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An atom is much like a Solar system, in that the electrons (like planets) orbit around the nucleus (like the sun), and most of the space between the nucleus and the electrons is empty. To get an idea of the scale, if you were to magnify an atom until it was a mile across, the nucleus would be roughly an inch across, and the electrons would be even smaller. So why don't atoms pass through each other equally easily? To put it in a sentence, because like charges repel (while unlike charges attract)! Let me explain in two simple steps.
First, the nucleus of an atom is positively charged, while electrons are negatively charged. So the electrons are attracted to the nucleus, which is what keeps them in orbit (in the Solar system, this attraction is due to gravity). However, if two electrons came close to each other, they would repel, since they are both negatively charged. Second, the electrons move around the nucleus very, very fast; for example, the single electron around a hydrogen nucleus goes around the nucleus about six trillion times a second!! In fact, they move so fast that they're effectively a blur; all we see is a "cloud" of negative charge shielding the nucleus from all around. So a truer picture of an atom would be a negatively charged cloud (or shell) completely shielding a positively charged nucleus inside.
When two atoms try to pass through each other, therefore, the electronic shells of the atoms meet each other first, and they repel fiercely since both shells are negatively charged. So despite having large empty spaces inside, two atoms simply cannot pass through each other because these shells are fairly impenetrable! And since both you and the brick wall are made of many, many atoms, there is a negatively charged "shell" covering both of you, and you cannot walk through a brick wall because these "shells" won't let you. This is the same reason why you don't fall through your chair (or through the floor, for that matter)!
Incidentally, now you can see why two Solar systems wouldn't have much trouble passing through each other; gravity is never repulsive, and the planets are not moving fast enough to create a "cloud" around the sun. Of course, other things would happen because now everything in the two Solar systems would be attracting each other, but that's material enough for another column by itself!
Now if you were reading attentively, you might have noticed a glitch; if atoms can't come close to each other because their electronic clouds repel, then how come so many atoms clump together to form a brick wall in the first place? The short answer - because atoms sometimes like to come just close enough to form solid objects, but not any closer. The long answer? Ask your chemistry teacher!!
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