What is the temperature and the pressure necessary for hydrogen to fuse to helium?
This answer is courtesy of Peter Giles, SOI, Stanford University.
It is a simple question, but, unfortunately the answer is a little bit complicated!
Let's start at the beginning of a fusion reaction. Most of the energy that is produced in the sun starts with a very simple nuclear reaction, where a hydrogen nucleus joins with another hydrogen nucleus. Eventually, a helium nucleus is formed. This reaction releases energy, and that energy heats the sun. In turn, the sun heats the earth and all the other planets.
Now, you know that there is a lot of hydrogen right here on earth. In fact, all of the water in the world is made up of just hydrogen and oxygen. Imagine if all of the hydrogen on earth started joining together to make helium! Things would get very hot! Lucky for us, this doesn't happen. Why not? Well, as you seem to know already, it has to be very very hot for a hydrogen nucleus to join with another hydrogen nucleus.
You see, a nucleus always has a positive electric charge. And two positive charges don't like to be too close together. When they get near each other, they repel one another; that is, they are forced apart. (This is a lot like gravity, only in reverse.) So, all of the hydrogen in the ocean is safe; the positive charges keep each hydrogen nucleus a safe distance away from all the others.
So now I can get back to your question. In order to force two hydrogen nuclei together, we need to have a very high pressure, or a very high temperature, or both. A high pressure helps because it causes all the hydrogen nuclei in the sun to squeeze into a smaller space. Then there is more chance of one hydrogen bumping into another. A high temperature helps because it makes the hydrogen nuclei move faster. They need this extra speed so that they can get close together and join. It's as if the nucleus has to break through a barrier, and so the faster it is moving, the greater chance it has.
So, at the "normal" temperature and pressure on earth, a hydrogen nucleus has basically no chance of ever joining with another hydrogen nucleus. Now we know that in the middle of the sun, where the temperature is about 16 million degrees, and the pressure is 250 billion atmospheres, hydrogen nuclei will sometimes have enough energy to join together. (An atmosphere is the "normal" pressure of the air here on earth. A pressure of 250 billion atmospheres is like having a large mountain piled on top of you!)
But did you know that even at the center of the sun, an average hydrogen nucleus has to wait about 6 billion years before it joins together with another!! (There are about 6 billion people on Earth, so this is one year for each person on earth!) That means that each nucleus has a pretty small chance of becoming a helium nucleus. But because there is so much hydrogen in the sun, there are always some nuclear reactions happening, and enough energy is produced to keep the sun going.
I hope this explains a little bit of how hydrogen becomes helium. There isn't really any one temperature and pressure where the nuclei start joining together, but the chance of it happening depends very strongly on the temperature and pressure.
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Special Thanks to P. Giles.