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Do We Take Atoms for Granted?

New optical microscopes are capable of visualizing objects down to 1 nanometer in size. The photo above shows a conventional micrograph of dispersed oil droplets alongside a simultaneously taken optical image showing features 1 nm across (approximately 5 atoms in width). Advances like this could allow direct visualization of viruses and lead to storage media which hold much more information than present ones.

As you may already realize, we possibly take atoms (and molecules) for granted. For instance, we possibly don't even think twice about the conceptual evolution of ideas that led to the notation H2O. In light of what was said previously, it is perhaps useful to give a thumbnail sketch of the ideas and struggles that resulted in the acceptance of the atomic hypothesis as probably the "way things are", not just as "one of several useful theoretical frameworks". This will hopefully give us a greater appreciation for "counting atoms" and the mole concept - which necessitates the reliable measurement of (relative) atomic weights. An appreciation of the central role of atomic weight determination in the development of chemical ideas may allow you to better appreciate the technological advances that afforded precise measurements of these weights. .

As recently as 1910, Alexander Smith - a Professor of Chemistry at the University of Chicago, stated the following "The atomic hypothesis provides a convenient form of speech, which successfully describes many of the facts in a metaphorical manner. But the handy way in which the atomic hypothesis lends itself to the representation of the characteristic features of chemical change falls short of constituting proof that atoms have any real existence."

In 1802, John Dalton proposed the "modern form" of the atomic hypothesis. However, 58 years later the best chemists in the world still thought that the correct molecular formula for water was "HO". Some questions should be stated:

As Alexander Smith alluded to above, how do we know that atoms exist at all? As mentioned previously, even the alchemists had a "reductionist" view of matter as being ultimately composed of Air, Earth, Fire, and Water. It turns out that Antoine Lavosier and John Dalton (among others) demonstrated that the fundamental units in chemical reactions are atoms.

Aristotle - ironically - gave a decent definition of an element "Everything is either an element or composed of elements... An element is that into which bodies can be resolved, and which exists in them either potentially, or actually, but which cannot itself be resolved into anything simpler, or different in kind." This above statement does not make it clear, however, how to recognize an element when we "see" it. Even so, Robert Boyle (you will investigate the results of his J-tube experiments with air for homework) gave a definition in his book - "The Skeptical Chymist (1661) that is not too far from our current view "And, to prevent mistakes, I must advertize you, that I now mean by Elements, as those chymists that speak plainest do by their principles, certain primitive and simple, or perfectly unmingled bodies; which not being made of any other bodies or of one another, are the ingredients of which all those called perfectly mixed bodies are immediately compounded, and into which they are ultimately resolved." However, during the time of Boyle and his contemporaries, there was no way to determine if a chemical substance (element or compound) was - with certainty - pure. Chemical reactions were fairly carefully being investigated however. For instance, when mercuric oxide (red powder, modern formula = HgO) is heated, oxygen gas (O2) is evolved and a silvery mercury liquid (symbol=Hg) remains that weighs less than the original oxide. [Today, we would write the "balanced" chemical reaction: 2HgO(s) goes to 2 Hg(l) + O2(g).] However, it was not until about a century later that Lavosier performed careful weighing experiments on the decomposition in a closed flask to demonstrate that there is no overall loss of weight during the reaction, i.e. mass is conserved. Hence, according to Boyle, mercuric oxide could not be an element (because it had decomposed into pieces) but mercury could (unless observation demonstrated that it could be further decomposed). As the chemist Justus von Liebig stated in 1857 "The elements count as simple substance not because we know that they are so, but because we do not know that they are not." Some of the first elements to be recognized were metals (c.f. Gold (Au), Copper (Cu), Tin (Sn), Iron (Fe), Platinum (Pt), Lead (Pb), Zinc (Zn), Mercury (Hg), Nickel (Ni), Tungsten (W), and Cobalt (Co)). Of the non-metals: Helium (He), Neon (Ne), Argon (Ar), Krypton (Kr), and Xenon (Xe), were discovered in the mixture of gases that remained after air was depleted of all its nitrogen (N2) and Oxygen (O2).

