Avogadro and His Number

Avogadro’s Hypothesis

In 1811, Amedeo Avogadro (1776–1856) (born Lorenzo Romano Amedeo Carlo Avogadro di Quaregna e di Cerreto) looked at Gay-Lussac’s results and concluded that when they are at the same temperature and pressure, equal volumes of gas (like two balloons of the same size) contain the same number of “particles.” These particles can be individual atoms, molecules, or even a mixture thereof.

So, if two balloons are filled to the same size, say one filled with helium and the other is simply blown up (and therefore is a mixture of oxygen, carbon dioxide, nitrogen, and water vapor), the number of particles in each will be exactly the same, since both balloons are at the same temperature and pressure (room temperature and atmospheric pressure). Avogadro wasn’t the first to suggest this, but he was the first to formulate it into a very coherent and persuasive concept. Avogadro’s Hypothesis has some very interesting implications.

Avogadro Versus Dalton

If at constant temperature and pressure, the same number of particles is contained in a given volume, then the theories of Dalton and Avogadro are in disagreement. So, where’s the problem? Recall, Dalton thought different types of atoms come in different sizes. This was a reasonable assumption and today we know it to be true. However, Dalton also imagined the atoms of the gas to be directly touching each other, which followed from his refusal to believe in action at a distance. Therefore, for Dalton, a balloon filled with gas meant that the entire space inside was completely filled with atoms packed in tight against each other. Herein lies the problem. Let’s consider our analogy again.

Before, we were concerned with packing in the balls (either golf balls or basketballs) completely until the box was full. This time let’s see what happens if we don’t worry about this requirement. However, instead of one box we imagine two of the same kind. First, we place a basketball in the one, followed by a golf ball in the other. We continue to do this until at some point we can’t fit anymore of either ball into its respective box. To be sure, we find that the box containing the basketballs will fill up first (since basketballs are bigger than golf balls and the boxes are the same size), therefore, causing us to stop as promised.

In this way, we have ended up with two boxes of equal size (volume) that contain the same number of “particles”. Of course, atoms do behave differently than basketballs and golf balls, but in this case, (I assure you) our analogy is true to form. Therefore, if we could count the number of small and large particles placed into the two different balloons, we would find that at the point the balloons were the same size, not only would there be the same number of small particles as there are large particles in each balloon, but there’s also a lot of empty space left over.

You see the particles (atoms, molecules, mixture thereof) of a gas are not packed in tightly against one another. Rather, they are in motion, moving about in the empty space that surrounds them. Indeed, a model consisting of atoms along with empty space is already familiar to us from the ancient Greek atomic theories of Democritus and Epicurus. Avogadro’s Hypothesis of equal particles for equal volumes at constant temperature and pressure provided remarkable insight into the nature of gases. Avogadro never proved his theory nor was he able to determine what the actual number of particles would be given the temperature or pressure. However, Avogadro’s Hypothesis follows from the kinetic theory of gases.

The Foundation of Avogadro’s Hypothesis Comes from Kinetic Theory

From the kinetic theory of gases, it can be shown that the average total translational kinetic energy (KE) of an ideal gas at temperature T, containing N particles is given by the equation, KE=3/2NkT, where k is just a fixed number known as Boltzmann’s constant. The translational kinetic energy is the kinetic energy associated with movement in the x, y, or z (length, width, height) directions. This result, which can be directly calculated using Maxwell distribution, reveals the very interesting fact that the KE does not depend on the mass of the particles or their identity, but rather it depends only on the temperature. This means that at the same temperature, if two volumes of different gas contain the same number of particles both volumes will have the same KE.

Another result from the kinetic theory of gases is that the pressure P of an ideal gas at volume V is given by the equation, P=2/3V(KE), which is simply P=NkT/V, after substituting our above result. Consider two different gases denoted as 1 and 2. Their respective equations for the pressure will be P1=N1kT1/V1 and P2=N2kT2/V2. Therefore, if two different ideal gases are at equal volume, pressure and temperature, they will have the same number of particles since we then have P1=P2, were both the volumes and the temperatures are the same giving, N1kT/V=N2kT/V, or simply N1=N2, which is Avogadro’s hypothesis. However, real gases differ from ideal gases in that they do experience attractive and repulsive interactions. As a consequence, real gases only contain Avogadro’s number of particles when they behave “ideally,” which is at low pressures and/or high temperatures.

