PROPOSED ARTICLE

This paper discusses circumstances culminating in the development of Flint's methodology and illustrates its potential utility. For 100% of the 22 CRC‑listed⁸ mono‑atomic positively‑charged elemental ions with Z = 1-19 or 55-80, plus OH‑, observed conductivities are within 10% of calculated values, assuming either maximum or zero hydration as per Flint (Table 1); calculations for both H+ and OH‑ ions involve the assumption of zero hydration, indicating that this and not "quantum effects"⁴ may be primarily responsible for their high conductivities.

Also discussed, and based on this same algebraic methodology, are agreement with alternately‑determined relative hydration values (e.g. as in Table 2), approximations of both linear size and conductivities for two sets of ions (Table 3), correlations with relative energies of solution (Table 4), patterns corresponding to Mendeleev's periodicity (Table 5), and a proposed explanation for the six‑cornered snowflake.

Background

While he is best known for his work on the periodicity of the elements, Dimitri Mendeleev was also, by 1887, an early and strong supporter of the hydrate theory of solutions⁹. Bredig's subsequent use of the hydrate concept in 1894 to explain why the chlorine ion is not more mobile than the heavier iodine ion, attributing this to a larger hydration shell around the Cl‑ ion that migrates with it¹⁰, has remained essentially unchanged up to the current time¹¹.

In 1899 Abegg and Bodländer sought to explain why mobility of the K+ ion is greater than that of the smaller and lighter Na+ ion, and the latter greater than that of the even smaller and lighter Li+ ion, with the suggestion that the hydration potential of these ions may vary inversely with their anhydrous weights¹². This would allow for the hydrated Li+ ion to be the heaviest and least mobile of the three ions, and the much‑less‑hydrated K+ ion to be the lightest and most mobile. In accord with this same premise, they offered that the large conductivities of H+ and OH‑ might be attributed to an absence of hydration. Over the next few decades, other investigators similarly invoked the concept of hydration in discussions of conductivities^13-16; however, it was not until 1932, when Flint called on Graham's law¹⁷ to deliniate the relationship between hydration and conductivity, that a mathematical framework for Abegg and Bodländer's suggestions was advanced, exposing an apparent algebraic relationship between solute ions and their water solvent.

In accord with Graham's law of diffusion (or effusion, of

gases), the mobility of a gas varies inversely with the square root of its (density or) molecular weight¹⁷, a relationship required by the more recent law of kinetic energy¹⁸ and one which may be derived from the right side of Newton's equation characterizing resistance as varying "directly as the squares of velocity and ... as the quantities of matter"^19a. (Graham himself credited Professor John Robison with having deduced this "pneumatic law" directly from the pre‑Newtonian theorem of Torricelli on the velocity of efflux of fluids¹⁷.) As solute behavior had been shown by van't Hoff to be analogous to gaseous²⁰, and conductivity was recognized as an index of mobility^21,22, Flint was able to view conductivity values as a measure (inverse‑square‑root) of relative hydrated weights of solute ions. In essence, Flint's work rendered the Wheatstone bridge, which measures conductivities, a "scale" for "weighing" solute ions.

(Van't Hoff's successor at the Royal Prussian Academy in 1914, Albert Einstein²³, had as early as 1907 used the relationship embodied in Graham's law in conjunction with solutions, characterizing velocity as varying with the inverse-square-root of mass for particles in colloidal platinum solutions²⁴. But Einstein apparently did not attempt to extend this characterization to considerations of conductivity, although he did refer to conductivities of H+ and K+ ions in a 1908 discussion of "displacement"²². Nor did Walter Nernst¹⁵, who was eulogized in 1933 by Einstein as "having ascended from Arrhenius, Ostwald and Van't Hoff, as the last of a dynasty which based their investigations on thermodynamics, osmotic pressure and ionic theory"; Nernst was specifically remembered by Einstein for his "witty" use of the Wheatstone Bridge²⁵.)

