1.  Compare the elemental percent composition of urea using modern atomic weights with Wöhler's values for theory and experiment. What do you notice?

         Answer

The "experimental" % composition of Prout agrees better with modern atomic weights than the calculated composition based on the "new atomic weights of Berzelius" does. This would seem to underline the primacy of experiment over theory and to teach the following key lesson:

Lesson I: A good experiment carefully observed and recorded will retain its value forever, while results that are tinkered with to make them conform with current expectations are of transient, if any, utility.

This is the answer the question was designed to elicit. One might go further to infer that it would have been better to base atomic weights on Prout's analysis than on Berzelius's experimental data. This is presumably the thinking behind the prevailing view among historians of chemistry that Prout was "one of the finest analysts of his or any other day."

(A. J. Rocke, "Organic analysis in comparative perspective: Liebig, Dumas, and Berzelius, 1811-1837," in F. L. Holmes, T. H. Levere, Instruments and Experimentation in the History of Chemistry, Cambridge: MIT Press 2000, p. 280 )

But deeper detective work teaches another two valuable lessons.

Just how good were Prout's experiments? That he gets the right percentage for Nitrogen to 4 significant figures makes it look like he was accurate to better than 0.1%. Accuracy of a part in a thousand would be fabulous. However, in fact he determined Nitrogen by the volume of N₂ gas that was evolved, and he reported a volume of "6.3 cubic inches", presumably accurate to only one part in 63 or +/- 1% (To judge by a drawing of Prout's apparatus, there would seem to be no way he could measure the gas volume to better than +/-1%). Thus it would appear that in being correct to within 0.1% he was in fact only lucky. This would teach an equally important second lesson:

Lesson II: Luck is a useful thing to have on your side in science.

And there is a further twist. It appears the numbers for Prout's "experimental" elemental percentages, as reported by Wöhler, are not truly experimental! Prout was resorting to "dry labbing," a temptation to which Lavoisier naively succumbed, and from which teachers always try to wean beginning students.

At the time he analyzed urea, Prout was pushing Prout's Hypothesis that all elements are composed of hydrogen, meaning that all atomic weights should be integral multiples of the atomic weight of hydrogen. He used H = 1, C = 6, N = 14, O = 8, Cl = 36. (Note that Prout's carbon and oxygen were half of ours. We'll see Couper correct the carbon, but not the oxygen, in 1858.)

Prout's Hypothesis is not such a terrible idea in the sense that most of the mass of atoms comes from the protons and neutrons in the nuclei, which have nearly the same mass as hydrogen does (although there is a tiny conversion of mass to energy on forming the nucleus). When an element has several abundant isotopes there is a problem, e.g. Cl = 35.45 (where the dominant isotope of natural Cl is 35, but there is also 24% of the 37 isotope). (see Isotope Ratio Mass Spectrometry problem)

For natural C, N, O the relative atomic masses do truly involve integers to within less that 0.1% and H is off by only 0.8%. So for compounds containing only these atoms Prout's theoretical atomic weights work better than Berzelius's experimental atomic weights (see Question 2).

In fact the "experimental" weight percentages that Wöhler took from Prout's analysis of urea are theoretical. Although he doesn't say so, Prout calculated them by using his integral atomic weights and an atom proportion (CH2NO, using his atomic weights) that he inferred from his approximate analytical results. Because relative atomic weights from Prout's Hypothesis are pretty good, his "results" agree with modern expectations to within 0.1 to 0.7%, even though his actual experiments were correct to within only 1.5 to 4%. The same is true for two of three other compounds whose analyses Prout reported in the same paper. With the fourth compound it seems he just messed up.

Lesson III: Juggling experimental results to make them agree with expectations may succeed in some cases, but only if you're really lucky.

My advice: Don't fudge data.  We now recognize that it is dishonest and a recipe for disaster. Worst of all, adjusting the data pretty much assures that you'll never learn anything really new, because you make everything fit the Procrustean bed of preconceived notions.

2. How good are Berzelius's values for atomic weights?

    Answer

If oxygen were 100.000, then N should be 100 * 14.01 / 16.00 = 87.56 and Berzelius's value of 88.518 is 1% too heavy. Similarly his C is 2% too heavy and his H is 1% too light.

3. Liebig and Wöhler found that salts of fulminic acid (HONC) and cyanic acid (HOCN) were isomeric. How accurate would elemental analyses have to be to reveal that "a liquid hydrocarbon" (say C₁₂H₂₆) and "the olefiant gas" (C₂H₄) are not isomeric? What if, by the liquid hydrocarbon Wöhler meant an olefin, such as butylene (C₄H₈)?

   Answer

 C₁₂H₂₆ is 12 * 12.011 / (12 * 12.011 + 26 * 1.008) = 84.6% by weight Carbon. C₂H₄ is 85.7% by weight Carbon. So the amount of CO₂ from combustion would have to be correct within 1.1/85.7 = 1.3%, which is less than the error in Berzelius's atomic weights.

However, the hydrogen percentages are 15.4 and 14.3, respectively. So the amount of H₂O would have to be correct only to within 1.1/15.4 = 7%. It would be possible to tell this difference with the kind of precision Berzelius obtained.

Since C₂H₄ and C₄H₈ have the same C:H ratio they could not be distinguished by elemental analysis, only by determination of molecular weight, a process which was not agreed upon by organic chemists until after 1860.

4. After consulting the page on Liebig's method of analysis, reconstruct how the elemental analysis of urea could have been done experimentally. (You need also to know that when heated with cupric oxide and copper many substances released their nitrogen as N₂ which could be measured by volume.) In light of this approach examine the theoretical and experimental results in Wöhler's paper carefully for errors in logic or arithmetic or both. What do you conclude?

The reported results show that in these early days of analysis Wöhler, a very intelligent person but until recently a student, was so naïve as not to notice that components should sum to 100%. This glimpse of a glimmer of his frailty helps us establish a fellow feeling with Wöhler, Prout and the other giants who developed organic chemistry and makes it clear that you don't have to be perfect to make an important contribution, just honest, diligent, perceptive, and lucky.

There is an addition error in the "experimental" case. The elements should sum to 99.945, not 99.875. This addition error was corrected in a subsequent French version of the paper. Most of the remaining difference from 100 is due to two cycles of round-off error in Prout's two-stage calculation of weight of the elements in 4 grains of urea. Prout's round-off errors were partially offset by Wöhler's mistake in reporting the % of hydrogen as 6.670 rather than 6.650. (Remember that these results were not truly experimental, as explained in the answer to Question 1 above.)

In the calculated case for ammonium cyanate the numbers were added correctly, but 20.198 had been rounded down to 20.19, rather than up, and more importantly the last two digits had been transposed in 26.24. These errors were not corrected in the French, so even when Wöhler (or an editor) went over the numbers, he didn't notice the problem of failure to total 100%.

We can't blame Wöhler too much for this error. After all this was early days, and quantitative precision didn't have a lot to do with the point of the paper. Still, especially if analysis is all you have, it is important to do it right. It is remarkable that no one else (as far as I know) from these later, more sophisticated days has commented on the arithmetic errors in this widely quoted (if not carefully read) paper of great historical significance. Analyses in papers of Berzelius, Wöhler's teacher, did total 100%.