Everyday physics: pressure cookers.

Most graduate students out there can at times feel a lot pressure to perform. Heck, that is not just graduate students. Pressure at work seems to be one of those things that most people can relate to and feel all the time. It gives rise to a lot of stress. I don’t have any good recipe for combating stress that is guaranteed, but cooking sometimes helps. Seeing as what I just wrote is not too funny, maybe I should get to the point and talk about Pressure cookers and as a bonus you get a cooking recipe (no peeking). When in doubt, pick a gadget and explain it, yes sireee. Besides, the explanations that I found on the internet about how pressure cookers work teeter on the edge of crackpottery. Not all are bad, but most don’t really say anything other than water boils at higher temperature at higher pressure which is correct, but unsatisfying. It’s like saying it just works that way.

My 3d rendition of a pressure cooker

Ok. So lets turn this problem into a physics problem. What is pressure? Apart from your boss looking over your shoulder and telling you to get everything done by tomorrow, in physics it is something that we label with the letter . A gas or liquid pushes on the walls of a container. If you take a little bit of area of the container you will find a force exerted on it, usually outwards. Take twice the area, you measure twice the force, three times the area, three times the force, well, you get it. The force is proportional to the area and we call the constant of proportionality pressure. This is

The pressure is defined as the constant of proportionality between force and area. For a gas or liquid, the pressure is a useful concept and when you heat up a gas the pressure changes. As such, it is usually called a thermodynamic variable.

Now, a pressure cooker is essentially a fancy machine to boil water. That does not explain why it has all those parts so neatly drawn in the graph.

You will notice that it is not a conventional boiling pot because the water is sealed in. There is also a weight (cap) on the top. It prevents the water steam from escaping unless it has enough pressure to lift the cap. Usually it weighs a few grams (30?), and the escape hole is a couple of millimeters squared in area. If you divide the weight by the area, you can get a big number (seeing as the area is so small). That is the extra pressure needed to lift the cap. So the cap lets you increase the pressure of the inside system up to a point and that regulates the pressure. You can even control the weight to get different pressures inside.

So what’s the rub? It’s not like having more pressure will compress your vegetables or anything mechanical like that. It will not produce flattened chicken soup either.

What is fun is that when you increase the pressure, you get higher boiling temperatures for liquids. A phenomenological explanation is in order, (phenomenological here indicates that we look at phenomena and don’t necessarily bother to explain them in detail from first principles). If you take a typical ideal gas, the ideal gas law is of the form

This is called an equation of state. In it is the temperature, is the volume, is the number of moles of material, is the pressure, and is a constant, called very appropriately, the ideal gas constant. This can be derived from a completely microscopic description, so is really telling us the units of measuring temperature.

It relates the pressure and volume to the temperature. At fixed volume, higher implies higher pressure. However, ideal gases do not become liquid. We need a better model for what the equation of state looks like if you want to include the fact that gases can turn into liquids. This is provided by the Van-der Waals gas law, which is somewhat similar, but it has a couple of correction terms added to the ideal gas law.

The Van-Der Waals gas law predicts that at fixed temperature and pressure, one can have more than one volume. The small volume could be said to be the liquid state, and the large volume the gas state.

The equation of state of a Van-Der Waals gas is slightly messier:

It has two extra phenomenological variables to describe the system. These are and . The second one, is the minimum volume that one mole of the liquid can have when you apply as much pressure as you possibly can. If we have moles, we just multiply by the number of moles to say that we have times more volume. Seeing as liquids are hard to compress, this is a convenient way to model that fact. Now, the other piece modeled by tells you that there is some type of attractive interaction between the particles of the gas. We know this because when we set the external pressure to zero at finite temperature the liquid does not disperse completely (the volume remains finite).

You will notice that if you fix , then you have to solve a cubic equation to calculate the volume . This is how one sees that this model can predict more than one volume given some pressure. However, in the region of parameters where that happens it turns out that the equation is not exactly the best description of the system (you have three real roots of a cubic, not two). So in the region where you have these three solutions you have to do something different. Instead of applying the math in a silly way, you have to think that you can have some amount of liquid and gas and that these are not mixing, but that they are in equilibrium. This is the Maxwell’s equal area trick: you replace the S-shaped curve you get when you graph at constant $n,T$ and you replace it by a straight line where remains fixed, but can vary between two values. I’ll just give you a reference. What that means is that at constant pressure you can have a lot of different volumes: they are different ratios of liquid and gas. Now, if you fix the total volume and increase the temperature at fixed the pressure increases. Both the liquid and gas part predict this. When we reach the regime where there is a mix of gas and liquid, we call that regime the boiling point of the liquid, and it is clear that the boiling temperature depends on pressure.

Thus, in a pressure cooker, the pressure serves as a temperature regulator: if you closed the system completely, the liquid would keep on boiling, the pressure would keep on increasing and the temperature would go higher and higher and eventually you would burn the food and you would be setting yourself up for a big explosion when the container fails. This is the reason why there is a fail-safe valve: in case the cap gets stuck or something, the valve will open and let the gas out. This still causes some damage to the pressure cooker, but will not ruin your whole kitchen when all the bits of the pressure cooker fly out hitting everything.

The benefits are that things cook faster (higher temperature means you can transfer more heat faster). These are especially good if you live in high mountains: the air pressure is so low that the boiling point of water goes down quite a bit and it can take forever to cook a potato. This is also good for things that need to absorb lots of water for cooking and you forgot to put them to soak the night before: think beans and other legumes.

Now, I’m going to give you a recipe, since my mouth is starting to water. This is a recipe for hummus (if you don’t know, that just means chickpeas in Arabic or Hebrew). I usually don’t measure, but instead I have developed this intuition for roughly how much should make it work. You should experiment a little.

Step one: boil chickpeas in a pressure cooker with some salt. Usually three times more water than chickpeas. For me it works best at around 30 minutes.

Step two: take one cup of cooked chickpeas, add one heaping tablespoon of tahini (this is adjusted to taste), one clove of raw garlic (not too big) and put it in a food processor. Add water sparingly until you get the desired creamy consistency (you can use the leftover broth from cooking the chickpeas).

Step three: take the mush and put it in a bowl. Add the juice of one or two limes (depending on size, and how much juice you can extract and also adjusting for taste) and some high quality olive oil. Blend it with a spatula.

Step four: decorate the mush with a sprinkle of good olive oil, paprika and pine nuts. You can also add zahatar (I’ve been told this is some type of wild marjoram blended with other spices) and it makes it really nice.

Serve and eat (usually with Pita bread). I have cooked giant bowls of this stuff when I have parties and it has been quite successful. Also, you can use canned chickpeas, but the quality goes down enormously. There are emergencies however and it is good to be prepared for those.

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