In 1932² Flint cited Nernst conductivity values for K+, Na+ and Li+ of 65.3, 44.4, and 35.5, resp., as listed W. Bayliss's 1915 physiology textbook¹⁴. (On an adjoining page of this text were listed W. Bousfield's proposed hydration numbers 4, 8 and 16 for these same ions, a self‑proclaimed "attractive looking series"²⁶.) Inspired by Abegg and Bodländer^3b, Flint sought to determine hydration numbers that would allow for an optimum fit with the Nernst data, via Graham's law. He found that presumed hydration numbers of 4, 12, and 20, each multiplied by 18 (for the weight of each water unit) and added to conventionally‑ appraised atomic weight values of (approx.) 39 for potassium (K), 23 for sodium (Na) and 7 for lithium (Li), resp., yielded totals (111, 239 and 367) whose relative inverse‑square‑roots (.0949, .0647 and .0522) correlated remarkably well with the Nernst data.

As his initially‑hypothesized hydration numbers (4, 12, and 20) were integers, each of which when added to its associated atomic number (19, 11 and 3 for K, Na and Li, resp.) equalled 23, Flint was induced to attempt to use integral values based on atomic numbers in his conductivity calculations^3c. He then found that adjusted atomic‑number‑equivalent values (Z') which incorporated a shift equal to one unit Z per unit valence (C) "permitted an integration of observational data with a satisfying convincing nicety"^3d (equation 1):

Z' = Z+C, (1)

where Z=atomic number and C=valence. (Flint's use of values based on atomic numbers is consistent with contemporary recognition of the primacy of the atomic number²⁷ as established by Moseley²⁸; contemporary science also embraces the concept of a shift in atomic‑number‑equivalent values with ionization in the case of an increase in the atomic number of some radioactive elements resulting from the loss of a nuclear electron²⁹.)

Noting that the number 23 equals one‑fourth of the number (92) of naturally‑occurring elements, Flint projected four equal periods, in each of which the maximum hydration number (Hmax) decreases from 23 to zero with increasing atomic number^2,3e (equation 2):

Hmax = 23n ‑ Z', when Hmax = 23 to 0, and n = 1 to 4 (2)

(for Z' = 0 to 23, n=1;

for Z' = 23 to 46, n=2;

for Z' = 46 to 69, n=3;

for Z' = 69 to 92, n=4).

In his calculations Flint characterized water units involved in hydration as negatively‑charged ions (H2O‑)^3e, allowing for calculation of the relative hydrated Z' value (Z'h) for a given solute ion as the sum of Z' for the ion plus a value of 9 for each associated H2O‑ unit (equation 3)^3a:

Z'h = Z' + 9H, (3)

where H is the actual number of associated H2O‑ units.

Approximations of Ionic Data

Table 1 displays observed conductivities for the 41 monoatomic elemental ions through Z=80, plus OH‑, listed in CRC⁸ and error values (E) for approximations derived as the relative inverse‑square-root of Z'h, under one of four hydrational assumptions ‑ maximum, zero, half‑maximum and one‑fourth‑maximum hydration levels. (Throughout this paper, "maximum" or "full" hydration levels are maximum levels as proposed by Flint.) Maximum and zero hydration were the primary assumptions employed by Flint, but he also introduced the concept of shared water ions within discussions of hydrational bonding and utilized fractional‑maximum-hydration levels in associated calculations^3f.

The specific impetus for the use herein of a speculative half‑maximum-hydration assumption is the observation that, as measured against presumed maximum hydration for the base ion Na+, calculations of both conductivity and diameter for the Cl‑ ion are optimum using an hydration number of 3.5 (as in Table 3), a value first proposed by Bousfield in 1906³⁰. It was noted that this value is equal to half the maximum hydration number as per Flint, and further that a half‑maximum-hydration assumption allows for agreeable conductivity calculations for F‑ and for the 6 divalent "transition‑series" ions Mn++, Fe++, Co++, Ni++, Cu++ and Zn++. Thus conductivities for all but 7 of the 42 ions in Table 1 are within 10% of calculated values, assuming either maximum, zero or half-maximum hydration levels. A single fourth (speculative) assumption of a one‑fourth-maximum hydration level allows for approximations of conductivities for 5 of these 7 with error values less than 10%.

