You can see that the Cp line is very much like the one you saw on the alkali-metal slide. It rises rapidly from 0 at 0-atomic weight to asymptote at the reciprocal of the slope (1/0.1513). Please note that there is no inflection point in this curve. Now to explain why boron and carbon don't work.
The graphic below shows Lewis plots of Debye's theoretical heat capacity equation where Cv now is plotted vs the log of the absolute temperature.
In my new equations, we are looking only at data from the curves taken at 298°K (above graphic). Let's imagine for a moment that these four curves are one family. We'll have carbon and lead adopt aluminum and silver. Now the three specific heats calculated from the top three (Pb, Ag, Al) are taken from their respective curves from the same side of the inflection points whereas the data point for carbon is from the opposite side of its inflection point. Since my Cp equation has no inflection point it cannot accommodate data like this. Consequently, the carbon point is not linear with the rest of its family. Only boron and the two allotropes of carbon have their inflection points to the right of 298°K, and this is why they do not join the other members of their respective families in a linear plot. If this argument is valid, then we should be able to choose a temperature higher than those where diamond and graphite have their inflections and see if carbon then joins the others in linearity.
At 700°K, the infection points of all the solid elements occur at temperatures less than 700°K, and these family members (lead & tin are liquids and excluded) are lined up. Another more empirical way of looking at this is to say the my Cp equation is only good when the heat capacities are about 4 calories and above. The inflection point of the Be curve is lower than 298°K but close; thus, Be gives a tiny bit of trouble since it is just starting to line up. At 400°K, the alkaline earths are lined up better.
In the following table, I give the summary to show that I did all the groups from IA to VIIA and the noble gases. Don't bother trying to make notes because a copy of the slide is available after the talk. To keep the number of entries down and for completeness, I lumped the transition metals, lanthanides, and actinides together here (although the result is less than exciting). The first column shows the group in question; the second shows any elements omitted; the third the radioactive elements for which predictions are made; the fourth the slopes; the fifth the constants (only two decimal places needed to make calculations for predictions); and the sixth the S%D to show you how well we are doing.
Transition series ##
8 (noble gases)
Fr = 7.52
Ra = 6.78
Ra = 6.39
Ac = 6.82
Po = 6.24
At = 8.77
# Beryllium is included in this calculation to show the extent of error.
## The transition metals, lanthanides and actinides were lumped together because I wanted to deal with each of the first 92 elements. For better correlation, each file should be treated separately. However, since the lanthanides and actinides form groups of two elements each only, it is really not possible to treat these series.
I thought it would be interesting to see if I could take the original data of Dulong and Petit, put it into the computer and come out with a straight line with small positive ordinate intercept. The result was disappointing since the line essentially passed through the origin and had a slope of almost exactly 1/6. What had happened?
Dulong & Petit had a very strong expectation that equal numbers of atoms of different elements should absorb the same amount of the caloric fluid on being heated one degree (molar heat capacity = constant in today's parlance). They reported on 13 elements, and I surmise that, after measuring them and getting the average value (for SpHt x AtWt), they went back and remeasured the ones which were too low or too high until they got what they wanted. This is indicated since they measured a specific heat for Te twice too high to make up for the fact that the atomic weight was thought to be 1/2 of today's value. The same thing happened with Co which was thought to have 2/3 of today's atomic weight and for which they dutifully measured a specific heat 1.5 times higher than today's value. See my table of the Dulong-Petit errors. They were close to a law, and their report did stimulate much work in the future - and the time was 1819.
Now we have come full circle from my opening remarks. We have seen how experimental expectation while generally helpful sometimes gets in the way. So be prepared for the unexpected, too.
In a plug for all the local libraries in the country, I'd like to thank Lynn Tesar of the Village of Ridgewood, NJ library who gets many of the hard-to-find references for me from other libraries. In particular, she scoured the countryside for the original 1819 Dulong & Petit paper finding it at Eastman Kodak who supplied a photocopy free. I'd also like to thank Professors Robert Shine, Gab Rubenstein and Carol Frishberg at Ramapo College and Professor Richard Laity at Rutgers University for the helpful comments and suggestions on an early write-up of this material. And finally, I thank you for your attention.
Deriving and using the derivative of
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