In the Classroom
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Origin of the Formulas of Dihydrogen and Other Simple Molecules Andrew Williams University Chemical Laboratory, University of Kent, Canterbury, United Kingdom
One of the triumphs of 19th century chemistry was the development of the concept of the molecular formula and its elucidation for molecules of the fundamental compounds.1 This is all the more remarkable because this was done using non-mathematical logic and experimental data obtained from the most basic of laboratory equipment. The arguments for the formulas and structures of these fundamental compounds are left out of current introductory courses, probably for reasons of space, and the formulas and structures of such molecules1 are taken as axiomatic. The student is thus deprived of the fascinating logic upon which the subject is based. This is a pity as chemistry then appears as a descriptive “gee whiz” subject not possessing any deductive excitement. Knowledge of the structures and formulas of molecules of the fundamental compounds was vital for the development and calibration of the current physical methods of structural analysis (1).2 The modern student is left to puzzle over the logical fallacy that the structures and formulas of fundamental compounds were determined in the 19th century well before the physical methods of structural analysis were invented. It is almost 200 years since Avogadro (2) and Ampère (3) hypothesised that equal volumes of gases at the same pressure and temperature contain equal numbers of molecules. Their conclusion was based on the validity of Dalton’s atomic theory (4) and on Gay-Lussac’s law of combining volumes,3 which had been published a few years previously (5) (see the Supplemental MaterialW). Half a century elapsed before Clausius provided a logical proof of the hypothesis from the kinetic theory of gases (6) and chemists were ready to accept Avogadro’s and Ampère’s hypothesis and to use it to determine unambiguous molecular and atomic weights. Molecular formulas and structures came thick and fast after Cannizzaro (7) strongly publicized the hypothesis, known simply as Avogadro’s hypothesis, at the Karlsruhe Chemical Congress in 1860.4 The estimation of atomic weights dates from Dalton’s early work and several successive lists of data were compiled by Berzelius (1814, 1818, and 1826) (8). However the lists have significant ambiguity owing to wrong assumptions being made regarding some formulas.4 It was Cannizzaro’s work in the 1860s that resolved this problem and pointed the way to unambiguous atomic weights and molecular formulas. The atomic weight of an element and the molecular weight of a compound were originally defined relative to that of an atom of the lightest element, namely, hydrogen (taken as 1.00).5 Hydrogen6 gas was used as the standard for molecular weights and if its molecule were to possess z atoms of hydrogen the relative molecular weights of gas molecules (as well as those of vapors) could readily be determined by using z and comparison of the vapor densities with that of the standard (at the same temperature and pressure). The relationship between the relative density and molecular weight www.JCE.DivCHED.org
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follows from Avogadro’s hypothesis (eq 1).7
z × vapor density (1) hydrogen gas density To determine z Cannizzaro compared the chemical analyses for hydrogen for a library of hydrogen-containing compounds (Table 1) assuming that several or at least one molecule of the library would possess only a single atom of hydrogen. Cannizzaro formulated a law to determine atomic weights of elements: the different quantities of the same element contained in different molecules are all whole multiples of the atomic weight. The hydrogen contents of these molecules are the smallest for the hydrogen chloride, hydrogen bromide, hydrogen iodide, and hydrocyanic acids and are half of that of the hydrogen molecule. If the atomic weight of the hydrogen atom were set at 1 then the atomic weight would be represented by the smallest mass of the element in all of its compounds8 and z9 would then equal 2. This is confirmed by the observation that the smallest difference in relative masses is also 1 when z = 2. It is therefore highly probable that the formula of hydrogen gas is H2 and the molecular weights may be calculated accordingly. molecular weeight =
Table 1. Library of Compounds for Determining the Molecular Formula of Dihydrogen Compound a
Mol. Wt
Hydrogen gas
00z
b
H (% by mass) 100
Mass b,c of H
No. d of H
1z
02
Hydrogen chloride 18.25z
2.74
0.5z
01
Hydrogen bromide 40.5z
1.235
0.5z
01
Hydrogen iodide
64z
0.7813
0.5z
01
Hydrocyanic acid
13.5z
3.704
0.5z
01
Water
09z
11.111
1z
02
Hydrogen sulfide
17z
5.882
1z
02
Formic acid
23z
Ammonia
08.5z
Phosphine
17z
Acetic acid
4.348
1z
02
1.5z
03
8.824
1.5z
03
30z
6.667
2z
04
Ethene
14z
14.286
2z
04
Ethanol
23z
13.043
3z
06
Methane
08z
25.000
2z
04
Ethyl ether
37z
13.514
5z
10
17.65
a
The relative masses of the constituent elements are determined by b gravimetric or volumetric analysis. Relative to that of a hydrogen atom c set at 1.00. Mass of hydrogen per molecule = (H% x mol. wt.)/100, d e.g., H in HI = (0.7813 x 64z)/100 = 0.5z. Defining the smallest mass 0.5z = 1.00. The number of hydrogen atoms in a molecule of hydrogen gas = z.
