Statistical weight factors

Finally, we have applied equation (10.14) to a collection of harmonic oscillators. But it can be applied to any collection of energy levels and units of energy with one modification. Equation (10.14) assumes that each level has an equal probability (as in a harmonic oscillator), and this is true only if g, the degeneracy, is one. The quantity g, is also known as the statistical weight factor. If it is greater than one, equation (10.14) must be multiplied by the g, for each... 

Statistical weight factors are present in rroi and re eci. The multiplication involving these partition functions still works, since the total statistical weight factor is the product of the statistical weight factors for the individual units. Thus g, =g,.,rans g, g,.Mb g,. elect... 

Calculation of Thermodynamic Properties We note that the translational contributions to the thermodynamic properties depend on the mass or molecular weight of the molecule, the rotational contributions on the moments of inertia, the vibrational contributions on the fundamental vibrational frequencies, and the electronic contributions on the energies and statistical weight factors for the electronic states. With the aid of this information, as summarized in Tables 10.1 to 10.3 for a number of molecules, and the thermodynamic relationships summarized in Table 10.4, we can calculate a... 

Debye heat capacity equation 572-80 Einstein heat capacity equation 569-72 heat capacity from low-lying electronic levels 580-5 Schottky effect 580-5 statistical weight factors in energy levels of ideal gas molecule 513 Stirling s approximation 514, 615-16 Streett, W. B. 412... 

The nuclear spin statistical weight factors for " NH3 are determined in Section 8.4.1 of Ref. [3] and we do not repeat the derivation here. The results are summarized in Table 2. The 24 nuclear spin functions (see Chapter 8 in Ref. [3])... 

The characteristic ratio changes from 1.3 to 2.8 with o changing from 0 to 1, when the virtual bond is used. On the other hand, when each bond of the phenylene group is taken into account individually, the two extreme values are 3.41 and 7.40. By assuming all the statistical weight factors to be unity, which corresponds to the freely-rotating chain, the characteristic ratio is 1.60 when the virtual bond is used, and 4.22 if it is not. 

Mean-square unperturbed dimensions a and their temperature coefficient, d tin 0) I d T, are calculated for ethylene-propylene copolymers by means of the RIS theory. Conformational energies required in the analysis are shown to be readily obtained from previous analyses of PE and PP, without additional approximations. Results thus calculated are reported as a function of chemical composition, chemical sequence distribution, and stereochemical composition of the PP sequences. Calculations of 0 / nP- are earned out using ( ) r r2 = 0.01, 1.0, 10.0, and 100.0, (ii) p, = 0.95, 0.50, and 0.05, liii) bond length of 153 pm and bond angles of 112°for all skeletal bonds, iv) = 0 and 10°, and (v) statistical weight factors appropriate for temperatures of 248, 298, and 348 K. Matrices used are ... 

The first term represents a statistical weighting factor, the second term a solid angle factor for the probability of getting the angle 0, and the third term is a symmetric top wave function. 

While in the photodissociation of H2O through the AlB state the two possible A-doublets of OH(2n) are populated in a highly nonstatistical way, the two spin-orbit states, 0H( n ) and 0H( n3/2), are perfectly statistically populated. Unlike for the A-doublets there is a priori no geometrical reason to expect a difference in the spin-orbit states other than that given by the 2j + 1 statistical weighting factor. Since j = N + 1/2 for 2n3/2 and j = N — 1/2 for 2n1/2, the statistical weighting factor is (N + l)/N. Therefore, the population ratio 2n3/2 / 2n1/2 multiplied by N/(N-1-1) must be 1 for a statistical distribution as it is indeed measured in the bulk, in the beam, as well as in the dissociation of single rotational states of H2O (Andresen and Schinke 1987). The reason for the statistical... 

The statistical weight factor of each electronic level, normal or excited, is equal to 2j + 1, where 7 is the so-called resultant quantum number of the atom in the given state. The expression for the electronic factor in the partition function is then given by... 

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