Restricted Rotator Thermodynamic Properties from the Old Quantum Theory

Accurate partition functions and free energies are calculated for the hindered internal rotator from the old quantum theory based upon the quantization rule of Wilson and Sommerfeld. The description of the system is incomplete and the energy levels may even be incorrectly placed, but the partition functions are surprisingly close to those now accepted as correct. In the course of the development the lowest energy state of the harmonic oscillator turns up in its correct position at the half‐quantum level.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70358/2/JCPSA6-15-9-645-1.pd

Entropy of the Monomeric Forms of Formic Acid and Acetic Acid

A recent measurement of the entropy of gaseous formic acid at its equilibrium vapor pressure is combined with vapor density data to give 60.0±0.3 as the entropy of the monomer at 25° and one atmosphere. The corresponding dimer entropy is 83.1±0.3 e.u. Pauling's residual entropy of R ln 2 per dimer weight for random orientation of hydrogen bonds in the crystal has been included in the above figures in order to allow a reasonable entropy for the torsional motion of the hydroxyl group in the monomer. The same correction has been applied to a previously published result for the acetic acid monomer to yield 70.1±1.0 as a revised value. It is highly probable that only a single significant potential minimum of undetermined breadth and depth occurs in the rotational cycle of the hydroxyl group.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70884/2/JCPSA6-10-9-582-1.pd

Thermodynamic Properties of the Internal Rotator. Double Minimum with Attractive Forces

A previously outlined general method of calculating thermodynamic properties for an unsymmetrical internal rotation is applied to a double minimum with attractive forces, for which the potential energy has two maxima at the same height and two minima at different heights. The entropy is lower than that of the system of two equal potential valleys or that for a single valley, and can be lower than the former by as much as R ln2 when the potential barriers are high. This decrease tends to disappear in the region of close approach to free rotation, when the barriers are lower than RT. For very high potential barriers, if one of two equal minima is raised, the system acquires the properties of a simple harmonic oscillator.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70066/2/JCPSA6-16-6-560-1.pd

Normal Frequencies of a Simple Cubic Lattice. III

For a simple cubic crystal, with forces resisting displacement along and perpendicular to the connecting line between closest neighbors, the secular equation for the internal motions is rigorously broken down to ultimate factors of the formν2=νx2+α(νy2+νz2),in which νx, etc., are the normal frequencies of the one‐dimensional crystal.The corresponding frequency distribution shows no indication of the ``infinities'' found by Montroll for a model with central forces between first and second neighbors.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70300/2/JCPSA6-20-5-822-1.pd

Comparative Thermodynamic Properties of the Restricted Rotator and the Harmonic Oscillator

The differences between the thermodynamic properties of the restricted internal rotator and a defined harmonic oscillator are expressed within narrow limits as a function of the single variable V/RT, where V is the height of the restricting potential barrier. As a result, a single table with one column for each of the four thermodynamic functions can be used in conjunction with standard harmonic oscillator tables to give the restricted rotator properties. This approximation is accurate within 0.02 cal./mole/deg. for nearly all cases to which it might be applied, and it is shown to be improbable that rotators will be found for which the error would rise appreciably above this limit. The harmonic oscillator chosen for comparison has a slightly lower frequency than the limiting one approached by the restricted rotator at very high values of V/RT.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/71278/2/JCPSA6-15-6-364-1.pd

The Hydrogen Isotope Exchange Reaction

The possibility of deriving the equilibrium constants for the higher deuterium substitutions from the isotope exchange coefficient is examined with the aid of calculations on the ammonia and methane reactions with water. If the experiment is carried to solutions at equilibrium with water containing as much as 25 percent D2O and the error in a single measurement does not exceed 1.0 percent the second constant is determined to within 10 percent of the correct value. As a standard of comparison against which the experimental results may be examined to advantage, an equation is developed showing the relation between the equilibrium constant for the first substitution and the exchange factor at higher deuterium fractions, subject to the assumption that each successive substitution of D for H produces the same frequency shift. The evaluation of frequency shifts from exchange data is considered.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70397/2/JCPSA6-8-3-243-1.pd

Contribution of the Deviation from Perfect Gas Behavior to the Entropy and Heat Capacity. Water and Benzene

The properties of water vapor are used to illustrate the point, previously found true for ethyl alcohol, that the modified Berthelot equation of state may not give a satisfactory evaluation of the deviation of thermodynamic properties from those of the hypothetical perfect gas. The entropy deviation for benzene, however, is accurate enough for most purposes, and the heat capacity is not too far out of line.On limited information, it appears that the Berthelot equation may be satisfactory for the vapors of ``normal'' liquids up to limited pressures. It is recommended, however, that some attempt be made to check the properties by means of vapor densities, obtained directly or by the Clapeyron equation, before results based upon the Berthelot equation are accepted.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/71070/2/JCPSA6-17-4-405-1.pd

Energy Levels and Thermodynamic Properties of the Internal Rotator

To facilitate the calculation of the thermodynamic properties of internal rotators lying outside the limits of available tabulations, methods of determining the energy levels are studied for the purpose of finding the most expeditious route to accurate results. It proves to be unnecessary in any case to calculate many levels from the wave mechanics. A partition function is set up in terms of the correct ground state and a series of approximate levels derived from the old quantum theory, and this is corrected to constancy by the successive substitution of correct levels, starting with the first excited state. A set of levels obtained in this way which gives the correct partition function always leads to correct values of all the thermodynamic properties.To obtain accurate energy levels, the continued fractions of Koehler and Dennison are recommended as most convenient.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70165/2/JCPSA6-18-4-444-1.pd

Stretching Force Constant of the Carbonyl Bond in Unconjugated Ketones

The secular equation derived for the normal frequencies of a Y‐shaped C2CO model can be used with fair accuracy to obtain the carbonyl bond force constant from the measured carbonyl frequency of any symmetrical ketone if the carbon‐carbon force constant and the bond angles are known or can be reasonably assigned.The narrow spread of carbonyl frequencies observed for unconjugated ketones, with assumed ``normal'' angles and carbon‐carbon force constant, indicates a nearly constant carbonyl stretching force constant in the range 10.2±0.3 (×105 dynes/cm). This value should be equally reliable for unsymmetrical unconjugated ketones.In formaldehyde, the calculated force constant is usually above 12.5 units.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70330/2/JCPSA6-24-4-830-1.pd

Force Constant of the Association Bond in the Formic and Acetic Acid Dimers

The entropies of the formic and acetic acid dimers are used to find a minimum value for the force constant of the association bond. With the aid of a normal coordinate treatment the probable value is found to be 4×104 dynes/cm. This agrees with a rough estimate from the heat of dimerization. A Raman band in the formic acid spectrum at 200 cm−1 may be due to the association vibrations.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/69883/2/JCPSA6-14-6-395-1.pd