Cold process is a method of making soap (saponification) which does not use an external heat source. For this reason, this process is often used by home soapmakers. Lye, either sodium hydroxide or potassium hydroxide, is mixed with fats and oils to start the saponification process to become soap. This process lets the chemicals do the work, and the result in a few days is soap.
Hot process takes this a step further and the raw soap is heated to force the chemical reaction to occur in a shorter period of time.
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