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Hair, skin and nails may look completely unrelated, but they are all built using the same protein building block: keratin. Yes, your hair, your skin and your nails are all basically the same thing. Your body uses the keratin differently to create each part, but they are still the same. So a chemical you put on your hair will affect your skin in much the same way, hence the burning.
Bleach lightens hair through several chemical reactions. First, the bleach starts breaking down the keratin cells that form the outer, protective layer of your hair. Once it breaches this defense, it pulls color molecules out of your hair's cortex -- the middle layer. Bleach doesn't know the difference between your hair and your skin -- heck, it will even bleach a carpet. When it gets on your skin, it goes to work, breaking down the keratin cells that make up your skin. It's slower than acid, but it's the same concept.
The Developer Equation
Bleach as a powder doesn't do much. It's only when you mix it with developer that it starts working. Stylists use different strengths of developer to get the job done. The darker the hair, the stronger the developer. Stronger developers are more powerful and they work quicker. This is why there is more burning associated with high-lift bleach -- platinum blond -- than a color stripper, which is a milder form of bleach.
If your scalp is burning, your stylist needs to go back to the basics. Bleach should never touch the scalp. Most of the time it is placed on frosting caps or inside foils to keep it from touching the skin. In all-over bleach jobs, the bleach is put on all the hair, but it is kept at least 1/8 inch from the scalp. A little might get on your scalp here or there, but it's not supposed to be everywhere. If you're burning in a particular spot, your stylist can clean the bleach off your scalp and stop the burning.
If it's left on the skin for a long time, bleach can cause first-, second- and third-degree burns. Chemical burns burn just as badly as burns caused by fire. They are also treated the same way. As with all burns, chemical burns have a high risk of infection and they are extremely painful. Don't ever sit and let the bleach burn. You might be able to withstand the discomfort, but you might walk away with a nasty burn for your effort.