To understand how chemicals dissolve things, you first need to know that some molecules are polar, having electrically positive and negative sides, while nonpolar molecules are electrically neutral. "Like dissolves like" is a phrase chemists use, meaning polar liquids generally dissolve polar solids, while nonpolar liquids dissolve nonpolar solids. Gasoline, acetone and a few other nonpolar organic solvents eat through styrofoam, a nonpolar solid.
Polystyrene, the plastic used to make styrofoam, contains benzene, a strong organic solvent. Benzene will dissolve styrofoam. Though benzene is a hazardous, carcinogenic substance, it is useful as an industrial chemical, helping to produce plastics, rubbers, detergents and drugs.
Toluene, another organic solvent, also dissolves styrofoam. A chemical cousin of benzene, toluene sees use as paint thinner, an octane booster in gasoline and a chemical feedstock used to produce other substances. Since it is less toxic, toluene has replaced benzene in most nonindustrial uses.
The same gasoline you put in your car will dissolve styrofoam, though it will not dissolve some other plastics, such as polythene. Unlike toluene and benzene, which are pure substances consisting of a single molecule, many different compounds go into making gasoline, including hydrocarbons and additives.
Acetone, an organic solvent found in nail polish remover, will dissolve styrofoam. The acetone does not chemically destroy the polystyrene; it simply causes the tiny bubbles in the foam structure to come apart. The polystyrene remains, though dissolved in the acetone. If you let the acetone evaporate, it will leave a clump of solid polystyrene behind.
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