Thus, many chemists in the 1700's spent much of their investigations preparing and describing pure compounds - and subsequently decomposing them into their pure elements. Henry Cavendish discovered hydrogen (H2) in 1772 and Joseph Priestly invented carbonated water (CO2(aq)) and identified nitrous oxide (N2O - "laughing gas"), nitric oxide (NO), carbon monoxide (CO), sulfur dioxide (SO2), hydrogen chloride (HCl), ammonia (NH3), and oxygen (O2). In 1781, Cavendish proved that water consists only of hydrogen and oxygen [after he had witnessed Priestly explode the two gases in a "random experiment to entertain a few philosophical friends."] After oxygen was discovered, Antoine Lavosier was able -through careful quantitative measurements - to overthrow "phlogiston theory".

Conservation of Mass - Phlogiston Theory refuted

Lavosier discovered that mercuric calcx (HgO) lost weight when it was heated and that the free mercury and a gas were produced. After measuring the volume of the gas released, he showed that when mercury was converted back into calcx, the same volume of gas was re-absorbed and that the weight increase was equal to the earlier loss. As a result of careful weighing experiments such as these, Lavosier hypothesized that combustible materials burn by adding oxygen and thus increase in weight.

Chemical Compounds & Constancy of Composition: Bethollet & Proust

As a result of the researches of Lavosier, chemists began an intensive study of masses in chemical reactions. A feud developed between Berthollet - who believed that the elemental composition of compounds could have a continuous range of proportions - and Proust - who insisted that compounds had fixed composition. This principle of Proust became the Law of Constant Composition. The debate did cause chemists to return to their laboratories and to perform many quantitative experiments.

Equivalent Proportions: Richter & Dalton

Between 1792 and 1802, Jeremias Richter made a discovery that was simultaneously very important and yet almost totally ignored - the Law of Equivalent Proportions. The same relative amounts of two elements that combine with one another will also combine with a third element (if the reaction occurs at all). Some examples: Thus, according to Richter, if carbon and oxygen combine, they should do so in a ratio of 3 to 8 (by weight); and if carbon and chlorine combine, they should do so in a ratio 3 to 35.5 (by weight). Thus, the combining weight for an element can be defined as the weight that combines with 1 gram of hydrogen. The problem arose in the observation that many elements appear to have more than one combining weight. Take for example carbon - which can form more than one oxide. Humphrey Davy himself reported three oxides of nitrogen with nitrogen mass percents of: 29.50 percent, 44.05 percent, and 63.30 percent.

Theory of Atoms & John Dalton

John Dalton - after spending a lot of time reviewing combining weights - published his Atomic Hypothesis:
  1. All matter is made up of atoms. These particles are indivisible and indestructible.
  2. All atoms of a given element are identical, both in weight and in chemical properties.
  3. Atoms of different elements have different weights and different chemical properties.
  4. Atoms of different elements can combine in simple whole numbers to form compounds.
  5. When a compound is decomposed, the recovered atoms are unchanged and can form the same or new compounds.

Dalton - as had Lavosier - paid careful attention to the quantitative apsects (weights). Hence, he developed the chemical reaction symbolism and chemical formula notation - except that he used "pictures" for the elements rather than letter symbols. Nevertheless, his symbols did not represent arbitrary amounts of a given element - it represented either one atom or a standard specified number of atoms.

Dalton then tried to understand why the ratios of elements in compounds were fixed. His hypothesis was: A compound consists of a large number of identical molecules - each one of which is built up of the same small number of atoms - arranged in the same way. This was termed his Law of Constant Composition or Fixed Composition. Dalton still had the problem of determining what that number proportion was. For example, in binary (two-element) compounds such as the oxide(s) of carbon and water - what was the simplest formula. Dalton thus established a guide - "the rule of simplicity". He reasoned that the simplest binary compound between two elements "X" and "Y" should be XY. Next would be XY2 or X2Y, etc. Unfortunately his first attempt at writing down a formula (for water) proved a mistake. He presumed (relative to 1 for hydrogen) that the combining weight of oxygen was equal to its atomic weight. Hence, Dalton postulated that oxygen had an atomic weight of 8 (relative to 1 for hydrogen). Thus, Dalton's formula for water was "HO". Bad data collection by Dalton further exacerbated the problem. Anyhow, Dalton had developed a way to relate numbers of atoms with their masses and hence could determine (relative) atomic weights!! The table below exemplifies the application of these ideas to the oxides of carbon and also to the oxides of nitrogen.

