Melting and Boiling Points, and Latent Heats, as Functions of Atomic Number

The melting points (mp) and boiling points (bp), and the latent heats of fusion and of evaporation (lhf and lhe), of each of the chemical elements are often held up as being fine illustrations of periodicity. By plotting each of these values against atomic number, repeating patterns of peaks and troughs emerge, with one cycle per period of the periodic table.

P-block

Indeed, one of the most notable events within each period is the abrupt jump in value amongst the p-block elements. It is so abrupt, in fact, that it bears a striking resemblance to the asymptotic change that occurs in the y=-1/x function. It is tempting to suppose, therefore, that each of the four parameters, (mp(Z), bp(Z), lhf(Z) and lhe(Z)) can be expressed as a function of atomic number, based on terms of the form A/(Z+B).

So, starting with the p-block elements in the second period of the periodic table, we can suppose that the boiling point function might contain a term involving A/(Z+B) where B is a constant (around -6.5) that puts the asymptote just after Z=6, and A is a scaling coefficient to get the magnitude right for the values for the boiling points.

It turns out to be convenient to rewrite this as Ka2p/(Z+Kb2p-10) where Ka2p is A, and Kb2p is 3.5 (that is, 3.5 elements before the end of the p-block of the current period).

The next step is to include a similar term for each of the other periods:

bp = Ka2p/(Z+Kb2p-10) + Ka3p/(Z+Kb3p-18) + Ka4p/(Z+Kb4p-36) + Ka5p/(Z+Kb5p-54) + other terms

The idea is that when Z has the value 14, say, the value of the overall expression is dominated by the contribution from the Ka3p/(Z+Kb3p-18) term, with only a small residual contribution from the other terms.

D-block

Looking at the elements in the d-block, the value of the boiling points rises on one side of the mid-point, and then falls away again on the other side, fairly symmetrically. This is somewhat reminiscent of the curve for y=K/(x²+a²) which could also be written as y=(K/a²)/(1+x²/a²).

This leads to the following expression:

bp = Ka4d/(1+Kc4d*(Z+Kb4d-30)^2) + Ka5d/(1+Kc5d*(Z+Kb5d-48)^2) + other terms

(It is, of course, just a naming convention that leads to these coefficients being called Ka4d, Kc5d, etc., where the integer (4 or 5) refers to the number of the current period in the periodic table, rather than the name of the electron shell, which has an integer one less than this (3d and 4d).)

S-block

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

Complete function definitions

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.

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.

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.

Conclusion

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.