**Level**: introductory (ex. 1), advanced (ex. 2), introductory+ (ex. 3)

**References**:
Alexis-Thérèse Petit & Pierre-Louis Dulong, "Research on some important aspects of the theory of heat," Annales de Chimie et de Physique **10**, 395-413 (1819).

**Notes**: Pierre-Louis Dulong (1785-1838) discovered nitrogen trichloride, whose explosiveness cost him a finger and the sight in one eye. He is best known, however, for his researches on heat, and principally for the law that is the subject of these exercises. Alexis-Thérèse Petit's life and career were tragically short (1791-1820): he died of tuberculosis before the age of 30. The law of constant molar heat capacities is also his best-known work.

The researchers noted that the product of atomic weight and specific heat was very nearly constant. Furthermore, they recognized that the product represents the atomic heat capacity (or molar heat capacity, as we would say). As they put it, "The atoms of all simple bodies have exactly the same capacity for heat." The law was never established to be quite so general or exact as Dulong and Petit presented it. Indeed, the law proved to be only approximate. For one thing, several non-metals have atomic heat capacities that deviate from the law. For another, heat capacities are temperature dependent, with different temperature dependences--a criticism of the law that dates back at least to John Dalton. The law was to receive attention and prominence after the deaths of its discovers. Stanislao Cannizzaro's "Sketch of a Course in Chemical Philosophy", which was widely influential in establishing a consistent set of atomic weights and formulas, made extensive use of it. Statistical mechanics eventually explained why the law holds even approximately at relatively high temperatures and why the law breaks down at low temperatures.

Exercise 1 involves practice with unit conversions. Conversion of calories to joules is straightforward unit conversion. The key to the atomic weight conversion is to recognize the conversion factor from the unnamed units. The value of oxygen is 1 in the given unit and 16.00 g/mol in the desired unit. So the conversion factor is 16 g/mol = 1 DP unit. (By the way, Petit & Dulong did not call their heat units calories. They employed specific heat units in which water has a value of 1; I recognized those units to be equal to cal/g.)

Exercise 2 is an exercise in data analysis. The object here is to find the relationship between the two variables. The fact that the variables are specific heat and atomic weight is not important. The same is true of exercise 3.

Alert readers may notice that the atomic weights obtained in exercise 1 for Co and Te are not close to the values accepted today. In fact, they are about 2/3 and 1/2 respectively of the accurate values. There are compensating errors in the specific heats of these elements so that the approximate constancy of molar heat capacity that is at the heart of the law of Dulong & Petit still holds. Atomic weights that are incorrect by factors involving the ratios of small integers are entirely explicable given early 19^{th}-century uncertainties in molecular formulas. The compensating errors in specific heat, however, are not easily explained. This fact and the fact that the molar heat capacities reported by Dulong & Petit are much more nearly constant than modern molar heat capacities have led to questions of possible data fabrication. For details, see Carmen Giunta, "Dulong and Petit: a case of scientific misconduct?" Bulletin for the History of Chemistry **27**, 62-71 (2002).

__Further information__: A detailed summary of key primary literature on multiple proportions, the atomic hypothesis, and atomic weights, including some quantitative treatment of data may be found in Leonard Nash, "The Atomic-Molecular Theory," in James Bryant Conant, ed., Harvard Case Histories in Experimental Science, vol. 1 (Cambridge, MA: Harvard, 1957), pp. 215-321.

**Solutions**: To download solutions, go to:

http://web.lemoyne.edu/giunta/classicalcs/DP.doc

To download worked spreadsheet, go to:

http://web.lemoyne.edu/giunta/classicalcs/DPfinal.xls

Copyright 2003 by Carmen Giunta. Permission is granted to reproduce for non-commercial educational purposes.

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