of crystallinity with the elongation and temperature, and the retractive force at crystallization equilibrium with the elongation at constant temperature. The reciprocal of the absolute temperature for incipient crystallization is predicted to decrease linearly with a simple function of the elongation and the average number of chain segments between cross linkages. Only moderate degrees of crystallinity are predicted at equilibrium. In conformity with requirements of the second law of thermodynamics, equilibrium crystallization decreases the tension in the stretched specimen.Apparent discrepancie...

Theory of the Increase in Rigidity of Rubber during Cure

Abstract: The rigidity of rubber, considered as a network of flexible molecules with Gaussian configuration functions, can be calculated for a particular sample if one is given a complete description of the molecular network or certain types of statistical description. In particular, it is sufficient to know the distribution of lengths and vector‐mean extensions of the segments of the network, or only the latter distribution if it is of Gaussian form. The assumption that the vector‐mean extensions have a Gaussian distribution corresponds to a similar postulate in the theory of Wall, and has been applied to a simplified network theory by Flory. In the complete theory it leads to calculation of the same proportionality between the rigidity of the material and the number Ga of segments, per unit volume, in the ``active'' part of the molecular network. However, consideration of the process of cure shows that this assumption cannot be expected to be correct, though it does lead to results of the right order of magnitude. An alternative approach to the problem is based on a study of the increase in rigidity of the material as cure proceeds. It uses a more realistic picture of the process of cure than those hitherto employed, and uses Gaussian distribution functions only where they can be logically justified. The final result is of the same form as that given by the previous theory, except that Ga is replaced by Ba, the number of bonds formed within the molecular network during cure. Since Ba is smaller than Ga by a factor of about two or more, the improved theory leads to the prediction of somewhat lower rigidities per bond formed. Available experimental results of Flory on Butyl rubber vulcanizates do not appear to give an adequate check on the quantitative features of the theory.

The Statistical Mechanical Theory of Transport Processes II. Transport in Gases

Abstract: By means of the methods developed in the first paper of this series, SMTI, the Maxwell‐Boltzmann integro‐differential equation, underlying the well‐developed Chapman‐Enskog theory of transport phenomena in gases of low density, is derived from the principles of statistical mechanics. The derivation supplements and clarifies the usual physical argument employed to establish this important equation.

Statistical Properties of Networks of Flexible Chains

Abstract: A Gaussian network is defined as a network of flexible chain segments, linked to each other and to a system of fixed points, in which each unbranching chain segment can take on a number of configurations which is a Gaussian function of the distance between its ends. Real molecular networks, such as those of rubber, can under certain circumstances be treated as Gaussian networks. The present paper carries out a systematic mathematical discussion of the statistical properties of Gaussian networks: the total number of possible configurations of the network as a function of the fixed‐point coordinates, the probability of finding a given element of the network in a given position, or of finding two elements of the network in given relative positions, and so on. All probability‐density functions appear as exponentials of quadratic forms, with constants explicity expressible in determinant form. An explicit reduction to a sum of squares is given for all quadratic forms occurring in the theory of coherent Gaussian networks, and an explicit general formula is found for integrals of the form ∫ −∞+∞dX1 ∫ −∞+∞dX2··· ∫ −∞+∞dXq exp{− ∑ in ∑ jnγijXiXj}. There is described a mechanical analog of a Gaussian network, by consideration of which the statistical properties of the Gaussian network can be determined. The method is applied to the discussion of the statistical properties of a Gaussian network with the connectivity of a regular cubic lattice.

The Adsorption of Hydrogen on Tungsten Powders

Abstract: An analysis of the extensive experimental data of Frankenburg on the adsorption of hydrogen by metallic tungsten powder between −194 and 750°C has been made, using the Fowler‐Guggenheim treatment of adsorption phenomena as the mode of approach. It has not been found possible to interpret the data in terms of interaction between the adsorbed species on a uniform adsorbent surface. It is shown that the experimental data can be interpreted on the basis of a non‐uniform surface without interaction, the heterogeneity of the surface being specified in terms of an exponential distribution of sites given by the distribution function N=ce−χ/χm, where χ represents the energy difference between the lowest energy state of the gas and the lowest energy of the adsorption complex. Such a distribution function yields the experimentally observed relation d lnp/d lnθ=constant at constant temperature. Such a heterogeneity is especially significant in the case of tungsten metal which hitherto has been treated as an essential...