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Table 2. Library of Compounds for Determining the Atomic Weight of Oxygen Compound a
Oxygen % by mass
Oxygen gas
100.00
16
32 = 2 x 16
32 = 32 x 1
88.9
09
18 = 2 x 9
16 = 18 x 0.889
Water
Gas Density b
Mol. Wt. c
Mass of Oxygen c
–– 02(H)
Carbon monoxide
57.1
14
28 = 2 x 14
16 = 28 x 0.571
12(C)
Carbon dioxide
72.7
22
44 = 2 x 22
32 = 44 x 0.727
12(C)
Sulfur dioxide
50.0
32
64 = 2 x 32
32 = 64 x 0.5
32(S)
Sulfur trioxide
60.0
40
80 = 2 x 40
48 = 80 x 0.6
32(S)
Nitric oxide
53.3
15
30 = 2 x 15
16 = 30 x 0.533
14(N)
Nitrous oxide
36.4
22
44 = 2 x 22
16 =44 x 0.364
28(N)
a The relative masses of the constituent elements are determined by gravimetric or volumetric analysis. standard = 1.00. cRelative to that of a hydrogen atom set at 1.00.
The atomic weights of other elements (in this case oxygen, nitrogen, carbon, and chlorine) can be determined in similar fashion. The relative mass of the oxygen in the library of compounds (Table 2) is determined from the percentage of oxygen and the molecular weight obtained from the relative density of the gas as shown (the worksheet in the Supplemental MaterialW also illustrates this). The smallest relative mass for oxygen occurs in water, carbon monoxide, nitric oxide, and nitrous oxide and this, 16, is thus taken as the atomic weight of oxygen; this value is confirmed because 16 is the smallest difference between the relative masses of oxygen in the compounds. There needs to be confirmatory evidence because the above arguments depend on the library of compounds selected containing at least one where the element is present as a single atom; this is not strictly known. In the case of the standard, hydrogen atom, Odling (9) provided alternative arguments based on chemical reactions of water, ammonia, and methane that these molecules possess 2, 3, and 4 hydrogen atoms, respectively, thus providing an alternative calibration of z in Table 1. In water we can replace one half or two halves of its hydrogen by sodium but not any other fractions: H2O + Na Na + NaOH
NaOH + 1 2 H2
(2)
Na2O + 1 2 H2
(3)
The hydrogens in ammonia can be displaced in three consecutive steps by an alkyl halide: NH3 + CH3I NH2CH3 + CH3I NH(CH3)2 + CH3I
NH2CH3 + HI
(4)
NH(CH3)2 + HI
(5)
N(CH3)3 + HI
(6)
In the case of methane hydrogen can be substituted by chlorine in four consecutive steps: CH4 + Cl2
CH3Cl + HCl
(7)
CH3Cl + Cl2
CH2Cl2 + HCl
(8)
CH2Cl2 + Cl2
CHCl3 + HCl
(9)
CCl4 + HCl
(10)
CHCl3 + Cl2
That z = 2 also follows from the fact that hydrogen chloride is a monobasic acid with only one replaceable hydrogen. 1780
Other Atom Mass c
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b
Relative to hydrogen gas as
The number of oxygen atoms in the formula of water can now be obtained from its molecular weight, 18, the knowledge that it possesses two hydrogen atoms, and that the atomic weight of oxygen is 16. We cannot say for certain that the atomic weight is 16—it could be some sub-multiple such as 8 (in which case the formula of water would be H2O2) but the relatively large library of oxygen-containing compounds utilized in Cannizzaro’s technique (Table 2) renders 16 as the most likely number.10 The formula is consistent with Gay-Lussac’s law for the combination of hydrogen gas and oxygen gas: 2 volumes 1 volume + hydrogen gas oxygen gas