(It may be noted that each of the four hydrational assumptions as utilized in conductivity calculations in Table 1 is associated with a distinct grouping of atomic‑number‑equivalent values [Z'], which groupings tend to correspond with Flint's periodicity: The assumption of maximum hydration is associated with all six CRC‑listed⁸ univalent positive ions with Z'=4 to 20, all within Flint's first hydrational period; half‑maximum with all six divalent "transition‑series" ions with Z'=27 to 32, all within Flint's second hydrational period; one‑fourth‑maximum with all three CRC‑listed⁸ trivalent "transition‑series" ions with Z'=24 to 29, all within Flint's second hydrational period; and zero hydration with all of 16 CRC‑listed⁸ ions with Z'=52 through 82, values falling within Flint's third and fourth hydrational periods.)

Calculations of relative mobilities of ions, aside from playing an indispensable role in Flint's initial work on hydration, were subsequently used by Flint in discussions of the concept of "Ph" and development of a "tentative" acidity-alkalinity index^31,3g.

Although efforts to determine hydration numbers have yielded a wide range of values, the ratio of values for Na+ v.s. Li+ per Flint (11/19 = .58) correlates reasonably well with the average (.61) for 14 sets of values by 9 investigators as listed by Amiss³². Even more striking is how Gluekauf's relative hydration values³³ for the series H+, Li+, Na+, K+, Rb+ and Cs+, adjusted to a base value of 11 for Na+ as in Table 2, correlate precisely with Flint's assumed maximum (H+, Li+, Na+ and K+) or zero (Rb+ and Cs+) hydration levels; except for H+, all adjusted values in Table 2 are identical with hydration values used in Table 1.

Seeking to validate his work through the approximation of specific gravities, Flint proposed that the relative volume of a solute ion (VZ) may be approximated as the quotient of (a) Z'h and (b) 1 + Z'/Z'h (equation 4)^3h:

VZ = Z'h/(1+ Z'/Z'h) (4)

Flint used this formulation in deriving theoretical values for comparison with specific gravity data^3h, results of original osmosis experiments³ⁱ, gas solubility data^34a, and densities of elemental solids^35a, but is not known to have used it in conjunction with linear ionic size.

Table 3a lists observed relative diameter values for a set of 8 solute ions of physiological interest³⁶, observed conductivities, and approximation-error values for both sets of data. Diameters are approximated as the cube-root of VZ, relative to a base value of 1.00 for Na+; conductivities are approximated as the inverse-square-root of Z'h, relative to 50.08 for Na+ as in Table 1.

As shown in Table 3a, observed diameters for 7 of the 8 listed ions (all except HPO4‑‑), and observed conductivities for all 8 ions, are within 11% of calculated values, assuming a "maximum" hydration level in all cases except (a) a half-maximum-hydration assumption used in calculations of both diameter and conductivity of Cl‑, and (b) a zero hydration assumption used in calculation of conductivity of SO4‑‑. In the case of HPO4‑‑, conductivity has been approximated using the assumption of maximum hydration, but diameter may be approximated (as relative cube-root of VZ) only if ionic size is assumed to be even larger -- approximately equal to that of two fully hydrated ions. (Such an hypothesized double ion might then be projected as splitting under electrical stress into separate, "fully" hydrated ions.)

Table 3b lists observed crystal ionic radii³⁷ and approximation error values for the series Li+, Na+, K+, Rb+ and Cs+. Radii are approximated as the relative cube root of theoretical volume, as were diameters in Table 3a, except that for all ions in Table 3b approximations of radii involve the assumption of zero hydration. While the order of agreement for approximations of linear size is less satisfactory in Table 3b than in Table 3a, it may be noted that the trend in error values in Table 3b for Li+, Na+ and K+ radii is consistent with Flint's discussion of how the attractive force mediating hydration may also act as a relative cementing force in solids^35a; this might allow for the Li+ ion, with a greater hydrational/attractive potential than Na+, to be more compressed in crystal form, and the K+ ion, with a lesser potential than Na+, to be less compressed. (Conductivity data and approximation error values from Table 1 are duplicated in Table 3b for convenient reference.)

