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If an atom is mostly empty space why can't you see the empty space as part of an object?

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When the space inside an atom is described as "empty," what that refers to is mass. Most of the mass of the atom is concentrated in the tiny center, the nucleus. So most of the space inside the atom is devoid of mass. However, that does not mean there is nothing inside the atom.
The "empty space" inside an atom is not actually empty. It is filled with extremely strong electric fields produced by the positive nucleus and the negative electrons. These fields interact with the electromagnetic radiation that corresponds to visible light. This interaction makes it impossible for light to travel through most atoms without being absorbed or deflected. Therefore, you can't see through most substances.
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