Perplexing claim in Chang and Overby's "General Chemistry"

Question

I'm reading the chapter on gases in Chang and Overby's book "General Chemistry: The Essential Concepts" [1, p. 145], and the following passage is causing me some confusion:

The work of the Italian scientist Amedeo Avogadro complemented the studies of Boyle, Charles, and Gay-Lussac. In 1811 he published a hypothesis stating that at the same temperature and pressure, equal volumes of different gases contain the same number of molecules (or atoms if the gas is monatomic). It follows that the volume of any given gas must be proportional to the number of moles of molecules present; that is,

$$V \propto n$$

$$V = k_4 n \tag{5.7}$$

I don't quite understand the emphasized sentence; in particular, I don't understand how the emphasized claim "follows" from the preceding statement. In fact, I don't actually think it does.

For example, suppose instead that the volume of any gas at a given temperature and pressure was inversely proportional to the number of moles of the gas present; that is, suppose the relationship between volume and amount was instead

$$V \stackrel{?!}{=} \frac{k}{n},$$

for some constant $k$. Then it would certainly still be true that equal volumes of different gases contain the same amount of gas. For example, if I had $\pu{5 L}$ of $\ce{H2}$, then, according to the above "false law", I would have $k/5$ moles of $\ce{H2}$. Similarly, if I had $\pu{5 L}$ of $\ce{N2}$, I would have $k/5$ moles of $\ce{N2}$ — the same amount!

So the claim that the proportionality follows from the fact that equal volumes of different gases contain the same number of molecules seems illogical. In fact, it seems to me that the only thing one could logically deduce from the first hypothesis is that there is some relationship between volume and amount which holds for all gases (that is, the relationship doesn't depend on the gas itself).

Am I missing something, or is the statement in the book an error?

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Added much later:

To clarify what I'm asking here (though I thought it was pretty clear), consider the following two statements:

Statement A: at a given temperature and pressure, equal volumes of different gases contain the same number of molecules.

Statement B: at a given temperature and pressure, the volume $V$ of a gas is related to the number of molecules by $V = kN$ (where $k$ depends on temperature and pressure).

My point is simply that these two statements are not logically equivalent. B implies A, but not the other way around.

So why is it that the two statements are (seemingly quite often) presented as logically equivalent?

References

1. Chang, R.; Overby, J. General Chemistry: The Essential Concepts, 6th ed.; McGraw-Hill: New York, NY, 2011. ISBN 978-0-07-337563-2.

Answer 1

The other answer is, I think excellent.

Your question is basically "but what if there isn't a direct proportion between number of gas molecules and volume?" Instead, that was Avogadro's hypothesis in a nutshell: the volume of the gas is determined exactly proportionate to the number of molecules in it.

You are right in principal that you could have such a "false law" where the number of molecules is inversely proportionate to the volume and still have equal number of molecules for $\ce{H2}$ and $\ce{N2}$.

Let's follow your course of logic. That would mean if I get 10 L of $\ce{H2}$, I would then have $k/10$ molecules. Let's imagine I add more and more volume, to get 100 L then 1000 L of gas. At some point, I must necessarily divide the molecules in fractions in your false law. Certainly a molecule must occupy some indivisible volume, since Avogadro believed in the atomistic theory.

Instead, the obvious solution is that the number of molecules must be a measure of the amount of "stuff." So if we have more "stuff," we need more volume for it to occupy. The expression must be directly proportional between the volume and the number of molecules.

Answer 2

You must be familiar with the ideal gas equation: PV = nRT. Thus, an ideal gas can be completely characterised by 3 state variables (if you believe, in the ideal gas equation)

Say, you fix pressure and temperature (say, at STP). You get V = cn (where c is some constant, and n is no. of moles).

Thus it is indeed true that equal moles of gas, have equal volumes (under a given set of conditions). In fact, one calculates this volume as nearly 22.4 L (for one mole, at STP)

In fact Avogadro was the first person to make this observation. (and even though, I started my explanation with the ideal gas equation, it was formulated afterwards)

Of course, this is an idealisation. A picture that is closer to reality is given by the van der waal's equation of state, which takes into account inter-molecular interactions. $[P + a(\frac{n}{V})^2][\frac{V}{n}-b]= RT$ But this departure from ideal behaviour is usually not significant at low pressure, high volume, dilute conditions.

Answer 3

There is a big difference between mathematical possibility and physical plausibility

You are right to point out that one claim does not lead to other logically or mathematically. But the claim makes sense physically and makes sense of the previous observations chemists had made.

The way these concepts are taught now makes it looks as though chemists leapt from one idea to another using just some form of logic. But that is not what happened. Possible mathematical relations could have been different, but chemists were not choosing between different mathematically derived ideas: they were trying to explain already observed phenomena.

They already know that twice as much stuff would give a gas with twice as much volume. They were not completely sure why. Nor did they understand that different gases did not all consist of the same components (we know that oxygen is diatomic and carbon dioxide triatomic, for example: they did not). But they knew that certain masses of reactants would combine in fixed mass ratios. They also know that gasses weighed different amounts and the volume ratios of certain gas reactions. They didn't have a unifying theory to tie these observations together.

Sure, the relationship between the composition of gases and the volume could have been more complex to a mathematician. But the idea that volume and the number of things making up the gas is simple and proportional has two benefits. It is the simplest possible theory; and it explains previous observations (not least that using twice as much of a solid in a gas producing reaction produces twice the volume of the gas). More complex theories are possible, but they won't explain that observation.

So the simple story of how the leap was made misses out some other facts relevant to the basic idea. The real theory was not a logical finger-in-the-air but a deduction based on other well-known physical observations. And one that didn't emerge from mathematical speculation but from its ability to explain many phenomena. Some modern explanations leave that story out and, as a result, sound like an imperfect logical leap.