What’s the deal with Avogadro’s number?


Dear Straight Dope:

I don't have a problem with Avogardo's number, anyone who can go down in history for something so completely random is fine by me.  (Egomaniacs who must have their name seared into the public consciousness usually just open up car lots.)  But what about the principle? I don't get how a given volume of molecular hydrogen (a small molecule) can have roughly the same number of molecules as an equal volume of, say, ammonia (a relatively larger molecule). If the number of molecules is the same, shouldn't the bigger ones take up more room? 

Son of Dex replies:

First off, let’s clear up a few misconceptions:

Misconception 1: We’re talking about Avogardo’s number and principle.

We’re not. It’s Avogadro. Sometimes the fight against ignorance really is this banal.

Misconception 2: Avogadro was an egomaniac who needed to have his name seared into the public consciousness.

Far from it. Egomaniacs want it all and they want it now. Avogadro’s work wasn’t fully appreciated in his time. We here at the Straight Dope know just how he felt.

The full import of Avogadro’s principle went ignored for almost 50 years. In 1811, Avogadro suggested that his gas laws could be used to nail down the atomic weights of the elements, some of which were still up in the air. No one really listened. It wasn’t until 1860, four years after Avogadro’s death, that a fellow named Cannizzaro convinced his colleagues that the old coot was right. Ain’t many guys opening car lots to get their name seared posthumously.

Misconception 3: Avogadro’s number is some random thing.

Avogadro’s number is not random. It’s extremely specific. It’s just very large and somewhat abstract. When you say "random" what you really mean is, "I don’t get it."

Cheer up, though. You’re not alone. Avogadro’s number (also called "a mole") is roughly 6.022×10^23. The number is precisely calculated so that when you have one mole of atoms of a given molecular weight M, it weighs M grams.

The chemists are still with me, but they aren’t the ones asking the question. So let’s take an example. Hydrogen gas, H2, has a molecular weight of 2 g/mol. Thus, if you had one mole, (6.022×10^23 molecules) of hydrogen gas, it would weigh two grams. Similarly, one mole of carbon dioxide, with a molecular weight of 44 g/mol, would weigh 44 grams.

So there we go. Avogadro’s number lets you keep track of how many molecules you have, which is pretty important in any kind of chemical work. It only appears random to the uninitiated.

Misconception 4: The volume of gaseous molecular hydrogen is the same as the volume of gaseous ammonia.

Ha-ha. Joke’s on you. It isn’t.

"Now wait a minute," you cry! "I distinctly remember my high school chemistry teacher drilling this in to me over and over and over again." Yeah, you do. Sort of.

What you remember is the ideal gas law, PV = nRT, where P stands for pressure, V for volume, n for the number of moles of gas, R for a special constant, and T for temperature. The ideal gas law does pretty concisely show that when pressure, temperature, and the number of moles are the same for two gases, regardless of their respective sizes, they occupy the same volume.

The catch, of course, is that this is the IDEAL gas law. Sadly, this ain’t an ideal world. If it were, we here at the SDSAB would have little to do (or we’d get paid more, take your pick). The ideal gas law is a convenient simplification of the behavior of real gases. There are a couple of equations that better model these, my favorite being the van der Waals equation. They don’t generally teach that ’cause no one but us chemists really wants to know about it.

The van der Waals equation contains two factors that are missing in the ideal gas law. They get left out because they usually contribute more to the complexity of the equation than they do to the accuracy of the answer. One of these factors is intermolecular forces of attraction, which tend to pull molecules closer together. The other factor is, you guessed it, the size of the actual gas molecules.

The thing to keep in mind is that a gas is mostly empty space between widely separated molecules. The molecules are so small compared to the overall volume of the gas that they may as well take up no space at all.

Let’s again compare hydrogen gas to carbon dioxide. At standard temperature and pressure (STP, 0 degrees Celsius and 1 atmosphere), one mole of an ideal gas occupies 22.4 liters. The molecules of hydrogen themselves only occupy a volume of 0.0266 liters, while the carbon dioxide molecules occupy 0.0427 liters. Thus, carbon dioxide, the larger molecule, does indeed have a larger volume. But it’s still less than 0.2% of the total volume of the gas. That’s small enough that we can ignore it completely and still be, as my high school chemistry teacher was fond of saying, close enough for government work.

SDSTAFF Karen Lingel observes:

I have a Ph.D. in physics, but I was in college before I stopped saying Agravado’s number.

SDSTAFF Ed can’t resist adding:

Don’t feel bad. Half the time in the original of the above Son of Dex spelled it Avagadro’s number.

Send questions to Cecil via cecil@straightdope.com.


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