Mass Possible Possible
Ratio Formula #1 Formula #2
Oxide "A" 3/4 CO C2O
Oxide "B" 3/8 CO2 CO
Atomic weight of C (if O=8) 6 3
Atomic weight of C (if O=16) 12 6
Mass Possible Possible Possible
Ratio Formula #1 Formula #2 Formula #3
Oxide "A" 3.5/8 NO NO2 NO4
Oxide "B" 7/8 N2O NO NO2
Oxide "C" 14/8 N4O N2O NO
Atomic weight of N (if O=8) 3.5 7 14
Atomic weight of N (if O=16) 7 14 28

Dalton and the Law of Multiple Proportions -
"Universal Acceptance" of Atomic Theory

The Law of Multiple Proportions - because it finally rationalized all the data published about combining masses, gained acceptance. We state it first in terms of Combining Weights and then in more modern terms.

In terms of Combining Weights, we can state the Law of Multiple Proportions as: "If an element shows more than one combining weight, these weights will differ among themselves by ratios of small whole numbers. Checking back to appropriate spots in the table on the previous page will easily show this for the oxides of carbon and of nitrogen. In more modern terms, we state it again: If two elements form more than one (binary) compound, then the different weights of one which combines with the same weight of the other are in the ratio of (small) whole numbers. In other words, Dalton showed - by analyzing large amounts of data from many sources - that the Combining Weights differ by small whole numbers because the atoms combine in small whole numbers. The acceptance of atomic theory was rapid and widespread.

Equal Numbers & Equal Volumes: Gay-Lussac & Avogadro

Unfortunately, as chemists tried to deduce more and more formulas for compounds, flaws began to appear in Dalton's determination of atomic weights based on his rule of simplicity. Joseph Gay-Lussac and Amedeo Avogadro provided the "two-step" key to Dalton's problems.

Gay-Lussac began to study the volumes of reacting gases. He found the following results:

Gay-Lussac found that these experiments - which were usually explosions done in closed containers - showed that gases react in simple whole number units of volume - provided that after the explosion the products are brought back to the same temperature and pressure as the initial reactant gases. Gay-Lussac reportedly was a cautious person and hence did not draw any conclusions.

Dalton agreed that if the volume of HCl gas (for example) is twice the volume of either hydrogen or of chlorine, there must be half as many molecules per volume in HCl. Hence, the (number) density of HCl per unit volume is n/2 for every n molecules of hydrogen or n molecules of chlorine that react.

Amedeo Avagadro proposed, on the other hand, that equal volumes of (gaseous) molecules at the same temperature and pressure contain equal numbers of molecules. He published these ideas in 1811. One consequence of Avogadro's Hypothesis is that all molecules in the HCl reaction - since they are at the same temperature and pressure would contain the same number of molecules per unit volume. Since volume was proposed to be proportional to the number of molecules, stoichiometric coefficents of the chemical reaction notation first developed by Dalton actually were proportional to volume. Thus, according to Gay-Lussac's data, we would write the chemical reaction as:

Thus, by applying Dalton's own ideas of the conservation of elements (atoms), i.e. that atoms are neither created nor destroyed, one hydrogen molecule must contain two (2) hydrogen atoms and one chlorine molecule must contain two (2) chlorine atoms. In other words, both gaseous hydrogen and gaseous chlorine must be diatomic (i.e. H2 and Cl2). The correct (balanced) reaction would read:

Note that if hydrogen is now presumed to be diatomic and - because of the implied proportionality of number of molecules and volume - Gay-Lussac's data would predict the following reaction:

In order to conserve atoms - without changing the coefficients (since these are already determined by the Gay-Lussac's volume measurements) the formula for water must be H2) and the formula for oxygen must be O2, i.e. gaseous oxygen is also predicted to be diatomic. The predicted reaction, according to Avogadro, would be:

Because of this new predicted proportion of oxygen to hydrogen in water, Dalton's original assessment of the (relative) atomic weight of oxygen would be:
8 grams of oxygen combine with 1 gram of hydrogen, but the molecule contains two atoms hydrogen for each oxygen atom.
Thus: if the (relative) weight of hydrogen is still assigned a value of "1", then the (relative) weight of hydrogen in a water molecule is 2 and thus the (relative) weight of an oxygen atom - in order to maintain the weight ratio of 8:1 - must be 16. So, oxygen has not only replaced phlogiston but now has a new (relative) atomic weight of 16!

Unfortunately, Avogadro's hypothesis of equal volumes implying an equal number of molecules was judged "shaky" due, in part, to the prediction of diatomic - single element - molecules ( O2, H2, Cl2, etc.). At that time ideas of chemical bonding were based almost entirely on electrical attraction and repulsion. It was thus difficult for scientists to understand how two identical atoms could do anything but repel one another. Also, if they did attract one another why did not elemental hydrogen (for example) from larger molecules: H3, H4, etc.? Thus, a half a century of confusion existed in chemistry, due to the rejection of Avagadro's ideas.

So, by 1860, there was such widespread confusion about atomic weights, that a conference was organized in Karlsruhe, Germany to attempt to reach some agreement. Stanislao Cannizzaro suggested a rigorous method for finding atomic weights based on the rejected work of his countryman Avogadro. Cannizzaro's method is listed below:

  1. Assume that the atomic weight of hydrogen is 1.0 and that that hydrogen is made of diatomic molecules - Gay-Lussac's data suggested.
  2. Assume that Avogadro was correct in deducing that oxygen gas is diatomic (O2) and hence that the correct molecular formula for water is H2O. As mentioned previously, this would give rise to a (relative) atomic weight of atomic oxygen of 16 and the (relative) molecular weight of O2 as 32.
  3. If equal volumes of all gases contain equal numbers of molecules then the molecular weight (M) of a gas is proportional to the density (d) of the gas: M= kd. Use H2 and O2 to evaluate the proportionality constant "k". Before we determine the value of "k" we must make some comments about the units of molecular and atomic weights. One outcome of all this work was the realization that the number of molecules present in 2.0 grams of H2(g) or 32.0 grams of O2(g) would be the same and be fixed. This number of molecules - a very important number for chemists - is termed the mole. The number was not known at the time of these measurements - but whatever it was it was thus hypothesized to be fixed. Thus, the actual value was not needed - since proportions were used.

    So, the relative atomic and molecular weights discussed thus far are the masses of 1 mole of such atoms and molecules in grams. Therefore, the units of atomic or molecular weights can be considered as grams per mole (g/mole). Now, list the determination of "k".
    Gas Density (in g/l) Molecular Weight (M) (in g/mole) Constant (k) in L/mole
    H2 0.0894 2.0 22.37
    O2 1.427 32.0 22.42
    Average value: 22.4 L/mole
    Remember, one of the stipulations of Avogadro's hypothesis was that all molecules be at the same temperature and pressure. The measurements above represent a temperature of 0° C (273° K) and 1.00 atmosphere of pressure. This value of temperature and pressure is termed Standard Temperature and Pressure (STP). Thus, at STP the volume of gas - predicted by Avogadro - 22.4 Liters per mole.

  4. Back to Cannizzaro's procedure. Evaluate the molecular weights of a series of compounds containing the elements whose atomic weights are to be determined. Starting with percent composition (by weight) calculated from gas densities, calculate the weight of each element per molecular unit. Look over these weights for a given element to see if the numbers can be interpreted as integral multiples of some common factor. This factor may be interpreted as the atomic weight of the element.
Compound d M=kd Elemental mass comp. Weight per molecular unit Probable formula
(g/l) (g/mole) C H Cl C H Cl
Methane 0.0714 16.0 74.8 25.0 --- 12.0 4.03 --- CH4
Ethane 1.335 29.9 79.8 20.2 --- 23.9 6.04 --- C2H6
Benzene 3.47 77.8 92.3 7.7 --- 71.8 6.00 --- C6H6
Chloroform 5.32 119.1 10.05 0.844 89.10 12.0 1.01 106.2 CHCl3
Ethyl Chloride 2.87 64.3 37.2 7.8 55.0 23.9 5.02 35.4 C2H5Cl
Carbon tetrachloride 6.81 152.6 g/L 7.8 --- 92.2 11.9 --- 141.0 CCl4
Greatest common factors are 12.0 for C, 1.0 for H, 35.3 for Cl