Hydrational potentiality per Flint is correlated with components of energies of solution³⁸ in Table 4. Relative square‑roots of Z'h values associated with maximum hydration for Li+ and Na+ against half‑maximum hydration for F‑ and Cl‑ (assumptions used in Table 1) correlate reasonably well with "total solvent energy" values (/\Esol); whereas relative square‑roots of Z'h values associated with "maximum" hydration for all 4 ions yield comparably‑satisfactory approximations of the "bulk solvent" portion (ESX2) but not first shell portion (ESX1) of total solute‑solvent energy (ESX; ESX = ESX1 + ESX2). For the first shell (ESX1) a shift in assumptions for the Cl‑ ion, from a maximum to a half‑maximum hydration level, yields an improved correlation for Cl‑ relative to calculations based on full hydration for the other three ions.

Periodicity

In attempting to correlate Flint's periodicity with that of Mendeleev, it was noted that Z values associated with three (Mendeleev) Group 1 elements (Z = 3, 29 & 55) are associated with consecutive Z'h values (Z'h = 183, 182 & 181, resp.) Consequently, Z'h values for all values of Z from one through 92 have been calculated and exhibited in order of ascending Z'h values, as in Table 5. (Z'h values are calculated as per equation 3, assuming full hydration as per equation 2 and C=0 in equation 1.) These Z'h values fall into groupings of up to four consecutive values (e.g., 148, 149, 150, 151), each value 8 digits distant from the same position in the next grouping (156, 157, 158, 159).

Data in Table 5 is arranged in five columns with a difference between Z'h values horizontally of 64, which arrangement allows all atomic numbers from Z=2 through Z=57 that fall within Mendeleev's Groups 0 through VII, except Z=24 & 25, to fall into corresponding horizontal groupings. The Z value for hydrogen (Z=1) is found with Z values for elements in Mendeleev's Group VII, where hydrogen is sometimes categorized due to some recognized chemical similarities³⁹. (A unique sort of balance is noted within the horizontal grouping encompassing Z = 2, 10, 18, 36 and 54, atomic numbers associated with Mendeleev Group 0: The Z'h value for Z=18 is 63, and all other Z'h values in this horizontal grouping are approximate multiples of this value. Also noted among horizontal groupings in Table 5 is that, for Z=2 through 57, none of six prime-numbered Z values in Mendeleev's Group I is paired with a Z'h value which is also a prime, whereas two of three prime Z values within each of Mendeleev Groups III, V & VII are paired with Z'h values which are also prime numbers.)

Flint's periodicity also dovetails nicely, geometrically, with that of R. B. Fuller: As per equation 2, Flint's four (overlapping) hydrational periods of 24 digits each (including "0" and duplicates of "23", "46", and "69"), terminating in the number 92, bears an uncanny resemblance to R. B. Fuller's "four of the 24‑ness of the duo‑tet cube" (and "disappearing octa set" of one "expendable exterior octa" and "three expendable interior octa")⁴⁰. A prospective "disappearing octa set" is virtually gone when Flint's periodicity is characterized continuously and three‑dimensionally as a quadruple helix.

Conclusions and Speculations

Beyond arguing for consideration of Flint's methodology as may relate to a general understanding of solute phenomena, this paper specifically supports (1) Bousfield's hypothesis of an hydration number of 3.5 for Cl‑ (incident to measurement of conductivity and diameter as in Table 2), offering that this may represent a level equal to one‑half the maximum hydration level per Flint and further that conductivities of several other ions may similarly indicate hydration levels which may be derived as distinct fractions of respective maximum hydration levels; (2) the hypothesis that the sulfate ion, fully hydrated when measured for diameter, may shed its water complement under electrical stress incident to measurement of conductivity; and (3) the work of Abegg and Bodländer¹², and others^15,16, suggesting that the large conductivities of the OH‑ and H+ ions in solution are due to a relative absence of hydration, although these ions may otherwise be strongly hydrated as has been inferred from studies of specific gravities of acidic aqueous solutions^35b and relatively large heats of hydration⁵.

