CORRECTION TO THE DULONG-PETIT LAW

 Posted here: March 26, 2001 9:30pm Pacific Time    Latest Update: March 26, 2001

WARNING:

The reader will need some background in chemistry
and mathematics to understand this page!


The following is the approximate text of my talk on Thursday, April 18, 1986 at the 191st American Chemical Society National Meeting in New York City given to one session of the Division of Chemical Education. I wrote it down immediately after the talk and have edited it here for the web:



{Contribution from the R&D Laboratory of Pure Synthetics, Inc.
(Corporation of New Jersey, 1973-1995, dissolved)}


#209 "Heat Capacity - Atomic Weight Relationship"


By J. S. Paul Schwarz


Abbreviations used in the write-up:

SpHt = specific heat
AtWt = atomic weight
S%D = standard % deviation
Cp = molar heat capacity at constant pressure
Cv = molar heat capacity at constant volume
* = multiplication operator, "times"

Scientists form hypotheses and, from them, devise experiments to prove their expectations correct. These expectations serve the useful purpose of keeping us from doing experiments which might not lead anywhere. Occasionally, when expectation is too high, things go awry.

In the current book, "Surely You're Joking, Mr. Feynman!" the author tells us of the course of events following Millikan's disclosure of measuring the charge on the electron in his famous oil-drop experiment. It seems that Millikan's value was a bit low because he used a value for the viscosity of air which was not quite right. Workers repeating his work in the ensuing years felt satisfied if they measured a value close to Millikan's but would look very critically at any results which were too "high" and, maybe, find some reason to throw out high values. In this way, the value of the electronic charge only rose slowly over the years to the currently accepted value. This is how it goes with expectation: usually it helps us in the laboratory, but sometimes we are lead astray.

Needing a topic for a discovery class suitable for high-school students, I chose the Dulong-Petit Law.


DULONG-PETIT LAW

(SpHt = specific heat; AtWt = atomic weight; Cp = heat capacity at constant pressure)

For Solid Elements:

SpHt * AtWt = Cp = 6 calories/gram-atom

Alexis T. Petit & Pierre L. Dulong, Ann. chim. phys., 10, 395 (1819).


But I thought it would be too easy if I just presented the class with a number of molar heat capacities which were all about 6.0; consequently, I chose to use the alternate form of the law where SpHt times AtWt = 6.0 and have the class discover graphically that reciprocal specific heat is proportional to atomic weight.

1/SpHt = AtWt/6

Beforehand, I wanted to make up the requisite graph; thus, after choosing ten common metals for the euphony in the names, I plotted reciprocal specific heat (298 °K) vs atomic weight fully expecting (naively) to find a spray of points about a line through the origin with a slope of about 1/6.

Graph of 1/SpHt vs AtWt for 10 metals What I found (graphic to left) were 10 points in a pretty good straight line which DID NOT pass through the origin but gave a small positive ordinate intercept. This was intriguing for it meant that the equation of the line was not the Dulong-Petit equation but the equation given in the graph below. From this, one can derive the equation for the molar heat capacity and, Cp is not a constant 6 but a non-linear function (at 298°K) of atomic weight.


HEAT CAPACITY EQUATIONS

1/SpHt = slope * AtWt + constant

Cp = SpHt * AtWt

Cp        

=        

AtWt
--------------------------------------
Slope * AtWt + Constant


Investigating further, I found that, if I confined the graphs to only elements in the same file or family in the periodic table, I got good linear plots.

Graph for alkali metals The graphic at the left shows plots at 298°K for the alkali-metal family, lithium through cesium. One can see that the points lie very nicely along a straight line, the actual curve-fitting being done by computer. Originally, I began by using the least-squares method for this but found with low ordinate values, particularly with an element like lithium, that least-squares would give a rather high %-error when comparing the measured value with the estimation from the equation. This happens because least-squares fits curves by minimizing only the standard deviation. Therefore, I derived equations which minimize the standard-%-deviation (S%D), and I call this method "least error." If you are interested in using this method, I have a page with the derivation of the equations that you can pick up after the talk.

Using the least-error method with the alkali metals, one gets a straight line of slope, 0.1319; constant, 0.27; and S%D of 2.3. There is no correlation coefficient for this method yet. The points at the top of the slide on the curved line are the measured Cp values and the line is from my Cp equation from the above table using the given slope and constant. The Cp equation starts at 0 calories at 0 AtWt and rises rapidly going close to all the measured points finally asymptoting at the reciprocal of the slope (1/0.1319). Before getting this far it goes through the principal atomic weight of francium giving a predicted value of 7.52 calories/mole. This value is not likely to be challenged any time soon since all known isotopes of francium have very short half-lives.

 

Below is your "Generic" Periodic Table :


Generic Periodic Table

I have already shown you the plots for the alkali metals. Although hydrogen is commonly put with these metals, I did not include it in the least-error calculation because it is a gas in a family of solids. Comparing only solids with solids or gases with gases, by plotting reciprocal specific heats vs atomic weight, one gets straight lines with small positive ordinate intercepts for every vertical file (three or more) in the table including the transition metals except, perhaps, the Mn, Tc, Re file where the intercept is negative using available data. The only other exception to this comes from the noble-gas family where all members have the same heat capacity 4.97 calories/moles which doesn't change with temperature as all the rest do. The noble gases form a very boring group with the line going through the origin.

Reiterating, as you go down a file the molar heat capacity increases with increasing atomic weight. As you go across a row from lower to higher atomic weight, at first the heat capacity decreases until you get to the transition metals where Cp changes erratically sometimes rising and sometimes falling. After the transition metals, the Cp again begins to fall. Travelling through the rare earths from lower to higher atomic weights, the Cp falls although the trip is bumpy.

In addition to the lone gas in a family of solids, I have had to exclude both boron and carbon since they do not fall in line with the other elements of their families at 298°K. I will, however, explain very shortly the reason for this.

Please go to the conclusion of the talk:


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