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The atomic weight of an element, is it accurate to say that another way to think of it is the expected value of that element's weight if we were to sample one at random from the environment?

Are man-made versions counted as part of the weight? I'm new to chemistry but for example let's say we had a typical carbon-12 and we just started removing/adding neutrons at will, resulting in a whole bunch of various isotopes of carbon. I don't know if it's even possible to do that but say we had carbon-6, carbon-7, carbon-8, ... etc etc etc and so on until whatever isotope of carbon can't hold any more neutrons. Would those be counted?

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Atomic weights in any periodic table are based on the natural abundance of different isotopes of the element. But sometimes the source matters and the results will be different.

The most important thing to note here is that creating new isotopes is hard. You can't just "add" neutrons to an atomic nucleus and often when you do it decays to a different element.

So most periodic tables quote atomic masses using the natural abundance of the elements as they are found in nature.

When there are exceptions, different atomic masses might be quoted for samples know to deviate from the natural abundance (or, more commonly, the specific ratios of the isotopes will be mentioned). Most elements abundance will be determined by the natural processes in stars (or the Big Bang) that created them in the first place. Occasionally natural nuclear processes will alter that ratio. For example, the radioactive isotope of carbon, 14C has a half-life of 5-6 thousand years but is produced naturally in the upper atmosphere. It is well distributed in the air and incorporated into plants. But if those plants die, the amount of the isotope will decay over time lowering the ratio of that isotope in the plant. This is both the basis of carbon dating and the reason why it is possible to tell whether alcohol is produced from fossil fuel sources (old enough that all the 14C has decayed) or from fermentation of sugars (produced from living things so incorporating the atmospheric level of the isotope).

It is even possible for nature to contrive conditions that mimic a nuclear reactor causing havoc with the natural abundance of heavy isotopes of uranium. In the uranium deposits in Oklo (in Gabon) the ratio of 235 was found to be far lower than the normal levels elsewhere because of a natural nuclear reactor that ran 2bn years ago.

But we don't change the numbers in the periodic table because of this. When we talk about specific samples of elements where the ratio is not normal, we specifically mention what the ratio is compared to the natural level.

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The atomic weight of an element should reflect the natural isotopic abundance. However there are statistically significantly different sampling variations. If you look at the nuclides the individual isotopes have masses known with much greater precision that the overall atomic weight for the particular element. For example Wikipedia lists atomic weight for lead as 207.2(1). However the individual isotopes of lead have masses known with much greater precision.

$$\newcommand{\d}[2]{#1.&\hspace{-1em}#2} \begin{array}{lrl} \hline \text{Isotope} & \text{% Abundance}&\text{Atomic Weight} \\ \hline \ce{^{204}Pb} & 1.4 & 203.9730436(13) \\ \ce{^{206}Pb} & 24.1 & 205.9744653(13) \\ \ce{^{207}Pb} & 22.1 & 206.9758969(13) \\ \ce{^{208}Pb} & 52.4 & 207.9766521(13) \\ \text{Avg Atomic Wt.} & & 207.2(1) \\ \hline \end{array} $$

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