The ease with which Flint's periodicity may be meshed algebraically with the eight‑fold symmetry of Mendeleev's periodicity invites speculation that (1) on the ionic level some form of simple attractive force, e.g. one related to total atomic‑number‑equivalent values for hydrated units calculated in accord with Flint's methodology, may underly aspects of chemical behavior; and (2) the helical structure of Flint's periodicity might relate to larger scale natural helical patterns^41-43.

The similar emphasis on the system of 92 naturally‑occurring elements in both Flint's and Fuller's periodic structures and the latter's relation to the number of spheres in the third layer of closest‑packed spheres (92)⁴⁴ draws attention to the coincidence that numbers of spheres in the first and second closest‑packed layers (12 and 42, resp.) are identical to Einstein's calculated "Z1" values for "pure gravitational" and "non-symmetric" fields^6b. (It may also be noted that the terminal number of the natural periodic system [92] falls within the first interval between prime numbers that exceeds five digits [90‑96 inclusive] and is at a distance [3 digits] from the previous prime number [89] which is approximately equal to the average of 22 intervals [2.9] between non‑consecutive prime numbers through 89.)

And what of the geometry of snowflakes? The open arrangement of atoms in ice crystals argues that hexagonal close‑packing is "irrelevant to an explanation of the hexagonal shape of snow crystals"⁴⁵. However, an alternate explanation of how the emerging snowflake might avoid what Kepler described as "the slippery slope into chaos"^46a may be found in Flint's projection of a role in the formation of water vapor for positively ionized molecular oxygen (O2+). Maximally hydrated with a complement of 6 H2O‑ units^3j, such an O2+ ion could release a water sextet when neutralized (presumably through contact with cold air) which would in the first instance, as per Kepler, "be essentially flat, incapable, that is, of combining with itself to form a solid body"^46b. If frozen in that shape, this could comprise Kepler's "6 points on a circle for 6 prongs to be welded on to them"^46c. Such an hypothesized role for the shadow of an O2+ ion in the formation of a snowflake might be viewed as consistent with Kepler's pondering "whether there is any salt in a snowflake and what kind of salt" which might account for its shape^46d.

(Another of Kepler's musings on the snowflake could similarly be viewed as anticipating the likes of Flint, the suggestion that "Some botanist might well examine the saps of plants to see if any difference there corresponds to the shape of their flowers"^46e. Three centuries later botanist Flint proposed to integrate studies of aqueous solutions, light sensitivity of plants and wavelengths of absorbed radiation of gases^47-49 in projecting chlorophyll structure^34b.)

Overall, the foregoing may be viewed as consistent with Einstein's desired "purely algebraic theory for the description of reality"^6a, and with Newton's suggestion that "the phenomena of Nature ... may all depend upon certain forces by which the particles of bodies by some causes hitherto unknown, are either mutually impelled towards one another, and cohere in regular figures, or are repelled and recede from one another"^18b. Spawned in the mainstream of classical scientific thought, Flint's proposed description of hydrational potentiality may come to serve as a fundamental tool in the quest for an understanding of the inner workings of the ponderable universe.

TABLE 1 Equivalent Ionic Conductivities

Z C Obs.(25⁰C)⁸ Calculation Error (by Hydrational Assumption)

(10^-4m² S mol^-1) (Hmax^2,3) (Hmax/2) (Hmax/4) (Zero^2,3)

