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Really basic question that's bugging me...

Speaking mathematically, 200 degrees of heat for 10 minutes should be the same as 400 degrees of heat for 5 minutes , but that's not the case, right?

So what's the difference if I cook something for 5 minutes at 450 vs. 350, as opposed to something at 350 for 3 minutes or 7 minutes?

Which "dial" (time / temperature) changes the outcome in what way? Why is this?

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This is an insanely general question and I don't think the simple answer you are looking for exists. It depends completely on the heating method (I'm assuming conventional oven, but you didn't specify), the size of the oven or vessel, but most importantly the type of food. How thick is it? How dense? How much surface area? Does it have a lot of water? Can steam escape? Does it have a crust? Are you using a braise, bain-marie, or anything else to control the internal temperature? Higher temperatures may activate entirely new chemical processes; Mathematics are a bit player in this story. – Aaronut Feb 2 '11 at 1:51
"Mathematics are a bit player in this story." and physics and chemistry! It isn't a basic question, because the input sum of energy over time can't easily be translated into a cooking effect. – tonylo Feb 2 '11 at 1:54
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Well, it's an interesting question, I'll give you that. Just a book or two might be needed to answer it fully. – Orbling Feb 2 '11 at 2:02
The basic answer has to do with penetration, as represented in a couple answers below. Which is why we're all saying it's too general, because penetration of the heat depends on the composition of the food being cooked a) as related to density and specific heat and b) as related to any chemical process specific to that food (i.e., that's temperature sensitive), surface area to mass ratio, flash point, just to name some things right off, certainly other considerations exist. – zanlok Feb 2 '11 at 14:15
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If there's a mathematical link, it would be based on the difference between the desired (cooked) temperature and the applied (cooking) temperature. Then we need to take into account various chemical changes at various temperatures. I could Sous-vide a steak at 141 degrees and produce something edible, but it wouldn't have any Maillard crust. I could also pull out the ox-acetelyne torch and make a steak that is charcoal on the outside and still raw on the inside. – Chris Cudmore May 8 '12 at 15:15
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5 Answers

Passing to a higher temperature (and cooking for less time) has the general effect of burning the outside of the meat, and getting the inside not perfectly cooked. Cooking for a longer time has then the effect of better mixing the flavors, and keep some type of meats tender.

In specific cases, it's probably possible to use a higher temperature, but simply cooking less is not the only thing that needs to be done; there is something else that needs to be corrected, or something that needs to be done and it was not necessary if you would have cooked at lower temperature.

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Many "things" happen in cooking a particular dish. These physical and chemical (even biological) processes require a certain optimal range of temperature (and humidity) and take a certain amount of time to be completed.

For example, when you bake bread, the yeast in the dough remains alive until the temperature rises high enough to kill it. It continues to produce gas as the heat begins to set the dough. The dough should set just as the bubbles are at their largest size for fluffy bread. If gas production peaks before the temperature is high enough, the bubbles can collapse; if the temperature rises too fast, the dough will set too early.

If I have a tough piece of meat, I might cook it for 12 hours at a low temperature and high moisture to tenderize it (and maybe in a braising liquid to add flavor). Then I can cook it for two minutes at a very high temperature to brown the surface without raising the overall temperature, so the inside stays rare. In general when dry-cooking meat you often want the inside to reach a certain temperature, without having the outside dry out too much. So it's a balance between two extremes. If you want an internal temperature of 150 to kill bacteria or parasites, you could imagine cooking for 12 hours until the whole piece reaches that temp, but then you lose a lot of moisture. You could turn it up to 500 and hope the inside heats up faster, but by the time the inside is ready, the meat on the outside gets way too hot and maybe even starts to blacken. Somewhere in between you get the interior done properly, with the exterior just a little browned and crispy.

If you are cooking seeds like rice or beans, it takes a certain amount of time for the seeds to absorb water and become soft enough to eat, and this happens faster if the temperature is high. While cooking in water you have a maximum temperature limit, at the boiling point.

