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|2||3Li 4Be||5B 6C 7N 8O 9F 10Ne|
|3||11Na 12Mg||13Al 14Si 15P 16S 17Cl 18Ar|
|4||19K 20Ca||21Sc 22Ti 23V 24Cr 25Mn 26Fe 27Co 28Ni 29Cu 30Zn||31Ga 32Ge 33As 34Se 35Br 36Kr|
|5||37Rb 38Sr||39Y 40Zr 41Nb 42Mo 43Tc 44Ru 45Rh 46Pd 47Ag 48Cd||49In 50Sn 51Sb 52Te 53I 54Xe|
The s-block elements pose a problem, since there are so few data points from which to discern a trend.
Indeed, it was decided not to model the first period (Z = 1 and 2), on the grounds that the model would involve defining more coefficients than there are data points.
However, it was considered useful to include terms for each of the four other periods, on the grounds of them being able to model effects that occur later in each period. In particular, to go some way towards modelling the occupancy of the 4s electron shell at Z=24 or 29, and of the 5s electron shell between Z=41 and 47.
Even so, it is still difficult to discern which expression to use for the terms, except for the observation that it needs to be an expression whose value reaches a peak at some central point, and tails off to have negligible effect away from that point.
Consequently, the y=K/(z2+a2) curve was chosen for the present, on the grounds that it tails off to positive values each side of the central point, as opposed to the A/(Z+B) term, that swings violently between positive and negative on either side of the central point.
This leads to the following expression:
bp = Ka2s/(1+Kc2s*(Z+Kb2s-4)^2) + Ka3s/(1+Kc3s*(Z+Kb3s-12)^2) + Ka4s/(1+Kc4s*(Z+Kb4s-20)^2) + Ka5s/(1+Kc5s*(Z+Kb5s-38)^2) + other terms
Putting these three expressions together (for the s, p and d-block elements of the 2nd to 5th periods) leads to following expression:
bp = Ka2s/(1+Kc2s*(Z+Kb2s-4)^2) + Ka2p/(Z+Kb2p-10) + Ka3s/(1+Kc3s*(Z+Kb3s-12)^2) + Ka3p/(Z+Kb3p-18) + Ka4s/(1+Kc4s*(Z+Kb4s-20)^2) + Ka4d/(1+Kc4d*(Z+Kb4d-30)^2) + Ka4p/(Z+Kb4p-36) + Ka5s/(1+Kc5s*(Z+Kb5s-38)^2) + Ka5d/(1+Kc5d*(Z+Kb5d-48)^2) + Ka5p/(Z+Kb5p-54)
The next job was then one of finding values for each of the coefficients.
This task was tackled by using techniques that are akin to those used in genetic algorithms. Indeed, the eventual aim is to use genetic algorithms to do the job, but up until now I have been content to implement the process manually: that is, to choose initial values for each of the coefficients, and then to experiment with varying each one in turn, keeping changes that lead to a better fit, and discarding those that lead to a worse fit (fortunately, the functions are all convergent, albeit with local minima dotted about).
The measure of the closeness of the fit is the traditional one of the variance: the sum of the squares of the difference between the predicted values and the actual values (for the boiling points, for example).
A decision was made to model, at least initially, the four functions (mp, bp, lhf and lhe) so that they had independent Ka and Kc coefficients, but so that they all shared the same set of Kb coefficients. Moreover, it was decided that the degree of fit for the particular choice of Kb coefficients should be the result of working on the simple sum of the four variances. This has the side effect of naturally weighting the influence of the four functions in the ratio 62500:120000:633:12300. Thus the influence of the bp values is greatest, followed by that of mp. This was considered appropriate at the time, but the decisions and weightings can always be changed, and the curve fitting process repeated, if it transpires that there is a more appropriate way to proceed.
In this way, the best-fit values of the Kb coefficients were determined to be as follows.
|Kb2p||3.7||Between 6C and 7N|
|Kb3s||-1.6||Between 13Al and 14Si|
|Kb3p||4.5||Between 13Al and 14Si|
|Kb4s||-2.3||Between 22Ti and 23V|
|Kb4d||2.9||Between 27Co and 28Ni|
|Kb4p||3.7||Between 32Ge and 33As|
|Kb5s||-5.1||Between 43Tc and 44Ru|
|Kb5d||5.2||Between 42Mo and 43Tc|
|Kb5p||3.97||Between 50Sn and 51Sb|
Similarly, the best-fit values of the Ka scaling coefficients were determined to be as follows.
Of course, a better fit could have been obtained if the Kb coefficients, too, had been determined independently (one set for mp, one set for bp, one set of lhf and another set of lhe). But this was not the aim of the exercise. Instead, the aim was to find a minimal set of coefficients that would give a sufficiently good fit. As a bonus, it was hoped that the values so derived might also give some indication of deeper mechanisms in the model.
In this vein, the best-fit values of the Kc coefficients were determined, as follows, and found to give a reasonable fit if taken in pairs (one set of coefficients for mp and bp, and the other set for lhf and lhe).
The one exception to this is Kc5s, whose values for lhf and lhe are so wildly different that they need to be determined independently.
|Coefficient||mp, bp||lhf, lhe|
Looking back at the other tables, it is not surprising that the Kc5s coefficient behaves differently. The Ka5s coefficient is very different to the other three KaNs coefficients, and so is the Kb5s coefficient compared to the other three KbNs coefficients. It all goes back to the explanation given earlier on this page: that the s-block coefficients were not included so much to model the s-block elements, as to handle s-shell occupancy later in the period.
Using these coefficients, expressions can be obtained for the melting point, boiling point, latent heat of fusion and latent heat of evaporation of the chemical elements as a function of their atomic number. The values so generated, when viewed globally, as functions in Z, have the right qualitative shape. However, their quantitative values are often out by a factor of 2 (or more) from the actual melting points, boiling points, and latent heats.
Indeed, the sum of all four sets of real values comes to 200,000, while the sum of all four sets of variances for the 52 elements comes to 15,000,000.
The expressions, therefore, have little value as a summary of the data tables. The results are offered here, nonetheless, for what they are worth. It was certainly instructive to see how many of the coefficients converged on similar values, thereby suggesting which ones could be combined.