Hence, the Cannizzaro method allows the determination of atomic weights for any element that appeared in compounds having measurable vapor densities. With these atomic weights, the percent composition of a new compound would lead unambiguously to the chemical formula. It was realized that a mole of any compound would have the same number of molecules - although the value of that number was not then known. This number was named Avogadro's number in honor of Amedeo's pioneering work. This number of particles (whether it be atoms, molecules, or apples) is also termed the mole. This number is now known to be 6.022 x 10² ³ . Thus, In hindsight it is not surprising that a mole of any compound contains the same number of molecules. This comes from the knowledge of the ideal gas law. It can be shown that the "k" (Cannizzaro's) constant is simply RT/P. Let's see:

        PV = nRT     Ideal Gas Equation of State

where P = pressure (usually in atmospheres)
      V = volume (usually in Liters)
      T = temperature (in Kelvin)
              remember T (in K) = t(in ° C) + 273.15
                                               mass of molecules in grams     g
      n = number of molecules (in moles) =   ----------------------------- = ---
                                               molecular weight in g/mole     M
where g = mass of molecules in grams
and   M = molecular weight of molecules in grams per mole
      R = ideal gas constant = 0.0821 Liter atmosphere/(mole ° K)
Remember that density (d) is defined as d = mass/Volume = g/V. Thus,
          g       MP         P
     d = ---   = ---  =   M ---  = M (k);
          V       RT         RT 
where the "k" from Cannizzaro's procedure is seen to be P/RT. We can now calculate the value of "k" at STP. The value is k = (0.0821 x 273)/1.00 = 22.4 L/mole the value that we arrived at before. Note that we can solve the ideal gas law directly for M:
      d R T           g R T
 M =  ------     =    ------
        P               P V

Let's look at a sample program that makes use of the Cannizzaro method in Maple. As usual, the Maple output is centered and not all Maple output is shown.

#Construct an array in which each row refers to the information
#for one of the (six) compounds.  Each row will consist of the
#data in the following order:
#  [density, mass of compound, mass of element #1 in comp'd,...]
#Since we are given elemental composition in mass percents, we can
#assume that we have 100.0 g of compound.
#Also, we will set Digits:=4 so that we can sell all of the
#entries aligned properly in the array.

a:= array([[0.714,100,74.8,25.0,0], [1.335,100,79.8,20.2,0],
[3.47,100,92.3,7.70,0], [5.32,100,10.05,0.844,89.10],

#Now we will determine our average value of our proportionality
#constant (k) by use of the relationship that molecular weight
#(M) equals k times the density (d), i.e. M = kd.  Thus,
#k = M/d.  So if we have some gaseous substances of known
#molecular weight and measured density under the prevailing
#conditions, we can determine k.  According to Cannizzaro,
#we presume that hydrogen (H) and oxygen (O) exist as
#diatomic gases, H2 and O2.  Given the atomic weight of
#H as 1.01 g/mole and O as 16.00 g/mole, we can determine
#k from the measured gas density of H2 and O2 and hence use
#the averaged k for our analysis.  The measured gas density
#of H2 and of O2 - at the prevailing conditions - are given
#as 0.0894 g/L and 1.427 g/L, respectively.  Remember, the
#molecular weight of diatomic hydrogen and diatomic oxygen
# are 2.02 g/mole and 32.00 g/mole, respectively.
k1:=2.0/0.0894; k2:=32.00/1.427;
#The averaged value of k is (k1+k2)/2 - as expected.
#Now, what we want to do is to multiply each density - in 
#column 1 of array "a" by k (i.e. M = k x d).  We can
#accomplish with a "for" statement.  We tell Maple to print
#the results.  Convince yourself that the units of M are
for i from 1 to 6 do print(k*a[i,1]) od;

#Now we will define another array, "b", that will contain the
#results of performing the Cannizzaro procedure on "a".  First
#we will set up the array - to get a look - and then we will
#"fill" it.  The array will have the same dimensions as "a".