So, cooking instructions are calibrated by trial and error (and educated intuition) to allow the different chemical and physical processes to happen in the conditions that produce best flavor and texture.

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One fundamental error in this question: 400 degrees is not twice as hot as 200 degrees. Temperature is a measure of the kinetic energy of the particles involved. The only scale on which you can do the kind of ratio you are imagining is Kelvins - you have to measure from absolute zero.

400 F = 477.59 K
200 F = 366.48 K

so the kinetic energy of the air in the oven is only about 477/366 = 1.3 times as high at 400 F as it is at 200. For simple cases, like how long it takes to evaporate a pan of water, 1.3x is probably pretty close to right, but as is pointed out above, there are a whole host of other variables that come into play with real food.

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And even once you're past this, double the heat flux for half the time is clearly not the same thing. – Jefromi Feb 2 '11 at 4:42
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Sure, go ahead and diss the Rankine scale just like that. ;) – Magnus Nordlander Feb 2 '11 at 14:11
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If we really apply this vein of logic, why things cook, then it's the difference in kinetic energy that matters. This obviates the idea of evaluating things in Kelvin. If something that would remain uncooked at 70°F cooks in X amount of time at 200°F, the energy imparted was due to the 130°F difference. And, there's a Y time for it to cook at 400°F from the 330°F difference. So, there is actually a 2.5x increase in the difference. Furthermore, this really should be thought of in terms of kinetic energy, which is a function of a square. Full calculations are beyond the scope of a comment. – zanlok Feb 3 '11 at 18:05
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@zanlok and the main reason for cooking something is to make chemical changes. A protein that denatures at 120C is not going to change at 37C however long you cook it for. – Martin Beckett Feb 4 '11 at 16:44
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@mgb, yes. You can reach a target internal temperature in an oven at that temperature by leaving it in there indefinitely. That's obviously going to take a long time, and possibly not yield edible or pleasant results. – zanlok Feb 4 '11 at 16:53

Michael at Herbivoracious pointed out that doubling the temperature does not double the heat. That's part of the problem, but you can correct for it, and you still won't get properly cooked food.

kiamlaluno pointed out that you will burn the outside before cooking the inside, which I think is more to your point. The reason behind it is that the heat takes some time to get to the inside of the food. If you were to have some sort of theoretical oven that could heat all of your food at precisely the same rate, then cooking at a higher temperature for a shorter period of time would get you the results you're looking for. Unfortunately, such a device doesn't exist. The heat transfer is described by Newton's Law of Cooling (dQ/dt = -h·AΔT)

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A rotisserie I randomly saw on QVC claims to cook 2x faster because they have you put a heating element through the inside of a chicken. – zanlok Feb 3 '11 at 18:10
Assuming you still had the original heating element outside the chicken that's almost right - the 'wall thickness' of chicken insulation is halved. – Martin Beckett Feb 4 '11 at 16:47

This us the question:

Speaking mathematically, 200 degrees of heat for 10 minutes should be the same as 400 degrees of heat for 5, but that's not the case, right?

To show that the two are different, all that is needed is a single counter example.

Consider the boiling of an egg. If you boil the egg in 105 Fahrenheit (40 Celsius) over an extended period of time, neither the egg yolk nor the egg white will set.

If you cook it at, say 160 Fahrenheit (70 Celsius) you will eventually get a hard boiled egg.

The egg white and egg yolk consists of proteins. When proteins are heated to a certain temperature, the proteins will denature. In the case of the egg the chemical reaction (denaturization) will simply not be activated at lower temperatures.

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You should probably add that this is because according to physics nothing can get hotter than the source from which it is gaining heat; once the temperature of the egg matches the temperature of the water, heat ceases to flow between the two. I've probably said that badly. – Yamikuronue May 9 '12 at 11:54

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