Avogadro’s Number

Today we recognize Avogadro’s Hypothesis with a fundamental constant called Avogadro’s number, which is the number of particles in one mole (often written as mol) of a substance. One mole of any substance is simply the amount (in grams) of the substance that is equal to its atomic weight. For example, the atomic weight of carbon is 12 grams per mole, and therefore one mole would be equal to 12 grams. Another example would be water, which has an atomic weight of 18 grams per mole, and therefore 18 grams of water (or about one tablespoon) would be one mole. Whereas Avogadro was simply talking about the number of particles in a volume of gas, Avogadro’s number is talking about the number of particles in a specific amount of substance (one mole) and applies to gas, liquid, and solid.

In 1909, Jean Baptiste Perrin (1870–1942) provided the first accurate experimental determination of Avogadro’s number from studying Brownian motion (we’ll discuss this in more detail later) to be 6.7 x 1023 particles/mol. He was the first to relate it to the mole amount of a substance and also suggested naming it after Avogadro. Today the value is more accurately determined at about 6.022 x 1023 particles/mol. An ideal gas at 32°F and atmospheric pressure would fill up a volume (like a balloon) to 22.4 liters, and would contain exactly 6.022 x 1023 particles.

Avogadro’s Hypothesis Was Ahead of Its Time

Avogadro’s Hypothesis was a whole new way of viewing the atoms in a gas. That gases at equal volumes, temperature, and pressure would have the same number of particles, whether atoms or molecules, also meant that there must be a significant amount of empty space in the gas, and that the atoms weren’t in constant contact with each other. Moreover, by proposing that the atoms of a gas can actually come together to form molecules challenged the popular idea that like atoms can only repel each other.

In 1811, these ideas weren’t too mainstream, and Avogadro didn’t offer up reasonable explanations either. He didn’t calculate or experimentally determine the number of particles occupying a given volume at constant pressure and temperature, showing that it was the same regardless of the particles. Consequently, Avogadro’s ideas remained neglected for almost a half-century.

The concept of the atom and the role it actually played in chemistry was still an open debate. In general, it was agreed that conceiving of matter as being made up of atoms provided a useful tool in chemical reactions and in visualizing the structure of the molecules they formed. As to whether this meant the true nature of matter consisted of ultimate, indivisible particles that should be denoted as atoms was another story. The lack of an unambiguous method for determining relative (and of course absolute) weights of atoms and molecules and their chemical formulas resulted in several incompatible atomic theories. Nonetheless, atomic theory maintained its foothold in chemistry in one form or another.

A much needed turning point came in 1858 (two years after Avogadro’s death) when Stanislao Cannizzaro published a pamphlet showing that Avogadro’s work could, aside from some minor exceptions to more general rules, provide the basis for the determination of relative weights for many substances existing in the gaseous state. Recall, that Dalton’s approach required knowing the amount of starting materials used in a chemical reaction to form the molecule of interest, and a guess as to how many atoms comprised the molecule. The approach proposed by Cannizzaro reduced finding relative atomic weights to the almost trivial measurement of determining relative gas densities. Unfortunately, Cannizzaro’s pamphlet had very little effect, but that was about to change.

In 1860, Cannizzaro spoke at an international chemical conference held in the German town of Karlsruhe. His talk made a lasting impression on the audience consisting mostly of prominent European chemists. Further, Cannizzaro’s friend Angelo Pavesi distributed Cannizzaro’s pamphlet to the attendees. Cannizzaro’s system, based off of Avogadro’s Hypothesis, was adopted soon afterwards.

Cannizzaro success at establishing Avogadro’s work as paramount in atomic theory was due in part to his much clearer account. However, perhaps the biggest factor was that, unlike Avogadro, he provided an (almost trivial) experimental means to test the hypothesis. Therefore, what was once mere speculation could now be readily verified and implemented. Further, the timing couldn’t have been better.

Atomic theory had changed since its early days. By now the first law had been established (as of around 1850), which put an end to heat being viewed as a fluid of particles (known as caloric); no more were atoms imagined (as proposed by Dalton) as being surrounded by a layer of caloric. Kinetic theory was coming into its own with the works of Clausius, Maxwell, and later Boltzmann.

Maxwell’s revolutionary publication of 1860 described atoms of gas as moving with velocities that occurred within a well-defined range, or distribution. Thus, not only are the atoms in a gas not fixed in place as Dalton envisioned, they also move at a variety of different velocities. Now, with a consistent system of relative weights it became possible to construct the periodic table of the elements, which organized the elements into groups or families thus revealing certain trends in their properties. An atomic theory that finally worked with the experimental data, provided chemists with a means of explaining chemical reactions, and writing chemical formulas seemed to have finally arrived.

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