Why do you add hexanes (after toluene) in the recrystallization of the adduct in a Diels-Alder reaction?
This will help induce crystallization by modifying the initial solvent with one of differing polarity, in this case a non-polar solvent (hexanes) are added to the slightly more polar toluene.
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Hexanes are added after toluene in the recrystallization of the adduct in a Diels-Alder reaction to dissolve any impurities that may be present in the crude product. Toluene is initially used to dissolve the crude product, but it may not effectively dissolve all impurities. Hexanes, being a nonpolar solvent, are added to further purify the product by dissolving impurities while leaving the desired compound unaffected. This selective solubility allows for the removal of impurities through filtration or decantation, resulting in a purer final product.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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