We'll start with saturation. It has to do with how much hydrogen there is in the fat. In general, animal fats are saturated with hydrogen, and vegetable fats are not. Saturated fats are thicker, often solid at room temperature, and have been linked to heart disease, possibly because they cling better to the sides of your blood vessels.
The unsaturated fats are therefore somewhat healthier, but it's a misnomer to call them "healthy". They're still full of calories, and too many calories makes you fat.
The monounsaturated fats (nearly saturated but missing only one hydrogen) are often considered healthier, but the reasons aren't well known. The main monounsaturated fat is olive oil, and it has been linked to good health, but the reasons aren't well known. It may have nothing to do with the fat itself and everything to do with the kinds of diets eaten by people who eat more olive oil.
Unsaturated fats are "kinked" where there is a double bond rather than a hydrogen. Those kinds often occur at the 3rd or 6th carbon, and we call those omega-3 and omega-6. Again, omega-3 isn't especially healthy, but people tend to eat far too much omega-6, which has been linked to health problems. Adding omega-3 isn't really a solution to the problem, since it just adds calories and makes you fat, but replacing omega-6 with omega-3 may be helpful.
Most vegetable seeds are high in omega-6, and the green parts have more omega-3. So oils are omega-6, and because we raise our meat animals on grains made with those same vegetables, they also tend to be high in omega-6. That includes all of the oils you mentioned (vegetable, soybean, corn, canola, peanut) except olive. People are encouraged to eat sources of omega-3 (especially fish) and animals raised to increase omega-3s (those raised on pasture rather than concentrated foods), as well as certain fruits and vegetables.
One other thing you didn't ask about: trans fats are what you get when you add saturation to unsaturated fats by hydrogenation. These are particularly problematic, for reasons that are not entirely clear. They're still unsaturated, but the trans shape of the fat seems to cause a lot of problems, perhaps because it doesn't occur in nature.
And as for the oils you mentioned: there's no real health benefit to corn, canola, etc. oil. Use whichever one happens to have the flavor for what you're trying to cook, and what's affordable. Peanut is expensive but has a nice flavor; canola is cheap but is nicely neutral for when you don't want your food to taste like corn or soybean oil. Vegetable oil is made of whichever of these happened to be cheap that day at the plant.
You also choose an oil based on its smoke point, if you're deep frying.
So, to sum up: none of those oils is especially healthy, except perhaps for olive, and even then you have to use it in moderation because it's full of calories. (They all have the same number of calories.) Eat more vegetables, lean meats, and whole grains, and you won't really have to worry about what kind of oil you use.
· 1 decade ago