9 OH ‑ 198 06

1 H + 349.65 ‑06

3 Li + 38.66 -03

4 Be ++ 45 08

9 F ‑ 55.4 ‑09

11 Na + 50.08 <BASE> 00

12 Mg ++ 53.0 -02

13 Al +++ 61 03

17 Cl ‑ 76.31 00

19 K + 73.48 -05

20 Ca ++ 59.47 -37

21 Sc +++ 64.7 05

24 Cr +++ 67 06

25 Mn ++ 53.5 08

26 Fe ++ 54 07

26 Fe +++ 68 06

27 Co ++ 55 07

28 Ni ++ 50 ‑04

29 Cu ++ 53.6 01

30 Zn ++ 52.8 ‑02

35 Br ‑ 78.1 ‑02 ‑14

37 Rb + 77.8 ‑09

38 Sr ++ 59.4 09

39 Y +++ 62 04

47 Ag + 61.9 -19

48 Cd ++ 54 19 ‑03

53 I ‑ 76.8 05

55 Cs + 77.2 09

56 Ba ++ 63.6 -08

57 La +++ 69.7 02

58 Ce +++ 69.8 03

59 Pr +++ 69.5 04

60 Nd +++ 69.4 04

62 Sm +++ 68.5 05

63 Eu +++ 67.8 04

64 Gd +++ 67.3 04

66 Dy +++ 65.6 01

67 Ho +++ 66.3 05

68 Er +++ 65.9 05

69 Tm +++ 65.4 05

70 Yb +++ 65.6 06

80 Hg ++ 63.6 09

───────────────────────────────────────────────────────────────── Key to Table 1:

Hmax = maximum hydration number^2,3; see equation 2.

Calc. conductivity = k/(Sq. Rt. of Z'h);² k=527.62553; see eq. 3.

Error (in percent) = 100([Observed/Calculated] - 1).

────────────────────────────────────────────────────────────────

TABLE 2 Hydration Values Compared: Gluekauf³³ v.s. Flint^2,3

────────────────────────────────────────────────────────────────

Gluekauf Values (X 11/2 =) Adjusted Relative Values

H+ 3.9 21 >= Hmax per Flint

Li+ 3.4 19 >= Hmax per Flint

Na+ 2.0 <<BASE>> 11

K+ 0.6 3 >= Hmax per Flint

Rb+ 0 0 >= H value in Table 1

Cs+ 0 0 >= Hmax per Flint

────────────────────────────────────────────────────────────────

──────────────────────────────────────────────────────────────────

TABLE 3a Solute Ionic Diameters & Equivalent Conductivities Diameter Conductivity

(Relative to Na+) (10^-4m² S mol^-1)

Z C Hmax Ass. Obs.³⁶ E Ass. Obs.⁸ E

17 Cl - 7 Hmax/2 .65 -08 Hmax/2 76.3 00

11 Na + <BASE> 11 Hmax 1.00 00 Hmax 50.1 00

19 K + 3 Hmax .68 ‑02 Hmax 73.5 ‑05

31 HCO3 - 16 Hmax 1.12 -01 Hmax 44.5 11

31 CH3COO - 16 Hmax 1.22 07 Hmax 40.9 02

49 H2PO4 - 21 Hmax 1.39 11 Hmax 33 -04

48 SO4 -- 23 0 1.25 -03 Hmax 80 03

48 HPO4 -- 23 Hmax 33 ‑01

2(HPO4 ‑-) 2Hmax 1.76 08

═══════════════════════════════════════════════════════════════════

TABLE 3b Crystal Ionic Radii & Solute Equivalent Conductivities

Ionic Radius Conductivity

(10^-10m) (10^-4m² S mol^-1)

Z C Hmax Ass. Obs.³⁷ E Ass. Obs.⁸ E

3 Li + 19 0 .60 -09 Hmax 38.66 -03

11 Na + <BASE> 11 0 .95 00 Hmax 50.08 00

19 K + 3 0 1.33 18 Hmax 73.48 -05

37 Rb + 8 0 1.48 06 0 77.8 -09

55 Cs + 13 0 1.69 06 0 77.2 09

─────────────────────────────────────────────────────────────────

───────────────────────────────────────────────────────────────────

Key to Table 3a and 3b:

Hmax = "Maximum" hydration number per Flint, as per equation 2.

Ass. = Hydrational assumption

Obs. = Observed

E = % Error = 100([Observed/Calculated] - 1)

Diameter/radius calculated as cube root of relative VZ;

VZ calculated as per equation 4.

Conductivity calculated as inverse-sq.-rt. of relative Z'h;

Z'h calculated as per equation 3.

TABLE 4 Energy of Solution Components (kcal/mol)³⁸;

approximated as Relative Square-Root of Z'h values.

