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I was reviewing some analytical chemistry and stumbled upon a section that explained the imperfection of using a salt bridge.

It said that the using dissimilar ions is a problem because in, for example, the case of KCl the K+ and Cl- have different mobility and so you get regions that are rich in K+ and others rich in Cl-. This means that some specific parts of solution (even though it's small) to be not electrically neutral i.e. electrically charged.

So I imagine in theory at least it must be possible to somehow extract these charge-rich areas and put them into a beaker. So what is all this I've been learning about solutions having to be electrically neutral?

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Because the electrostatic force is much stronger than you can imagine. Yes, you can get charge-rich areas in a cell. However, the amount of charge held here will be negligible. A rather small amount of charge can overcome the field near the plates and prevent more charge from approaching. The amount is quite small; it's not something that you would notice (and it won't give you a static shock1

1. Everyday static shocks are stronger by orders of magnitude because they deal with a lot of charge accumulated on the surface of an insulator, while here it is a relatively small amount of charge dispersed through the volume of a conductor

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