2 volumes steam
(11)
Applying Avogadro’s hypothesis to eq 11 indicates that 1 molecule of water is constituted from a hydrogen gas molecule and half an oxygen gas molecule. Cannizzaro’s technique can be applied to compounds of nitrogen (Table 3) and of carbon (Table 4) to arrive at the atomic weights of nitrogen and carbon (14 and 12, respectively). No smaller mass or mass difference other than 14 has ever been observed for nitrogen in any of its compounds so this value is taken as the maximum possible atomic weight of the element. Similarly the atomic weight of carbon is no greater than 12 (Table 4). Table 5 shows that 35.5 is the atomic weight of chlorine (as this is the smallest weight) in the molecules of a series of volatile chlorides. Avogadro’s hypothesis applied to the reaction 2 volumes ammonia
3 volumes 1 volume + hydrogen gas nitrogen gas (12)
indicates that ammonia is constituted from 3 half molecules of hydrogen and 1 half molecule of nitrogen. Similarly reaction of carbon with hydrogen gas 2 volumes + carbon hydrogen gas
1 volume methane
(13)
can be interpreted by Avogadro’s hypothesis as methane being constituted from 2 molecules of hydrogen gas and 1 equivalent of carbon. Utilizing the molecular weights of water, ammonia, and methane (18, 17, and 16, respectively) and the atomic weights of oxygen, nitrogen, and carbon leads to the respective formulas as H2O, NH3, and CH4.
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In the Classroom Table 3. Library of Compounds for Determining the Atomic Weight of Nitrogen Compound a
Nitrogen b Mass
Mol. Wt.b
Other Atom b Mass
Nitrogen gas
28
28
––
Nitrous oxide
28
44
16(O)
Nitric oxide
14
30
16(O)
Nitrogen sesquioxide
28
76
48(O)
Nitrogen dioxide
14
46
32(O)
Ammonia
14
17
3(H)
Hydrazine
28
32
4(H)
Nitric acid
14
63
1(H); 48(O)
Aniline
14
93
7(H); 72(C)
a
The relative masses of the constituent elements are determined by gravimetric or volumetric analysis. bRelative to that of a hydrogen atom set at 1.00.
Table 4. Library of Compounds for Determining the Atomic Weight of Carbon Carbon b Mass
Mol. Wt. b
Other Atom b Mass
Carbon monoxide
12
28
16(O)
Carbon dioxide
12
44
32(O)
Methane
12
16
4(H)
Ethene
24
28
4(H)
Propene
36
42
6(H)
Carbon disulphide
12
76
64(S) 6(H);16(O)
Compound
a
Ethanol
24
46
Acetic acid
24
60
4(H);32(O)
Methanol
12
32
4(H); 16(O)
Benzene
72
78
6(H)
Diethyl ether
48
74
10(H);16(O)
a The relative masses of the constituent elements are determined by b gravimetric or volumetric analysis. Relative to that of a hydrogen atom set at 1.00.
Table 5. Library of Compounds for Determining the Atomic Weight of Chlorine Compound
Chlorine Mass b
a
Mol. Wt. b
Chlorine gas
71
71
Hydrogen chloride
35.5
36.5
Mercuric chloride
71
271
Formula Cl2 HCl HgCl2
Ethyl chloride
35.5
64.5
C2H5Cl
Acetyl chloride
35.5
78.5
C2H3ClO
71
99
C2H4Cl2
106.5
117.5
Ethylene dichloride Boron trichloride
BCl3
Phosphorus trichloride
106.5
138.5
PCl3
Silicon chloride
142
170
SiCl4
Aluminium chloride
213
267
Al2C6
Tin chloride
142
259.6
SnCl4
a The relative masses of the constituent elements are determined by gravimetric or volumetric analysis. bRelative to that of a hydrogen atom set at 1.00.
