How could the ideal gas law be discovered from experiments on real gases?

Question

The gas laws, namely Boyle's Law, Charles' Law, Avogadro's Law and Gay-Lussac's Law, are all experimental laws. Combining these laws, we get the ideal gas law $pV=nRT$. Also, "real life" gases do not exactly follow this law, so there are more laws for "real life" gases: van der Waals' law, Dieterici equation, etc., which approximately describe these laws within certain boundaries of the gas parameters: pressure $p$, volume $V$ and temperature $T$.

But there seems to be an apparent logical flaw: the ideal gas law $pV=nRT$ was found by experimenting on "real life" gases, but these "real life" gases do not follow the ideal gas law. How could this be the case?

Answer 1

The ideal gas law is a very good approximation of how gases behave most of the time

There is no logical flaw in the laws. Most gases most of the time behave in a way that is close to the ideal gas equation. And, as long as you recognise the times they don't, the equation is good description of the way they behave.

The ideal gas equations assume that the molecules making up the gas occupy no volume; they have no attractive forces between them and their interactions consists entirely of elastic collisions.

These rules can't explain, for example, why gases ever liquefy (this requires attractive forces). But most of the common gases that were used to develop the laws in the first place (normal atmospheric gases like oxygen or nitrogen) are usually observed far from the point where they do liquefy.

As for the volume taken up by the molecules of the gas, consider this. A mole of liquid nitrogen occupies about 35mL and this is a fair approximation of the volume occupied by the molecules. at STP that same mole of gas occupies a volume of about 22,701mL or about 650 times as much. So, at least to a couple of decimal places, the volume occupied by the nitrogen molecules is negligible.

The point is that for gases not close to the point where they liquefy (and few components of the atmosphere are), the ideal gas laws are a very good approximation for how gases behave and that is what we observe in experiments on them. The fancy and more sophisticated equations describing them are only really required when the gas gets close to liquefaction.

Answer 2

You must consider this:

1. The question whether a physical system follows a particular law is not a "yes or no" question. There is always an error when you compare what you measure with what the law predicts. The error can be at the 17th digit, but it's still there. Let me quote a very insightful passage by H. Jeffreys about this:

It is not true that observed results agree exactly with the predictions made by the laws actually used. The most that the laws do is to predict a variation that accounts for the greater part of the observed variation; it never accounts for the whole. The balance is called 'error' and usually quickly forgotten or altogether disregarded in physical writings, but its existence compels us to say that the laws of applied mathematics do not express the whole of the variation. Their justification cannot be exact mathematical agreement, but only a partial one depending on what fraction of the observed variation in one quantity is accounted for by the variations of the others. [...] A physical law, for practical use, cannot be merely a statement of exact predictions; if it was it would invariably be wrong and would be rejected at the next trial. Quantitative prediction must always be prediction within a margin of uncertainty; the amount of this margin will be different in different cases[...]

The existence of errors of observation seems to have escaped the attention of many philosophers that have discussed the uncertainty principle; this is perhaps because they tend to get their notions of physics from popular writings, and not from works on the combination of observations.

(Theory of Probability, 3rd ed. 1983, §I.1.1, pp. 13–14).

2. The error can be different in different ranges of the quantities you're measuring.

3. When trying to guess a physical law, scientists often consider the simplest mathematical expression that's not too far from the data.

The "ideal gas law" is very accurate in the range where the temperature $T$ is high, the pressure $p$ is low, and the mass density $1/V$ (inverse volume) is low. In these ranges the law is very realistic. If you 3D-plot various measurements of $(p,V,T)$ for a fixed amount of a real gas, you'll see that a curved surface given by $pV/T=\text{const}$ fits the points having large $T$, low $p$, low $1/V$ very accurately. "Real" gases are very "ideal" in those ranges.

Moreover, the law involves simple multiplication and division: $pV/T$, so it's one of the first a scientist would try to fit the points.

Finally, the law wasn't just suggested by fitting, but also by more general physical and philosophical principles and beliefs. Take a look at

S. G. Brush: The Kind of Motion We Call Heat (2 volumes, North-Holland 1986)

for a very insightful presentation of the history of this and other laws.

Answer 3

I like @pglpm answer, but can't comment on it.
I would like to mention that the technology of those years was not precise enough for very exact measurements, and not good enough for measurements at "boundary" conditions (where differences from the ideal gas law increase greatly in magnitude), so the experimental/measurements errors would hide any deviation from the "ideal gas law".

Answer 4

Suppose you have a crate, and when you put one apple in the crate, the crate + apple weighs 1100 grams. The crate plus two apples weighs 1200 grams. The crate plus three apples weighs 1300 grams. Do you have any guesses as to what one, two, and three apples would weigh without the crate?

The gas laws are more complicated, but it's the same basic principle. If you hypothesize that $pV$ is equal to some interaction between an "ideal" function of $nRT$ and some complicating factor from real gasses, it's not too difficult to take a bunch of data points of different $p$, $V$, $n$, and $T$ for different gasses, and isolate out the "ideal gas component" from the rest of the effects, especially since you can take extreme cases where the ideal gas component dominates.

It's made easier by simply applying intuition to the situation. It's logical that the volume should be proportional to number of molecules. Increasing the temperature should also increase the volume. Increasing the pressure should decrease the volume. Much of the the ideal gas law can be worked out from first principles, and experiments then verify the models (as well as providing the value of $R$).