#To show that the array has been set up, we use the "evalm(...)"
#command to evaluate the array (or matrix).

#Recalling the Cannizzaro procedure, we want to find the mass of
#each element per mole of the compound.  Since, according to
#Dalton, there must be a whole number of atoms in each compound,
#i.e. a whole number of moles of atoms for each mole of compound,
#then these amounts should be a whole number of some smallest
#mass (Greatest Common Factor, GCF).  This smallest mass - for
#each element - should be the atomic weight of that element.
#First, to determine the grams of each element per mole of each
#compound, a simple dimensional analysis argument reveals that
#if we compute the number of grams of each element per gram of
#compound (i.e. entry a[i,3] or a[i,4] or a[i,5] divided by
#a[i,2]) and multiply this result by the molecular weight (M)
#of the compound (i.e. k*a[i,1] - as we did above) we will
#have the number of grams of the particular element per mole
#So, the Maple command is:
for i from 1 to 6 do for j from 1 to 5 do b[i,j]:=a[i,j]*
(k*a[i,1]/a[i,2]) od od;

#By inspection of each of the last three columns above, we
#look for an entry that is a multiple of the others, i.e.
#an entry that when multiplied by appropriate whole numbers
#will yield all of the other entries.  This is the Greatest
#Common Factor (GCF).  This GCF should be the smallest
#combining mass of the element in g/mole.  Convince yourself
#of this.  Hence, for C it is about 12.0 (11.90), for H it
#is about 1.0 (1.006), and for Cl it is about 35.4 (35.36).
#So, we divide each of the last three columns by the appropriate
#GCF and this will allow us to determine how atoms of each
#element appears in the compound, i.e. the probable molecular
#formula of the compound.
#So, we tell Maple to divide all of the entries in column 3 of
#"b" (i.e. b[i,3]) by the GCF of 11.90 (i.e. the "atomic weight"
#of C).  Similarly, we tell Maple to divide all of the entries
#in column 4 of "b" (i.e. b[i,4]) by the GCF of 1.006 (i.e. the
#"atomic weight" of H).  Finally, we tell Maple to divide all
#of the entries in column 5 of "b" (i.e. b[i,5]) by the GCF
#of 35.36 (i.e. the "atomic weight" of Cl).  We instruct
#Maple to print out only these last three columns - after
#they have been processed as stated above.  Remember, by the
#use of the Cannizzaro method, the GCF for a given element - for
#a range of compounds containing the element - is a good estimate
#of the probable atomic weight of the element.
for i from 1 to 6 do print((b[i,3]/11.90),(b[i,4]/1.006),
(b[i,5]/35.36)) od;

#Each of the entries above can be easily rounded to the nearest
#whole number.  These rounded whole numbers tell us - for each
#of the six compounds (represented by the six rows above) - the
#(whole) number of C atoms, of H atoms, and of Cl atoms in a
#molecule of each of the compounds.  In other words, we have
#determined the probable molecular formula of each of these
#compounds!  The rounded-off values agree with the results
#shown in the text.  They are listed below.
#Hopefully, you can easily apply this Maple-aided Cannizzaro
#method to your homework.

What about "heavy" elements - those that can't be vaporized?

Metals and heavy elements that cannot be prepared as easily vaporizable compounds cannot be subjected to Cannizzaro's procedure. A method that relates the atomic weights of metals to their ability to absorb heat and change temperature was put forth by Pierre Dulong and Alexis Petit and is called the Law of Dulong and Petit. More about that possibly at a later date!

Exercises 1-38 through 1-40 will allow you to use Maple for some sophisticated calculations ala Cannizzaro.

Next: The Most Useful Functions... Up: MATH AS A SECOND LANGUAGE Previous: Classification of Matter
Sat Feb 25 18:05:36 EST 1994