Dependence of Bond Order and of Bond Energy Upon Bond Length

Abstract: A simple inverse square relation of the form, N=aR−2+b, (where N is the bond order, R is the bond length, and a and b are constants characteristic of any given pair of atoms) has been found to agree satisfactorily with the available values for bond orders. For CC bonds a and b are found to have the values 6.80 and −1.71, respectively; for CN they are 6.48 and −2.00; for CO they are 5.75 and −1.85. The available data on bond energies suggest a relation of the same form, E=lR−2+m, between bond energy E and bond length R, where l and m represent the characteristic constants of a given atomic pair.

Calculation of the Equilibrium Composition of Systems of Many Constituents

Abstract: A computational procedure is developed for the calculation of the composition at chemical equilibrium of systems of many constituents. The general equations are applied to homogeneous systems and to homogeneous systems which exist in the presence of a single additional pure phase. A brief discussion is given of appropriate numerical methods of computation.

The Entropy of Solution of Molecules of Different Size

Abstract: The entropy of mixing two liquids whose molecules differ in size is expressed in terms which avoid the assumption of a lattice as an artificial frame of reference, and which is not limited to linear polymers. The equation obtained is ΔSR=N1lnV−N1b1−N2b2N1(v1−b1)+N2lnV−N1b1−N2b2N2(v2−b2), where N1 and N2 denote number of moles, V the volume of the solution, v1 and v2 the molal volumes of the pure components and b1 and b2 the sum of the actual geometrical volumes of 6×1023 molecules. With certain simplifying assumptions, equivalent to those used by authors who have analyzed the problem for linear polymers in a lattice frame of reference, the above equation reduces to the form obtained by the latter method. Various methods of obtaining experimental values for b are outlined.

The Study of Reaction Intermediates by Means of a Mass Spectrometer Part I. Apparatus and Method

Abstract: A method for coupling a reaction chamber to a Dempster‐type mass spectrometer in such a manner that short lived intermediates can reach the electron beam is described The presence of radicals is denoted by an increase in the ion current of the corresponding mass The sensitivity of the method depends on the difference between the appearance potential of the given ion produced by electron bombardment of either reactants or end products and the ionization potential of the free radical It is shown experimentally that excited molecules are not present in sufficient quantity to invalidate the assumption that free radicals are responsible for the changes in ion current The method has already extended more than tenfold the pressure range in which radicals are detectable, thus bridging the gap between mirror and spectroscopic methods so that low pressure combustion phenomena may be studied

The Mechanism of the Luminescence of Solids

Abstract: In this paper, effort is directed toward explaining various diverse luminescent properties of solids in terms of a simple model of potential energy versus configuration coordinate. Three states of different multiplicities—a normal, metastable and an emitting state—are involved. The luminescent process consists of the excitation of an electron to the metastable level, the activated release of the electron to the emitting level, and a forbidden transition between the emitting and the ground state. With high excitation energy, the metastable state is bypassed. At high temperatures, the electron in the metastable state surmounts a larger potential barrier to undergo radiationless recombination with the activator atom.Among those significant phenomena that have been measured experimentally and treated quantitatively by calculations based on this model are the temperature dependence of luminescent efficiency and the effect of type and wave‐length of excitation on this temperature dependence; the three types of ...

Length Changes of Activated Carbon Rods Caused by Adsorption of Vapors

Abstract: The length variations of two zinc chloride activated carbon rods caused by the adsorption of several vapors are reported. The relationship between percent extension of the rod and calculated spreading pressure of the adsorbed film suggested by Bangham has been tested. The relation agrees well with experiment for organic vapors above about ⅓ saturation of the monolayer capacity. It does not agree in the low adsorption region nor along the desorption branch of the water isotherm. A marked contraction of the rod below its original length has been noted when water is desorbed, and this contraction may be considered as a proof of the establishment of concave menisci in the capillaries along the desorption branch.

