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Until the fabric absorbs the chemicals and is a constant source against your skin until removed
Do you know that this happens or are you just speculating?
It’s pretty common knowledge for anyone who works with chemicals that the first thing you do when something spills on your clothes is to remove your clothing, otherwise it’s just keeping the chemicals in contact with your skin for it to prolong absorption.
It’s also common knowledge that protective clothing can prevent chemical contact with skin.
As long as it isn’t absorbative, so if you can find a tattoo sleeve made of nitrile or latex go ahead.
Lab coats are just normal textile. The point is you have to take it off immediately in case of accident. You do the same with the sleeves. No problem, much better than having the chemicals directly on your skin.