Both oil and fats are made up of hydrocarbons and are hydrophobic in nature. The key differences between oils and fats are the number of bonds between carbon atoms and hydrogen atoms, physical state at room temperature and origins. Fats are generally saturated in nature, which means that all the carbon atoms have maximum number of bonds with hydrogen atoms. Fats are generally solid at room temperature and are generally derived from animals (pork, chicken, bacon, etc.). Oils, on the other hand, are unsaturated since there are unsaturated carbon atoms (lacking in bonds with hydrogen atoms). Oils are generally liquid at room temperature and are generally derived from plants (e.g., mustard, olive, etc.). There are some exceptions though, fish oil is unsaturated and hence is oil and not fat.
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