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So electrons of specific atoms have a minimum amount of energy needed to escape the atom, called the work function, W. Now let's say that you emit a certain frequency of light, and $hf<W$. However, from my understanding, this work function is just a form of kinetic energy, and if an electron has enough kinetic energy, its inertia will be greater than the centripetal force and the electron will escape. So even though $hf<W$, the object being illuminated will still heat up from the light. This heating up means that the material is gaining internal energy, which means that the electrons are gaining kinetic energy. If the material gets hot enough and the electrons keep gaining speed, couldn't the they escape anyway even though $hf<W$? Thanks!

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There is a difference between an electron gaining energy and the whole system (atom) gaining energy. Electron orbital energy levels are quantized and make discreet jumps. For this reason "centripetal force" for electrons doesn't make much sense. There is an electron orbital binding energy though which is equivalent to the "work function" in your question.

The whole atom (nucleus and electrons) can absorbs energy without stripping any electrons off. Eventually the kinetic energy of the atoms (temperature) will be great enough that atom <-> atom collisions will have enough energy to ionize each other. This is what happens in a plasma.

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