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In the case of the formation of hydrogen chloride from chlorine and hydrogen gases the following equation holds
1 volume 1 volume + hydrogen gas chlorine gas
2 volumes hydrogen chloride
(14)
and the atomic weights of 1 and 35.5 together with the molecular weight of 36.5 indicates that hydrogen chloride has the formula HCl. Simple division of the masses of other elements in Tables 2–4 by the atomic weight of the element gives the number of atoms of that element in the molecules and hence their formulas. The Cannizzaro method for determining atomic weights depends on the existence of a large library of compounds containing the element under investigation and cannot be regarded as an “absolute” method; there is always the (remote) possibility that a compound with only one atom of the element has not been included in the library.11 An absolute approach to the problem of atomic weight (and hence to the elucidation of formulas) was taken by Petit and Dulong (10) who first showed that many solid elements, especially the metals had an atomic heat12 of 24.7 ± 0.8 J mol1. This observation was based on Berzelius’ atomic weights some of which were “corrected” in order to fit the law; however, the law was put on a secure foundation by Boltzmann using kinetic theory (11). This law yields approximate atomic weights that can be combined with accurately determined equivalent weights13 to obtain accurate atomic weights. Unfortunately the law breaks down for some of the low atomic weight elements.14 The availability of unambiguous molecular formulas from 1860 onwards (particularly of organic compounds) rapidly gave rise to ideas of connectivity between the atoms in molecules, valency, and, finally, to the arrangements in space of the constituent atoms—all this before the end of the century and before the current physical methods of structure determination were invented. These arguments are sufficient to convince the student that molecular formulas could be logically obtained using evidence available to 19th century chemists. These chemists had to deal with and invent new concepts such as atomic weights, molecular weights, and structures so it is not surprising that the elucidation of molecular formulas was not a straightforward process. Many mistakes were made and cul de sacs visited but the overall logic is valid. Emphasizing description and (rather heavy and incomplete mathematical) theory in introductory courses is not the best way of inculcating in students a long lasting interest in the intellectual pursuit of chemistry. The usual methods adopted to make chemistry attractive (explosions, colors, novel applications, and graphics) are liable to produce an ephemeral interest in the subject that soon wears off in the face of the current theory-based introductory courses. Stressing the strong non-mathematical logic of chemistry in introductory texts and courses seems to me to be an effective model for attracting and sustaining the interest of students compared with that of the usual suspects. Not only is it important that the modern student knows about the historical development of the subject (12) but the data and logic employed in the 19th century are still valid and would be readily understood by fourth-year high school and first-year undergraduate students.
