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When we talk about percentages of isotopes of an element in the atmosphere, is it percent by mass or volume or weight or something else?

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Short answer: it's the percent by number of particles.

Longer answer: It's the proportion of particles of a particular mass for that element. To chemists, a per particle basis is also commonly referred to as a per mol basis or a molar basis.

A way that it can be measured is by mass spectrometry - you pass the atoms through the apparatus which shows the mass to charge ratio. E.g. if you could pass $\ce{Cl}$ molecules, you'd see 2 peaks in the resulting graph - one is at 35 and the other at 37 marks which shows the mass/charge. First peak would be about 3 times larger (thus there were 3 times more particles). Thus chlorine would be about 75% isotope 35 and 25% isotope 37. So while this approach measures mass of each isotope, it also determines the relative molar amounts of each isotope.

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