/\Esol^38a ESX1^38b ESX^38b ESX2^38b

──────────── ────────────── ────────── ──────────

Obs. Error Obs. Error Obs. Error Obs. Error

Li+ -165+4 05 -138 -08 -231 -06 -92 -04

Na+ <BASE> -125+2 00 -119 00 -195 00 -76 00

F- -111+4 08# -129 -04 -212 -04 -83 -04

Cl- -80+5 -02# -77 -01# -23 -143 -13 -66 03

Key to Table 4:

Error = 100([Observed Energy Component³⁸]/[Approximation]) ‑1;

Approximation = Sq. Rt. of Z'h, relative to Na+ (BASE)

= (Sq.Rt. of [Z'h/111])(Obs. value for Na+);

Z'h calculated as per equation 3³.

"#" - Denotes calculation based on half‑maximum hydration level;

all other calculations based on maximum hydration level³. ───────────────────────────────────────────────────────────────────

──────────────────────────────────────────────────────────────────

TABLE 5 Hydrated Atomic‑Number‑Equivalents (Z'h) per Flint³

v.s. Mendeleev Periodicity

──────────────────────────────────────────────────────────────────

1 2 3 4 5 Mendeleev

Z'h (Z) Z'h (Z) Z'h (Z) Z'h (Z) Z'h (Z) Group

148(85) 212(77) 276(69)

┌─────────────┐

85(67) 149(59)│ 213(51) │------------(V)

┌──────────────────┘ ┌──────────┘

│ 86(41) 150(33) │214(25)

┌─────────────┘ │

│ 23(23) 87(15) 151( 7) │

└───────────────────────────────────┘

92(92) 156(84) 220(76)

┌─────────────┐

93(66) 157(58)│ 221(50) │------------(IV)

┌──────────────────┘ ┌──────────┘

│ 94(40) 158(32) │222(24)

┌─────────────┘ │

│ 31(22) 95(14) 159( 6) │

└───────────────────────────────────┘

100(91) 164(83) 228(75)

┌────────────────────────┐

101(65)│ 165(57) 229(49) │------------(III)

┌───────┘ ┌──────────┘

│102(39) 166(31) │230(23)

┌─────────────┘ │

│ 39(21) 103(13) 167( 5) │

└───────────────────────────────────┘

108(90) 172(82) 236(74)

┌────────────────────────┐

109(64)│ 173(56) 237(48) │------------(II)

┌───────┘ ┌──────────┘

46(46) │110(38) 174(30) │

┌─────────────┘ │

│ 47(20) 111(12) 175( 4) │

└───────────────────────────────────┘

116(89) 180(81) 244(73)

┌────────────────────────┐

117(63)│ 181(55) 245(47) │------------(I)

┌───────┘ ┌──────────┘

54(45) │118(37) 182(29) │

┌─────────────┘ │

│ 55(19) 119(11) 183( 3) │

└───────────────────────────────────┘

124(88) 188(80) 252(72)

┌─────────────┐

125(62)│ 189(54) │253(46)

┌───────┘ ┌──────────┘

62(44) │126(36) │190(28)

┌─────────────┘ └──────────┐

│ 63(18) 127(10) 191( 2) │-----------------------(0)

└───────────────────────────────────┘

132(87) 196(79) 260(71)

┌─────────────┐

69(69) 133(61)│ 197(53) │-----------------------(VII)

┌─────────────────────┘ ┌──────────┘

│ 70(43) 134(35) │198(27)

│ │

│ 71(17) 135( 9) │199( 1)

└────────────────────────┘

140(86) 204(78) 268(70)

┌─────────────┐

77(68) 141(60)│ 205(52) │-----------------------(VI)

┌─────────────────────┘ ┌──────────┘

│ 78(42) 142(34) │206(26)

│ │

│ 79(16) 143( 8) │207( 0)

└────────────────────────┘

─────────────────────────────────────────────────────────────────
Key to Table 5:

Z = Atomic number

Z'h = Z + 9(Hmax); Hmax calculated as per equation 2, after Flint³.

Bold numbers denote Z values through Z=57 which are prime numbers

and Z'h values paired with these Z values which are also primes.

─────────────────────────────────────────────────────────────────

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