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Molecular structures, deduced from molecular formulas, are also taken as axiomatic in introductory chemistry texts for many fundamental compounds (such as methane, glucose, benzene, and ethene). These structures were also elucidated in the 19th century before the advent of physical methods of structural analysis; how this was done is, however, another fascinating story of deductive reasoning! Notes 1. Examples of fundamental compounds are H2, H2O, Cl2, NH3, CH4, N2, and HCl. 2. Most first-year undergraduate chemistry students know that the molecular composition and structure of a compound can be readily determined by a variety of physical techniques comprising, in the main, mass spectrometry, UV and IR spectroscopy, NMR spectrometry, and diffraction crystallography. 3. Gay-Lussac stated the law of combined volumes as “When gases react together, they do so in volumes which bear a simple ratio to one another, and to the volume of the gaseous product of the reaction” (5). 4. Up to that time the literature was very confusing as several different atomic weight values were in use, for example, the atomic weight of carbon was variously 6 or 12 hence some formulas had “C2” where modern chemists would have “C”. 5. For ease of argument we take the atomic weight to be to the nearest integer except in the case of chlorine. Isotopes had not been discovered in the 19th century. In any case most atomic weights are close to integral numbers. 6. Dihydrogen in modern idiom. 7. The mass of gas in unit volume (n mol. wt.) is compared with that of hydrogen gas (n z) (where n = number of molecules in a unit volume). Thus (density of gas)兾(density of hydrogen gas) = (n mol. wt.)兾(n z); cancelling n and rearranging yields eq 1. 8. The atomic weight of an element is for the moment taken as the mass of its atom relative to that of the hydrogen atom and is also the least mass of that element (relative to the mass of a hydrogen atom) present in one molecular mass of all its known compounds. 9. The number of hydrogen atoms in a molecule of hydrogen gas = z. 10. Cannizzaro’s method depends on finding a sufficient number of volatile compounds to ensure that at least one molecule contains a single atom of the element in question. At a later date (1880) Raoult’s work on vapor pressures of solutions provided several methods for determining the molecular weights of non-volatile compounds. 11. The value of z is, of course, confirmed in Table 1 by Odling’s substitution method. 12. Atomic heat (molar heat capacity) is the specific heat capacity multiplied by the atomic weight. 13. The equivalent weight of an element is that mass in grams that combines with or displaces 1 g hydrogen, 16 g oxygen, or 35.5 g chlorine in chemical reactions.
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14. The explanation of this breakdown was advanced by use of quantum theory by Einstein (Ann. Phys. 1907, 22, 180, 800). W
Supplemental Material
Further explanations, questions, and worked answers to assist students and to help the instructor place the topic in the chemistry syllabus are available in this issue of JCE Online. Literature Cited 1. See, for example, Williams, D. H.; Fleming, I. Spectroscopic Methods in Organic Chemistry, 5th ed.; McGraw-Hill: London, 1995. 2. (a) Avogadro, A. Journal de Physique 1811, 58–76. (b) Causey, R. L. J. Chem. Educ. 1971, 49, 865. 3. Ampère, A. M. Ann. Chim. 1816, 2 (5), 105. 4. See, for example, The Atomic Debates; Brock, W. H., Ed.; Leicester University Press: Leicester, United Kingdom, 1967. 5. (a) Gay-Lussac, J. L. Mém. Soc. Arcueil 1809, 2, 207. (b) Hawthorne, R. M. J. Chem. Educ. 1966, 43, 411. 6. Clausius, R. J. E. Annalen der Physik 1857, 100, 353. 7. Cannizzaro, S. Il Nuovo Cimento 1858, 7, 321. 8. Partington, J. R. A Short History of Chemistry, 3rd ed.; Macmillan and Co. Ltd.: London, 1957; p 207. 9. Odling, W., Quart. J. Chem. Soc. (London) 1859, 11, 107. 10. (a) Dulong, P. L.; Petit, A. T. Ann. Chim. 1819, 10, 395. (b) Fitzgerald, R. K.; Verhoek, F. H. J. Chem. Educ. 1960, 37, 545. 11. Boltzmann, L. Wien Berichte 1870, 74, 553; 1871, 63, 712. 12. (a) Jensen, W. B. J. Chem. Educ. 1998, 75, 679. (b) Jensen, W. B. J. Chem. Educ. 1998, 75, 817. (c) Jensen, W. B. J. Chem. Educ. 1998, 75, 961.
Editor’s Note The approach to determining formulas, and encouraging students’ abilities to reason from data, espoused by Andrew Williams in this article has been developed using excellent pedagogy by John Hutchinson in the first chapter, The Atomic Molecular Theory, of a JCE LivText, Concept Development Studies in Chemistry. This is a recent addition to the JCE Digital Library collection (see http://www.jce.divched.org/JCEDLib/ LivTexts/genchem/ConceptDev/index.html#application, accessed September 2007). By using the data from tables in Williams’s article, similar approaches could be developed using different combinations of elements from those chosen by Hutchinson. If you would like to try this approach with your students, you could easily use Hutchinson’s material as the basis for a lesson on the 19th century chemical reasoning that was required to come up with both chemical formulas and consistent atomic weights.
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