Derivative Crystal Structures

Abstract: There exists a class of crystal structures which are derived from others by generalization. These are here designated derivative structures. A special case of derivative structure is the popular ``superstructure.'' The symmetry of a derivative structure is a subgroup of the symmetry of the basic structure. Here the term ``subgroup'' is used in a wider sense than common in mathematical crystallography, and includes subgroups with multiple cells. The methods of finding the derivative symmetries are discussed. There are two important kinds of derivative structures, here designated substitution structures and distortion structures. Substitution structures result when a set of different atoms is substituted for a set of like atoms in the basic structure. For a given kind of substitution, it is possible to predict the cell dimensions of the crystal resulting from the substitution, as well as all the possible symmetries it could have. Examples are cited of the applications of derivative structure theory to several branches of crystallography.

Heat-Capacity Lag Measurements in Various Gases

Abstract: The impact‐tube method of measuring relaxation times for the transfer of molecular energy from translational to internal degrees of freedom has been introduced in a previous report. This method has been applied to a series of gases to make measurements of relaxation times which would be difficult to measure quantitatively by sonic methods. Measurements of the relaxation times of the vibrational energy of H2O, N2, N2 catalyzed by H2O, and CCl2F2 are presented. The existence of a measurable lag in the adjustment of the rotational heat capacity of H2 has been confirmed, and measurements of its relaxation time are presented.

Statistical Mechanics of Multimolecular Adsorption. III. Introductory Treatment of Horizontal Interactions. Capillary Condensation and Hysteresis1 a

Abstract: The elementary treatment of interactions and first order phase changes in localized and mobile monolayers is reviewed. Calculations on multimolecular adsorption and capillary condensation, which take into account horizontal interactions in an approximate manner, are presented. Several other topics are discussed briefly, including the quasi‐chemical method and reversible hysteresis. It is suggested, on the basis of the capillary condensation calculations, that reversible hysteresis involves the use of metastable portions of the adsorption isotherm. The theory predicts, in agreement with experiment, that the adsorption curve should be below the desorption curve.

The Experimental Determination of the Intensities of Infra-Red Absorption Bands. III. Carbon Dioxide, Methane, and Ethane

Abstract: The absolute intensities of the chief infra‐red absorption bands of carbon dioxide, methane, and ethane have been determined experimentally to be (in cycles per centimeter at N.T.P.): Carbon dioxideMethaneEthane670 cm−1 band:560×10101306 cm−1 band:450×1010 820 cm−1 band: 73×10102350 cm−1 band:8600×10103020 cm−1 band:900×10101450 cm−1 bands: 230×10102950 cm−1 bands:2200×1010 These results are in satisfactory agreement with available data on infra‐red dispersion and atomic polarization. They may be interpreted in terms of dipole moments, μ′, of bonds between vibrating atoms, and their rates of change with internuclear distance, ∂μ′/∂r. When this is done for the CH bond, however, unexpected variations in these quantities are found for the different vibrations and molecules studied. The average values are about μ′=0.4×10−18, ∂μ′/∂r=±0.6×10−10. For the CO bond the values are considerably larger.

The Experimental Determination of the Intensities of Infra‐Red Absorption Bands II. Measurements on Ethylene and Nitrous Oxide

Abstract: The absolute intensities of the chief infra‐red absorption bands of ethylene and nitrous oxide have been determined experimentally to be (in cycles per centimeter at N.T.P.): Ethylene Nitrous oxide 950 cm−1 band: 1540×1010590 cm−1 band: 120×10101444 cm−1 band: 210×10101285 cm−1 band:1150×10103050 cm−1 band: 840×10102224 cm−1 band:5600×1010 From these intensities it is possible to calculate the dipole moments, μ, of bonds between vibrating atoms and also the rates of change of dipole moment with inter‐nuclear distances, ∂μ/∂r. The values of μ obtained are all of the order of magnitude of 0.5×10−18 c.g.s. unit, while values of ∂μ/∂r are of the order of 1×10−10 c.g.s. unit. The results are in agreement with available data on infra‐red dispersion and on atomic polarization within the limits of experimental error, which are estimated to be approximately ±10 percent.

