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Say that we have an atom which is inert, or stable. How would one go about adding one electron. Is it possible?

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    $\begingroup$ Yes. With an extra-electron such an atom/molecule would be "unstable". How long it would last depends on the conditions. In a gas ions can last for quite a while. In solids or liquids losing the extra electron would be almost instantaneous. $\endgroup$ – MaxW Nov 8 '15 at 0:13
  • $\begingroup$ Negative ions are not that hard to make for many elements. Some will auto-ionize fairly quickly, some are reasonably stable... $\endgroup$ – Jon Custer Nov 8 '15 at 1:03
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F-center defects in crystals are due to unpaired electrons and can be caused by ionizing radiation; these defects can accumulate over thousands of years. This can be used to find the age of minerals with retroactive inclusions, or to intentionally color gems.

An electron gun, as in a cathode ray tube (CRT), will also do the job. Older televisions had CRT's, and to prevent electrons building up on the front of the tube, they were siphoned off to ground. "Static" electricity on the faceplate of a CRT can be felt; it is due to the build-up of electrons.

Of course, one could question if electrons in the above examples are bound to a particular atom, are interstitial, or are free to move, but as mentioned above by @MaxW, an ionized gas (plasma) does have particles with excess electrons, as well as those with a deficiency. In interstellar space, an ion could last for years.

Also, radical anions are easily produced, for example the reduction of naphthalene by an alkali metal.

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