What is the Theory Behind Acid-Base Extraction?

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The discussion revolves around the theory of acid-base extraction, a method used to isolate compounds, particularly weak acids and bases. The process involves converting neutral acids or bases in a nonpolar solvent into charged ions, which are more soluble in water. This conversion is achieved through a neutralization reaction, allowing for the effective separation of these compounds. While the method is primarily aimed at isolating salts, it is also applicable for extracting other types of compounds. The explanation seeks to clarify the underlying principles of this extraction technique, which can often be complex and convoluted.
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I'm confused as to what the theory behind this is, I've checked on wikipedia, but it gives a rather convoluted explanation, can anyone clear this up for me?

http://en.wikipedia.org/wiki/Acid-base_extraction

What is this method trying to isolate, only salts? Or is this a valid method for separating other types of compounds?
 
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You may use it to extract any weak acid or base.

The idea is to convert undissociated (hence neutral) acid (or base) dissolved in nonpolar solvent, to charged ions (be it anions for acid or cations for base) which be well soluble in water. Conversion requires neutralization reaction.

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