Ignition of Explosive Gas Mixtures by Electric Sparks. I. Minimum Ignition Energies and Quenching Distances of Mixtures of Methane, Oxygen, and Inert Gases

Abstract: An apparatus and experimental procedure is described for measuring capacitances and gap voltages of condensed electric spark circuits for sparks just powerful enough to ignite explosive gas mixtures. Mixtures of methane, oxygen, and inert gases are investigated. From the measured capacitances and gap voltages the minimum ignition energies are calculated. These energies are found to be independent of gap voltage. With increasing gap length they attain a minimum at critical distances which mark the farthest penetration of the flame‐quenching effect of the electrode material. Above the quenching distances the energies remain constant over some range which is governed by mixture composition and pressure. Energies measured in this range may be regarded as absolute minimum energies, as defined in a subsequent paper. Data of such minimum energies and of quenching distances are presented for mixtures at room temperature and pressures ranging from 0.2 to 1 atmosphere.

Solubilization by Solutions of Long‐Chain Colloidal Electrolytes

Abstract: Increase in the concentration of a soap or other detergent does not increase the solubility of an oil above that in water until the critical concentration for the formation of micelles (cmc) is attained. Above this the solubility, designated as solubilization, increases and, in general, more rapidly as the soap concentration increases; i.e., per mole of soap the solubilization is greater in a 25 percent than in a 5 percent soap solution. For a homologous series the volume of oil solubilized at a constant temperature is to a first approximation inversely proportional to the molar volume. The polarity and shape of the molecules solubilized also play a role. Salts increase the extent of the solubilization; at low concentrations to an extent which may be accounted for by the increase in micellar area resulting from the depression of the cmc by the salt. At higher soap concentrations the increase in solubilization is greater than can be accounted for in this way.

A Raman Apparatus for Quantitative Polarization Measurements

Abstract: An apparatus for quantiative measurement of the depolarization factors of Raman lines is described. It is a modification of Edsall and Wilson's method, using polaroid cylinders. Short exposure times result from the use of eight exciting lamps. No apparatus corrections are needed. Alignment is not overly critical. Performance is reported on the Raman lines of CCl4, CHCl3, and C6H6.

Integral Equations between Distribution Functions of Molecules

Abstract: Integral equations are derived that relate variations in the potentials of average force between molecules of a system at two different densities or activities. These permit the calculation of the change in thermodynamic properties, or of the change in the distribution of molecules in space, in a liquid or crystalline phase, if either the temperature is varied, or if the the mutual potentials between the molecules is assumed to change. The equations are in a somewhat complex, but still distinguishable, matrix form. A matrix operates on the variations in potential occurring at one activity and transforms them into those occurring at a second activity. The matrix elements are combinations of the distribution functions at the second activity, to which the transformation is made, multiplied by powers of the difference of the two activities. The matrix approaches the unit matrix in value as this activity difference approaches zero. The product of the two matrices, one which transforms from activity α to activi...

Theory of Burning Velocity. II. The Square Root Law for Burning Velocity

Abstract: Previous papers by the authors having suggested that the diffusion of active particles (chiefly hydrogen atoms) from the flame front is the controlling factor in combustion in Bunsen‐type burners, an equation for burning velocity based on such a concept is derived. This equation shows that the burning velocity should be proportional to the square root of an expression in which the most important factor is the sum of the products piDi for all effective atoms or free radicals, pi being the concentration of an atom or radical at the flame front, and Di its coefficient of diffusion into cold, unburnt gas. The burning velocity equation is applied to the combustion of carbon monoxide and hydrogen, and excellent agreement between calculated and experimental values of the burning velocity is obtained. The equation is also used to account for the effect of pressure upon burning velocity.

Separation of the Nitrogen Isotopes by the Exchange Reaction between Ammonia and Solutions of Ammonium Nitrate

Abstract: The fractionation factor for the exchange reaction between ammonia and ammonium nitrate was determined as a function of the dissolved ammonia content. It was found that for a stock solution containing 59.3–59.6 percent ammonium nitrate α=−0.029 M+1.034, where α is the fractionation factor at 25°C and M the fraction (Moles NH3)/(Moles NH3+Moles NH4NO3), in solution. From these data, the equilibrium constant for the exchange reaction N14H4+(sol)+N15H3(gas)⇌N15H4+(sol)+N14H3(gas) was calculated to be 1.034, while that for the reaction N14H3(sol)+N15H3(gas)⇌N15H3(sol)+N14H3(gas) was found to be 1.005. Vapor pressure and density data were determined for the ammonia‐ammonium nitrate solutions as a function of the ammonia concentration.