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Chemically speaking, what are the biggest differences between saturated fatty acids and unsaturated fatty acids? What are the effects of those differences on the geometry of the molecule? And what is the effect of that on our body?

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Saturated fatty acids are fatty acids with no double bonds and unsaturated fatty acids are fatty acid with one or more double bonds. This is the biggest difference between a saturated and unsaturated fatty acid.

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There is two types of structural configuration for unsaturated fatty acids due to the presence of double bond; cis (kink/bend in fatty acid struture) and trans(No kink/bend in fatty acid struture).

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cis fatty acid (wikipedia)

A cis configuration means that adjacent hydrogen atoms are on the same side of the double bond. The rigidity of the double bond freezes its conformation and, in the case of the cis isomer, causes the chain to bend and restricts the conformational freedom of the fatty acid. The more double bonds the chain has in the cis configuration, the less flexibility it has. When a chain has many cis bonds, it becomes quite curved in its most accessible conformations. For example, oleic acid, with one double bond, has a "kink" in it, whereas linoleic acid, with two double bonds, has a more pronounced bend. Alpha-linolenic acid, with three double bonds, favors a hooked shape. The effect of this is that, in restricted environments, such as when fatty acids are part of a phospholipid in a lipid bilayer, or triglycerides in lipid droplets, cis bonds limit the ability of fatty acids to be closely packed, and therefore could affect the melting temperature of the membrane or of the fat.

Iodine value test is used to determine degree of unsaturation in a unsaturated fatty acids; that is to measure number of double bonds present in a unsaturated fatty acids.

This link will explain the health benifits of saturated and unsaturated fatty acids

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